Patent Publication Number: US-11397741-B2

Title: Heuristic inspection method to find causes of system abnormality based on dynamic uncertain causality graph

Description:
FIELD OF THE INVENTION 
     This invention is about an AI technology to process uncertain causality type information represented by Dynamic Uncertain Causality Graph (DUCG). Based on the technical scheme proposed in this invention, through computation with computer, one can rank the variables to be detected optimally, detect the ranked variables sequentially or group by group to find out their states, to diagnose causes of system abnormality as earlier and less cost as possible, and then enable to take effective actions to make system normal. 
     BACKGROUND OF THE INVENTION 
     There exist many cause events that may lead to abnormalities of industrial systems, social systems and biological systems, such as short circuit of coils, pump fails to run, failure of components, malfunction of sub-systems, blocking of pathway, entry of foreign matter, pollution, infection, damage, decrepitude of tissue or organ, etc. When system abnormality appears, people need to know the real cause event as soon as possible. Denote B n  or BX n  as these cause event variables indexed by n and B nk  or BX nk  as the event that B n  or BX n  is in state k. The difference between B n  and BX n  is that B n  represents root-cause variable without input while BX has inputs reflecting the influence of other factors. Usually, k=0 means that B n  or BX n  is in normal state; k=1, 2, 3 . . . means that B n  or BX n  is in abnormal state. Most of the states of B n  and BX cannot be or are hard to be detected directly. Furthermore, there are a large number of factors that have causal relationships to B n  or BX n , such as temperature, pressure, flow, velocity, frequency, various chemical or physical test results, investigation results, imaging examination results, acoustic examination results, and so on. Some factors may increase or decrease the occurrence probability of B nk  (k≠0), e.g. region, time, environment, season, religion, age, sex, skin color, career, blood relation, hobbies, personality, living conditions, working conditions, etc. The affected B nk  is BX nk . All these factors can be represented by event variable X i , while X ij  represents state j of X i . X i  and X ij  also represent cause or consequence of other variables/events. Usually, j=0 means that X is in normal state; j=1, 2, 3 . . . means that X is in abnormal state. By detecting the states of X i , people can diagnose the root cause (B nk  or BX nk  (k≠0)) of system abnormality, so that be able to take effective actions in time to get the system back to normal or reduce damage. Dynamic Uncertain Causality Graph (DUCG) is an AI scheme to represent the uncertain causal relationships among event variables graphically, and perform diagnoses based on the constructed graph and observed evidence E composed of known states of X-type variables X ij . For example, E=X 1,2 X 2,3 X 3,1 X 4,0 X 5,0 , where comma separates the variable index (the first) and state index (the second). In general, the more the state known X-type variables in E, the more accurate the diagnosis can be. Yet some state known X-type variables contribute little or none to the diagnosis, some contribute a lot. In practice, there exist cost to detect the states of X-type variables. Therefore, one may have to choose some to detect, or detect some earlier and some later. The problem to be solved in this invention is as follows: 
     Based on
         (1) the constructed DUCG,   (2) evidence E(y) observed by the time indexed with y (y=0, 1, 2, . . . ),   (3) possible cause events included in hypothesis set S H (y) diagnosed conditional on E(y),
 
which state-unknown and detectable X-type variable should be detected to update evidence E(y) as E(y+1)=E + (y)E(y), so that more accurate diagnoses S H (y+1) conditional on E(y+1) can be made as soon and with less cost as possible. Where, the new detected states of X-type variables are the new evidence denoted as E + (y), the new hypothesis set denoted as S H (y+1) is diagnosed conditional on E(y+1), and y=0 means the time no evidence received, that is, E(0) is a complete set denoted as E(0)=1.
       

     The possible cause event in S H (y) or S H (y+1) is represented as H kj , in which H k  is one or a set of variables indexed by k, for example, H 1 =BX 1 , H 2 =BX 2 B 4 , etc., and j indexes the state combination of the set of variables, for example, H 1,2 =BX 1,2 , H 2,3 =B 1,3 X 4,2 , etc. 
     An example of DUCG is illustrated in  FIG. 1 . In DUCG, B-type variable or event is drawn as rectangle, X-type variable or event is drawn as circle, BX-type variable or event is drawn as double line circle, G-type variable or event representing the logic relationship is drawn as gate, and D-type variable or event is drawn as pentagon representing the default or unknown cause of X-type variable/event. The {B−, X−, BX−, G−, D−}-type variable/events are also called nodes. 
     Variable G i  represents the logic combinations of input variables. It must have at least two inputs connected with directed arc  . The logic combinations are specified by logic gate specification LGS i . For example, in  FIG. 1  G 1  is specified by LGS 1 : G 11,1 =B 3,1 X 111,1 , G 11,2 =B 3,1 X 111,2 , G 1,0 =Remnant State that is defined as all other state combinations. 
     The textual descriptions of {B−, X−, BX−, D−, G−}-type variables and their states can be given according to their physical meanings. All {B−, X−, BX−, D−, G−}-type variable/event can be direct cause variable/event, and is called parent variable/event, and can be represented by V∈{B, X, BX, D, G} with the same index. For example, V 2 =X 2 , V 3,2 =B 3,2 , etc. Direct consequence or child variables/events are usually {X−, BX−}-type variables/events. 
     Directed arc   from parent to child is used to denote the weighted functional variable F n;i  representing the causal relationship between parent variable V i  and child variable X n  or BX n . F n;i  is an event matrix. F nk;ij ≡(r n;i /r n )A nk;ij  represents the causal relationship between parent event V ij  and child event X nk  or BX nk . Where r n;i &gt;0 quantifies the uncertain causal relationship intensity between V i  and X n  or BX n , r n ≡Σ i r n;i , and A nk;ij  represents the uncertain causal mechanism that V ij  may cause X nk  or BX nk . The probability of A nk;ij  is defined as a nk;ij =Pr{A nk;ij } usually given by domain experts. Define f nk;ij =Pr{F nk;ij }=(r nk;ij /r n )a nk;ij , in which f nk;ij  means the probabilistic contributions from V ij  to X nk , satisfying 
               P   ⁢   r   ⁢     {     X     n   ⁢   k       }       =       ∑     i   ,   j       ⁢       f     nk   ;   ij       ⁢   Pr   ⁢       {     V   ij     }     .               
F nk;ij , f nk;ij , A nk;ij  and a nk;ij  are member event in F n;i , f n;i , A n;i  and a n;i  respectively. Define v ij =Pr{V ij }, v∈{b, x, bx, d, g}. In general, the lower case letters represent the probabilities of the corresponding upper case letters that represent events or event variables.
 
     The weighted functional event matrix F n;i  can also be conditional on a condition event Z n;i , which can be drawn as dashed directed arc  . Conditional F n;i  is used to represent the conditional causal relationship between its parent event vector V i  and its child event vector X nk  or BX nk . The condition event Z n;i  determines whether F n;i  holds or not (discarded). Taken Z n;i =X 1,2  as an example, when X 1,2  is true, F n;i  is held, i.e.   becomes  ; when X 1,2  is false, F n;i  is not held, i.e.   is discarded. 
     For simplicity, the letter of a variable in each graphical node symbol can be ignored and only the indices are inside as shown in  FIG. 2 , in which the first number is the index of the variable and the second number is the state index of the variable. Since D-type variable has only one state, there is only one index inside the symbol. Non-D-type variables with known state can be indicated by colors, e.g. X 110,1  is blue in  FIG. 2 . If there is only one number in the symbol, the state of the variable is unknown. 
     With the received evidence E(y)=E′(y)E″(y), where E′(y) is composed of state abnormal events and E″(y) is composed of state normal events, the following rules can be used to simplify a DUCG: 
     Rule 1: If E(y) shows that Z n;i  is false, F n;i , is eliminated; if E(y) shows that Z n;i  is true, the conditional F n;i  becomes the ordinary F n;i , i.e.   becomes    3 . 
     Rule 2: If E(y) shows that V ij  (V∈{B, X}) is true while V ij  is not a parent event of X n  or BX n , directed arc F n;i  is eliminated. 
     Rule 3: If E(y) shows that X nk  is true while X nk  cannot be caused by any state of V i  (V∈{B, X, BX, G, D}), directed arc F n;i  is eliminated. 
     Rule 4: If E(y) shows that a state-unknown X-type node does not have any output and E(y) blocks its connection with B-type nodes, eliminate this X-type node. 
     Rule 5: If E(y) shows X n0  is true, and X n0  has no connection with B-type nodes unless through the nodes in E(y), eliminate X n0 . 
     Rule 6: If E(y) shows a group of state-unknown nodes that have no connection with E′(y), this group of state-unknown nodes are eliminated. 
     Rule 7: Given E(y), if G i  has no output, eliminate G i  and its input directed arc  ; if G i  has no input, eliminate G i  and its output directed arcs. 
     Rule 8: Given E(y), if a directed arc has no parent node or child node, eliminate this directed arc. 
     Rule 9: If there is a group of nodes and directed arcs that have no connection with E′(y), this group of nodes and directed arcs are eliminated. 
     Rule 10: If E(y) shows X nk  (k≠0) is true while X nk  does not have any input, add a virtual parent event D n  to X nk  as its input with a nk;nD =1 and a n′;nD =0 (k≠k′), where r n;D  can be any value. The added virtual D n  can be drawn as 
                         
Rule 11: Apply Rule 1-Rule 10 in any order, separately or together, and repeatedly.
 
     After applying the above rules, the DUCG is simplified, in which H kj =B kj  or H kj =BX kj  (j≠0) compose S H (y). 
     In the simplified DUCG, some state-unknown X-type variables are cause-specific, which means that the state observation of a cause-specific X-type variable can determine whether the corresponding hypothesis event H kj  (H kj ∈S H (y)) is true. 
     Technical references for this invention 
     
         
         [1] Q. Zhang and Z. Zhang. Method for constructing an intelligent system processing uncertain causal relationship information. Chinese patent: CN 200680055266.X, 2010. 
         [2] Q. Zhang and Z. Zhang. Method for constructing an intelligent system processing uncertain causal relationship information. U.S. Pat. No. 8,255,353 B2, 2012. 
         [3] Q. Zhang and C. Dong. Method for constructing cubic DUCG for dynamic fault diagnosis. Chinese patent: CN 2013107185964, 2015. 
         [4] Q. Zhang. “Dynamic uncertain causality graph for knowledge representation and reasoning: discrete DAG cases”, Journal of Computer Science and Technology, vol. 27, no. 1, pp. 1-23, 2012. 
         [5] Q. Zhang, C. Dong, Y. Cui and Z. Yang. “Dynamic uncertain causality graph for knowledge representation and probabilistic reasoning: statistics base, matrix and fault diagnosis”, IEEE Trans. Neural Networks and Learning Systems, vol. 25, no. 4, pp. 645-663, 2014. 
         [6] Q. Zhang. “Dynamic uncertain causality graph for knowledge representation and probabilistic reasoning: directed cyclic graph and joint probability distribution”, IEEE Trans. Neural Networks and Learning Systems, vol. 26, no. 7, pp. 1503-1517, 2015. 
         [7] Q. Zhang. “Dynamic uncertain causality graph for knowledge representation and probabilistic reasoning: continuous variable, uncertain evidence and failure forecast”, IEEE Trans. Systems, Man and Cybernetics, vol. 45, no. 7, pp. 990-1003, 2015. 
         [8] Q. Zhang and S. Geng. “Dynamic uncertain causality graph applied to dynamic fault diagnosis of large and complex systems”, IEEE Trans. Reliability, vol. 64, no. 3, pp 910-927, 2015. 
         [9] Q. Zhang and Z. Zhang. “Dynamic uncertain causality graph applied to dynamic fault diagnoses and predictions with negative feedbacks”, IEEE Trans. Reliability, vol. 65, no. 2, pp 1030-1044, 2016. 
       
    
     SUMMARY OF THE INVENTION 
     This invention discloses a technical scheme to rank the state-unknown X-type variables optimally, so that people can optically choose the state-unknown X-type variables to detect (state pending X-type variables) to obtain E + (y). Conditional on E(y+1)=E + (y)E(y), the more accurate S H (y+1) and more accurate Pr{H kj }, H kj ∈S H (y+1), can be found as earlier and with less cost as possible, while the danger degrees of H kj  are considered. 
     This invention is a subsequent invention and a further development of granted patents CN 200680055266.X, CN 2013107185964, and U.S. Pat. No. 8,255,353 B2. The technical scheme of this invention comprises: 
     1. The method to rank the state pending X-type variables with at least one CPU, according to the rank, part of or all of the state pending X-type variables&#39; states which form E + (y) are detected sequentially or parallel, in order to find the real cause H kj  that is in S H (y+1) and to rank the real H kj  as high as possible conditioned on E(y+1)=E + (y)E(y), detailed steps include: (1) determine the detectable state pending X-type variables whose index set is denoted as S X (y) based on the simplified DUCG conditioned on E(y); (2) the ranking ends if S X (y) contains only one element; (3) calculate the rank importance I i (y) of X i ; (4) rank X i  (i∈S X (y)) according to I i (y) and detect the states of X i  (i∈S X (y)) in reference to the rank; (5) if the ranking is still needing, increase y to y+1, and repeat the above step (1)-(5) until the diagnosis is satisfied or no state pending X-type variable available. 
     2. The method according to 1, wherein to determine the state pending X-type variables with at least one CPU, the detailed steps include: (1) Collect all possible H kj  based on the simplified DUCG conditioned on E(y), these H kj  form S H (y); (2) For each H k  in S H (y), search for the state pending X-type variable connected to H k  without state-known variables blocking them, where the indices of such X-type variables make up S X (y). 
     3. The method according to 1, wherein to determine the structure importance λ i (y)&gt;0 of X i  for calculating I(y) with at least one CPU, characterized in that: for the state pending X-type variables in the above 1(1), count the number of its connected different H k  (k∈S iK ) in S H (y) determined in the above 2, the number is written as m i (y), calculate λ i (y) based on m i (y) according to a method that features at that the bigger m i (y) is, the smaller λ i (y) is, such method includes but not limits to λ i (y)=1/(m i (y)) n  (n=1, 2, . . . ). 
     4. The method according to 1, wherein assign a value to the danger importance ω kj &gt;0 of H kj  for calculating I i (y) with at least one CPU, characterized in that: 
     For each possible cause event H kj  in S H (y), score all of the abnormal states according to their degree of concern, the score is called concern importance which is written as ω k , 1≥ω k &gt;0. The greater the concern is, the bigger ω k  is. The value of ω k  can be assigned when constructing DUCG or be assigned according to the concrete situation given H k  when S H (y) is known. 
     5. The method according to 1, wherein to calculate the probability importance ρ i (y) for calculating I(y) with at least one CPU, characterized in that: calculate the average variation of the conditional probabilities of H k  in S H (y) between the conditions E(y) and X ig E(y) over all abnormal states of all H k  connected with X without state-known variables blocking them, where g∈S iG (y), S iG (y) is the index set of possible states of X i , and X i  is non-cause-specific state pending X-type variables, based on the simplified DUCG conditioned on E(y), featuring at that the greater the average variation is, the bigger ρ i (y) is, such as but are not limited to: 
                 ρ   i     ⁡     (   y   )       =         1       m   i     ⁡     (   y   )         ⁢       ∑     k   ∈       S   iK     ⁡     (   y   )           ⁢       ω   k     ⁢       ∑     j   ∈       S   kJ     ⁡     (   y   )           ⁢       ∑     g   ∈       S   iG     ⁡     (   y   )           ⁢     P   ⁢   r   ⁢     {       X   ig     |     E   ⁡     (   y   )         }                 |       Pr   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   y   )           }       -     P   ⁢   r   ⁢     {       H   kj     |     E   ⁡     (   y   )         }         |     
     ⁢   Or                     ρ   i     ⁡     (   y   )       =       1       m   i     ⁡     (   y   )         ⁢       ∑     k   ∈       S   iK     ⁡     (   y   )           ⁢       ω   k     ⁢         ∑     j   ∈       S   kJ     ⁡     (   y   )           ⁢       ∑     g   ∈       S   iG     ⁡     (   y   )           ⁢       (     P   ⁢   r   ⁢     {       X   ig     |     E   ⁡     (   y   )         }     ⁢     (       P   ⁢   r   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   y   )           }       -     P   ⁢   r   ⁢     {       H   kj     |     E   ⁡     (   y   )         }         )       )     2                           Or                 ρ   i     ⁡     (   y   )       =         1       m   i     ⁡     (   y   )         ⁢       ∑     k   ∈       S   iK     ⁡     (   y   )           ⁢       ω   k     ⁢       ∑     j   ∈       S   kJ     ⁡     (   y   )           ⁢     ∑     g   ∈       S   iG     ⁡     (   y   )                   |       Pr   ⁢     {       H     k   ⁢   j       |       X   ig     ⁢     E   ⁡     (   y   )           }       -     P   ⁢   r   ⁢     {       H     k   ⁢   j       |     E   ⁡     (   y   )         }         |     
     ⁢   Or                     ρ   i     ⁡     (   y   )       =       1       m   i     ⁡     (   y   )         ⁢       ∑     k   ∈       S   iK     ⁡     (   y   )           ⁢       ω   k     ⁢         ∑     j   ∈       S   kJ     ⁡     (   y   )           ⁢       ∑     g   ∈       S   iG     ⁡     (   y   )           ⁢       (       P   ⁢   r   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   y   )           }       -     P   ⁢   r   ⁢     {       H   kj     |     E   ⁡     (   y   )         }         )     2                           Or                 ρ   i     ⁡     (   y   )       =         1       m   i     ⁡     (   y   )         ⁢       ∑     k   ∈       S   iK     ⁡     (   y   )           ⁢     ω   k         |         ∑     g   ∈       S   iG     ⁡     (   y   )           ⁢     P   ⁢   r   ⁢     {       H     k   ⁢   j       |       X   ig     ⁢     E   ⁡     (   y   )           }         -       1     m   i       ⁢       ∑       k   ∈       S   iK     ⁡     (   y   )         ,     j   ∈       S   kJ     ⁡     (   y   )             ⁢       ∑     g   ∈       S   iG     ⁡     (   y   )           ⁢     P   ⁢   r   ⁢     {       H     k   ⁢   j       |       X   ig     ⁢     E   ⁡     (   y   )           }               |           
E(y) and ω k  can be ignored, equivalent to being removed separately or together, that is to let E (y)=the complete set and ω k =1.
 
     6. The method according to 1 for calculating I(y) with at least one CPU, wherein to determine the cost importance, characterized in that: comprehensively assign a cost score for the state pending X i  (i∈S X (y)), or calculate the cost score as the sum of the weighted scores assigned for the difficulty of doing the detection (indexed by j=1), waiting time (indexed by j=2), price (indexed by j=3) and damage to target system (indexed by j=4) respectively, the weights σ ij  can be given when constructing DUCG or be given in an individual application, the bigger the cost score is, the smaller the cost importance β i  (1≥β i &gt;0) is, which including but not limited to: when the highest cost score is 100, β i = 1/100=0.01, when the lowest cost score is 1, β i =1/1=1, and the remnant values are between the highest and the lowest case, can be assigned when constructing DUCG or be given online for the state pending X-type variables included in the simplified DUCG conditioned on E(y). 
     7. The method according to 5 to determine the probability importance ρ i (y) with at least one CPU, characterized in that: based on the simplified DUCG conditioned on E(y), for each cause-specific state pending X-type variable connected to H kj  (H kj  ∈S k (y)), λ i (y) and ρ i (y) are not calculated according to the above 3 and 5, which means S Xs (y) is obtained by subtracting S s (y) from S X (y), but are calculated by selecting the maximum of i∈S s (y), which includes but is not limited to λ i (y)≥1 and 
                   ρ   i     ⁡     (   y   )       ≥       max     l   ∈       S     X   ⁢   s       ⁡     (   y   )           ⁢     {       ρ   l     ⁡     (   y   )       }         ,         
that means X i  in the above 5 is limited to i∈S Xs (y).
 
     8. The method according to 1 to comprehensively calculate I(y) with at least one CPU, characterized in that: the bigger λ i (y) or ρ i (y) or β i  in the above 2-7 is, the bigger I i (y) is, the detailed calculation formulas include but are not limited to: 
                 I   i     ⁡     (   y   )       =           λ   i     ⁡     (   y   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   y   )             ∑     i   ∈       S   X     ⁡     (   y   )           ⁢         λ   i     ⁡     (   y   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   y   )                       Or                 I   i     ⁡     (   y   )       =         λ   i     ⁡     (   y   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   y   )                   Or                 I   i     ⁡     (   y   )       =           λ   i     ⁡     (   y   )       ⁢       ρ   i     ⁡     (   y   )             ∑     i   ∈       S   X     ⁡     (   y   )           ⁢         λ   i     ⁡     (   y   )       ⁢       ρ   i     ⁡     (   y   )                       Or                 I   i     ⁡     (   y   )       =         λ   i     ⁡     (   y   )       ⁢       ρ   i     ⁡     (   y   )                   Or                 I   i     ⁡     (   y   )       =         β   i     ⁢       ρ   i     ⁡     (   y   )             ∑     i   ∈       S   X     ⁡     (   y   )           ⁢       β   i     ⁢       ρ   i     ⁡     (   y   )                       Or                 I   i     ⁡     (   y   )       =       β   i     ⁢       ρ   i     ⁡     (   y   )                   Or                 I   i     ⁡     (   y   )       =         ρ   i     ⁡     (   y   )           ∑     i   ∈       S   X     ⁡     (   y   )           ⁢       ρ   i     ⁡     (   y   )                     Or                 I   i     ⁡     (   y   )       =       ρ   i     ⁡     (   y   )                 Or                 I   i     ⁡     (   y   )       =         w   1     ⁢       λ   i     ⁡     (   y   )         +       w   2     ⁢       ρ   i     ⁡     (   y   )         +       w   3     ⁢     β   i                   Or                 I   i     ⁡     (   y   )       =           w   1     ⁢       λ   i     ⁡     (   y   )         +       w   2     ⁢       ρ   i     ⁡     (   y   )         +       w   3     ⁢     β   i               ∑     i   ∈       S   X     ⁡     (   y   )           ⁢       w   1     ⁢       λ   i     ⁡     (   y   )           +       w   2     ⁢       ρ   i     ⁡     (   y   )         +       w   3     ⁢     β   i                 
where w 1 , w 2 , and w 3  are three weights, and w i ≥0 (i=1, 2, 3), and w i =0 means this item is not considered, wherein the value of w 1 , w 2 , and w 3  can be assigned when constructing DUCG or be assigned according to the individual application situation.
 
     9. The method according to 1 to rank the state pending X-type variables X i  (i∈S X (y)) according to I i (y) with at least one CPU, characterized in that: in the ranking, when the top-ranking X-type variable is the only ancestor or descendant variable of the ranking lower X-type variable, the ranking lower X-type variable is eliminated from the current rank, also the X i  whose I i (y)=0 is eliminated from the rank. 
     10. The method according to 1 to determine whether the ranking ends or not with at least one CPU, characterized in that: the ranking ends if there is only one hypothesis event in S H (y), or if all the state pending X-type variables are state-known, or there is only one element in S X (y). 
     The chart of the above steps is as shown in  FIG. 3 . 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1 : Illustration of DUCG. 
         FIG. 2 : Illustration of a simple representation of DUCG. 
         FIG. 3 : Step chart of this invention. 
         FIG. 4 : The original DUCG of the examples. 
         FIG. 5 : The results of dividing and simplifying  FIG. 4  based on BX 1  when y=0. 
         FIG. 6 : The results of dividing and simplifying  FIG. 4  based on BX 2  when y=0. 
         FIG. 7 : The results of dividing and simplifying  FIG. 4  based on BX 3  when y=0. 
         FIG. 8 : The DUCG graph of example 1 after detecting the X-type variables ranking first 5. 
         FIG. 9 : The simplified DUCG when y=1. 
         FIG. 10 : The results of dividing and simplifying  FIG. 9  based on BX 1  when y=1. 
         FIG. 11 : The results of dividing and simplifying  FIG. 9  based on BX 2  when y=2. 
         FIG. 12 : The DUCG graph after detecting the states of X 8  and X 10 . 
         FIG. 13 : The simplified DUCG graph given E(2). 
         FIG. 14 : The results of dividing and simplifying  FIG. 13  based on BX 1 . 
         FIG. 15 : The results of dividing and simplifying  FIG. 13  based on BX 2 . 
     
    
    
     EXAMPLES TO IMPLEMENT THIS INVENTION 
     Suppose  FIG. 4  is the original DUCG graph whose parameters are as follows: 
                 b   1     =       (         -       0.08         )     T       ;       b   2     =       (         -       0.01       0.02         )     T       ;       b   3     =       (         -       0.05         )     T       ;       a     1   ;   1       =     (         -       -           -       1         )       ;                   a     2   ;   2       =     (         -       -       -           -       1       -           -       -       1         )       ;       a     3   ;   3       =     (         -       -           -       1         )       ;       a     12   ;   D       =     (         -           0.1         )       ;       a     13   ;   D       =     (         -           0.01         )       ;       a     14   ;   D       =     (         -           0.1         )       ;                   a     1   ;   12       =     (         -       -           -       0.7         )       ;       a     1   ;   13       =     (         -       -           -       0.8         )       ;       a     3   ;   14       =     (         -       -           -       0.6         )       ;       a     4   ;   1       =     (         -       -           -       0.9         )       ;                   a     4   ;   2       =     (           -           -         ⁢         -       -           0.5       0.5           )       ;       a     5   ;   1       =     (         -       -           -       0.7         )       ;       a     5   ;   2       =     (           -           -         ⁢         -       -           0.4       0.8           )       ;       a     5   ;   3       =     (         -       -           -       0.7         )       ;                   a     6   ;   2       =     (         -       -       -           -       0.6       0.3           -       0.4       0.7         )       ;       a     6   ;   3       =     (         -       -           -       0.2           -       0.8         )       ;       a     7   ;   4       =     (         -       -           -       0.7         )       ;       a     8   ;   4       =     (         -       -           -       0.8         )       ;                   a     8   ;   5       =     (         -       -           -       0.7         )       ;       a     8   ;   6       =     (         -       -       -           -       0.2       0.8         )       ;       a     9   ;   4       =     (         -       -           -       0.7         )       ;       a     10   ;   5       =     (         -       -           -       0.7         )       ;                   a     11   ;   6       =     (         -       -       -           -       0.3       0.8         )       ;       a     15   ;   3       =     (         -       -           -       1         )       ;       a     16   ;   3       =     (         -       -           -       1         )       ;         
r n;i =1 are assumed. Among all the X-type variables, X 14  is a non-detectable variable; other X-type variables are all detectable. X 15  and X 16  are cause-specific variables of BX 3,1 , that is when X 15,1  or X 16,1  is true, BX 3,1  must be true. Since {B 1,1 , B 2,1 , B 2,2 , B 3,1 } is equivalent to {BX 1,1 , BX 2,1 , BX 2,2 , BX 3,1 }, the B-type variables are not treated as the diagnosis object. In other words, H-type events are composed of only BX-type events.
 
     The parameter of the directed arc between X n  and D n  can be briefly written as a n;D . Our task is to detect the states of the state pending X-type variables as few and with less cost as possible, in order to minimize the size of the possible set of cause events S H (y) and to maximize the probability of real cause event. 
     Example 1: y=0 (with No Evidence) 
     According to 2, when there is no evidence (y=0), based on  FIG. 4 , S H (0)={H 1,1 , H 2,1 , H 2,2 , H 3,1 }={BX 1,1 , BX 2,1 , BX 2,2 , BX 3,1 }, and the state pending X-type variables are {X 4 , X 5 , X 6 , X 7 , X 8 , X 9 , X 10 , X 11 , X 12 , X 13 , X 15 , X 16 }. All these states are unknown and single-connect to H k  in S H (0) without state-known blocking variables, so that S X (0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16}. Since X 15  and X 16  are cause-specific variables of BX 3,1 , they are eliminated from S X (0), that means S Xs (0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. Accordingly, S 4G (0)=S 5G (0)=S 7G (0)=S 8G (0)=S 9G (0)=S 10G (0)=S 11G (0)=S 12G (0)=S 13G (0)={1} and S 6G (0)={1, 2}. 
     Based on  FIG. 4 , according to the DUCG algorithms in Ref [8], since the intersection of different H kj  is null set, which means different H kj  cannot occur simultaneously,  FIG. 4  can be divided according to H k  and can be simplified according to simplification rules, the results are shown in  FIG. 5 - FIG. 7 . 
     Since there is no evidence, E(0)=1 (complete set), the probability and weighting factor of each sub-graph are ζ i (y)=ζ i (0)=Pr{E(0)}=1 (i∈{1,2,3}) and 
                 ξ   i     ⁡     (   y   )       =         ξ   i     ⁡     (   0   )       =           ζ   i     ⁡     (   0   )       /       ∑   i     ⁢       ζ   i     ⁡     (   0   )           =         1   3     ⁢       ξ   i     ⁡     (   y   )         =         ξ   i     ⁡     (   0   )       =           ζ   i     ⁡     (   0   )       /       ∑   i     ⁢       ζ   i     ⁡     (   0   )           =       1   3     .                     
Based on  FIG. 5 - FIG. 7 , according to algorithms in the references, the state probabilities h kj   s (y)=h kj   s (0) of BX 1,1 , BX 2,1 , BX 2,2  and BX 3,1  are:
 
                       h     1   ,   1     s     ⁡     (   0   )       =       ⁢         ξ   1     ⁡     (   0   )       ⁢   Pr   ⁢     {       BX     1   ,   1       ❘     E   ⁡     (   0   )         }                   =       ⁢         ξ   1     ⁡     (   0   )       ⁢   Pr   ⁢     {     BX     1   ,   1       }                   =       ⁢         ξ   1     ⁡     (   0   )       ⁢   Pr   ⁢     {           r     1   ;   1         r   1       ⁢     A     1   ,     1   ;   1         ⁢     B   1       +         r     1   ;   12         r   1       ⁢     A     1   ,     1   ;   12         ⁢     A     12   ;   D         +         r     1   ;   13         r   1       ⁢     A     1   ,     1   ;   13         ⁢     A     13   ;   D           }                   =       ⁢         1   3     ×     1   3     ⁢     a     1   ,     1   ;   1         ⁢     b   1       +       1   3     ⁢     a     1   ,     1   ;   12         ⁢     a     12   ;   D         +       1   3     ⁢     a     1   ,     1   ;   13         ⁢     a     13   ;   D                       =       ⁢         1   3     ×     1   3     ⁢     (         -       1         )     ⁢     (         -           0.08         )       +       1   3     ⁢     (         -       0.7         )     ⁢     (         -           0.1         )       +       1   3     ⁢     (         -       0.8         )     ⁢     (         -           0.01         )                     =       ⁢   0.01756                               h     2   ,   1     s     ⁡     (   0   )       =       ⁢         ξ   2     ⁡     (   0   )       ⁢   Pr   ⁢     {     BX     2   ,   1       }                   =       ⁢         ξ   2     ⁡     (   0   )       ⁢   Pr   ⁢     {       A     2   ,     1   ;   2         ⁢     B   2       }                   =       ⁢         ξ   2     ⁡     (   0   )       ⁢     a     2   ,     1   ;   2         ⁢     b   2                   =       ⁢       1   3     ⁢     (         -       1       -         )     ⁢       (         -       0.01       0.02         )     T                   =       ⁢   0.003333                               h     2   ,   2     s     ⁡     (   0   )       =       ⁢         ξ   2     ⁡     (   0   )       ⁢   Pr   ⁢     {     BX     2   ,   2       }                   =       ⁢         ξ   2     ⁡     (   0   )       ⁢   Pr   ⁢     {       A     2   ,     2   ;   2         ⁢     B   2       }                   =       ⁢         ξ   2     ⁡     (   0   )       ⁢     a     2   ,     2   ;   2         ⁢     b   2                   =       ⁢       1   3     ⁢     (         -       -       1         )     ⁢       (         -       0.01       0.02         )     T                   =       ⁢   0.006667                               h     3   ,   1     s     ⁡     (   0   )       =       ⁢         ξ   3     ⁡     (   0   )       ⁢   Pr   ⁢     {     BX     3   ,   1       }                   =       ⁢         ξ   3     ⁡     (   0   )       ⁢   Pr   ⁢     {           r     3   ;   3         r   3       ⁢     A     3   ,     1   ;   3         ⁢     B   3       +         r     3   ;   14         r   3       ⁢     A     3   ,     1   ;   14         ⁢     A     14   ;   D           }                   =       ⁢         1   3     ×     1   2     ⁢     a     3   ,     1   ;   3         ⁢     b   3       +       1   2     ⁢     a     3   ,     1   ;   14         ⁢     a     14   ;   D                       =       ⁢         1   3     ×     1   2     ⁢     (         -       1         )     ⁢     (         -           0.05         )       +       1   2     ⁢     (         -       0.6         )     ⁢     (         -           0.1         )                     =       ⁢   0.01833               
The calculation result of rank probability
 
                 h   kj   r     ⁡     (   y   )       =         h   kj   s     ⁡     (   y   )           ∑       H   kj     ∈       S   H     ⁡     (   y   )           ⁢       h   kj   s     ⁡     (   y   )                 
is as follows:
 
     
       
         
           
               
               
               
             
               
                   
               
               
                 index 
                 H kj   
                 h kj   r  (0) 
               
               
                   
               
             
            
               
                 1 
                 BX 3, 1   
                 0.3995 
               
               
                 2 
                 BX 1, 1   
                 0.3826 
               
               
                 3 
                 BX 2, 2   
                 0.1453 
               
               
                 4 
                 BX 2, 1   
                 0.0725 
               
               
                   
               
            
           
         
       
     
     According to 3, based on  FIG. 4 , one can get S 12K (0)=S 13K (0)={1}, and S 4K (0)S 7K (0)S 9K (0)={1,2}, and S 6K (0)=S 11K (0)={2,3}, and S 5K (0)=S 8K (0)=S 10K (0)={1,2,3}. Accordingly, one can get m 12 (0)=m 13 (0)=1, and m 4 (0)=m 6 (0)=m 7 (0)=m 9 (0)=m 11 (0)=2, and m 5 (0)=m 8 (0)=m 10 (0)=3. Since BX 1  and BX 3  have only one abnormal state indexed by “1”, thus S 1J (0)=S 3J (0)={1}. And BX 2  has two abnormal states indexed by “1” and “2”, thus S 2J (0)={1,2}. 
     According to 4, assume ω kj =1 (k∈{1,2,3}). 
     According to 5, with the following formula: 
                 ρ   i     ⁡     (   y   )       =         1       m   i   ′     ⁡     (   y   )         ⁢       ∑     k   ∈       S   iK     ⁡     (   y   )           ⁢       ∑     j   ∈       S   kJ     ⁡     (   y   )           ⁢       ω   kj     ⁢       ∑     g   ∈       S   iG     ⁡     (   y   )           ⁢     P   ⁢   r   ⁢     {       X   ig     |     E   ⁡     (   y   )         }                 |       Pr   ⁢     {       H     k   ⁢   j       |       X   ig     ⁢     E   ⁡     (   y   )           }       -     P   ⁢   r   ⁢     {       H     k   ⁢   j       |     E   ⁡     (   y   )         }         |           
where i∈S Xs (0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13} and m′ i (y)=m i (y). Based on  FIG. 4 , since ω kj =1, k∈{1,2,3}, thus
 
                       ρ   4     ⁡     (   0   )       =       ⁢         1       m   4     ⁡     (   0   )         ⁢       ∑     k   ∈     {     1   ,   2     }         ⁢       ∑     j   ∈       S   kJ     ⁡     (   0   )           ⁢       ∑     g   ∈     {   1   }         ⁢     P   ⁢   r   ⁢     {       X   ig     |     E   ⁡     (   0   )         }               |                     ⁢         Pr   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   0   )           }       -     P   ⁢   r   ⁢     {       H   kj     |     E   ⁡     (   0   )         }         |                 =       ⁢       1   2     ⁢   P   ⁢   r   ⁢     {     X     4   ,   1       }     ⁢     (                    Pr   ⁢     {     H     1   ,   1            ⁢     X     4   ,   1         }     -     Pr   ⁢     {     H     1   ,   1       }                          +          Pr   ⁢     {     H     2   ,   1            ⁢     X     4   ,   1         }       -     Pr   ⁢     {     H     2   ,   1       }                   +            Pr   ⁢     {       H     2   ,   2       ❘     X     4   ,   1         }       -     Pr   ⁢     {     H     2   ,   2       }                      )                   =       ⁢       1   2     ⁢   P   ⁢   r   ⁢     {     X     4   ,   1       }     ⁢     (                  Pr   ⁢     {       BX     1   ,   1       ❘     X     4   ,   1         }       -     Pr   ⁢     {     BX     1   ,   1       }                        +            Pr   ⁢     {       BX     2   ,   1       ❘     X     4   ,   1         }       -     Pr   ⁢     {     BX     2   ,   1       }                          +            Pr   ⁢     {       BX     2   ,   2       ❘     X     4   ,   1         }       -     Pr   ⁢     {     BX     2   ,   2       }                      )                   =       ⁢       1   2     ⁢     (                  Pr   ⁢     {       BX     1   ,   1       ⁢     X     4   ,   1         }       -     Pr   ⁢     {     X     4   ,   1       }     ⁢   Pr   ⁢     {     BX     1   ,   1       }                        +            Pr   ⁢     {       BX     2   ,   1       ⁢     X     4   ,   1         }       -     Pr   ⁢     {     X     4   ,   1       }     ⁢   Pr   ⁢     {     BX     2   ,   1       }                          +            Pr   ⁢     {       BX     2   ,   2       ⁢     X     4   ,   1         }       -     Pr   ⁢     {     X     4   ,   1       }     ⁢   Pr   ⁢     {     BX     2   ,   2       }                      )                   =       ⁢       1   2     ⁢     (                      Pr   ⁢     {       BX     1   ,   1       ⁡     (           r     4   ;   1         r   4       ⁢     A     4   ,     1   ;   1         ⁢     BX   1       +         r     4   ;   2         r   4       ⁢     A     4   ,     1   ;   2         ⁢     BX   2         )       }                   -   Pr     ⁢     {     X     4   ,   1       }     ⁢   Pr   ⁢     {     BX     1   ,   1       }                            +                Pr   ⁢     {       BX     2   ,   1       ⁡     (           r     4   ;   1         r   4       ⁢     A     4   ,     1   ;   1         ⁢     BX   1       +         r     4   ;   2         r   4       ⁢     A     4   ,     1   ;   2         ⁢     BX   2         )       }                   -   Pr     ⁢     {     X     4   ,   1       }     ⁢   Pr   ⁢     {     BX     2   ,   1       }                              +                Pr   ⁢     {       BX     2   ,   2       ⁡     (           r     4   ;   1         r   4       ⁢     A     4   ,     1   ;   1         ⁢     BX   1       +         r     4   ;   2         r   4       ⁢     A     4   ,     1   ;   2         ⁢     BX   2         )       }                   -   Pr     ⁢     {     X     4   ,   1       }     ⁢   Pr   ⁢     {     BX     2   ,   2       }                          )                   =       ⁢       1   2     ⁢     (                  Pr   ⁢     {       A     4   ,     1   ;   1     ,   1       ⁢     BX     1   ,   1         }       -     Pr   ⁢     {     X     4   ,   1       }     ⁢   Pr   ⁢     {     BX     1   ,   1       }                        +            Pr   ⁢     {       A     4   ,     1   ;   2     ,   1       ⁢     BX     2   ,   1         }       -     Pr   ⁢     {     X     4   ,   1       }     ⁢   Pr   ⁢     {     BX     2   ,   1       }                          +            Pr   ⁢     {       A     4   ,     1   ;   2     ,   2       ⁢     BX     2   ,   2         }       -     Pr   ⁢     {     X     4   ,   1       }     ⁢   Pr   ⁢     {     BX     2   ,   2       }                      )                   =       ⁢       1   2     ⁢     (                  (       Pr   ⁢     {     A     4   ,     1   ;   1     ,   1       }       -     Pr   ⁢     {     X     4   ,   1       }         )     ⁢   Pr   ⁢     {     BX     1   ,   1       }                      +            (       Pr   ⁢     {     A     4   ,     1   ;   2     ,   1       }       -     Pr   ⁢     {     X     4   ,   1       }         )     ⁢   Pr   ⁢     {     BX     2   ,   1       }                        +            (       Pr   ⁢     {     A     4   ,     1   ;   2     ,   2       }       -     Pr   ⁢     {     X     4   ,   1       }         )     ⁢   Pr   ⁢     {     BX     2   ,   2       }                    )                   
In which
 
                     Pr   ⁢     {     X   4.1     }       =       ⁢     Pr   ⁢     {           r     4   ;   1         r   4       ⁢       A     4   ,     1   ;   1         ⁢           (               r     1   ;   1         r   1       ⁢     A     1   ;   1       ⁢     B   1                   +       r     1   ;   12         r   1         ⁢     A     1   ;   12       ⁢     A     12   ;   D                     +       r     1   ;   13         r   1         ⁢     A     1   ;   13       ⁢     A     13   ;   D               )       +         r     4   ;   2         r   4       ⁢     A     4   ,     1   ;   2         ⁢   B   ⁢     X   2         }                   =       ⁢     Pr   ⁢     {         1   2     ⁢       A     4   ,     1   ;   1         ⁢           (             1   3     ⁢     A     1   ;   1       ⁢     B   1                   +     1   3       ⁢     A     1   ;   12       ⁢     A     12   ;   D                     +     1   3       ⁢     A     1   ;   13       ⁢     A     13   ;   D               )       +       1   2     ⁢     A     4   ,     1   ;   2         ⁢     A     2   ;   2       ⁢     B   2         }                   =       ⁢         1   2     ⁢       a     4   ,     1   ;   1         ⁢           (             1   3     ⁢     a     1   ;   1       ⁢     b   1                   +     1   3       ⁢     a     1   ;   12       ⁢     a     12   ;   D                     +     1   3       ⁢     a     1   ;   13       ⁢     a     13   ;   D               )       +       1   2     ⁢     a     4   ,     1   ;   2         ⁢     a     2   ;   2       ⁢     b   2                     =       ⁢         1   2     ⁢     (         -       0.9         )     ⁢     (             1   3     ⁢     (         -       -           -       1         )     ⁢     (         -           0.08         )                   +     1   3       ⁢     (         -       -           -       0.7         )     ⁢     (         -           0.1         )                   +     1   3       ⁢     (         -       -           -       0.8         )     ⁢     (         -           0.01         )             )       +       1   2     ⁢     (         -       0.5       0.5         )                         ⁢       (         -       -       -           -       1       -           -       -       1         )     ⁢     (         -           0.01           0.02         )                   =       ⁢   0.0312               
Thus:
 
                       ρ   4     ⁡     (   0   )       =       ⁢       1   2     ⁢     (                  (       a     4   ,     1   ;   1     ,   1       -     Pr   ⁢     {     X     4   ,   1       }         )     ⁢   Pr   ⁢     {     BX     1   ,   1       }                      +            (       a     4   ,     1   ;   2     ,   1       -     Pr   ⁢     {     X     4   ,   1       }         )     ⁢   Pr   ⁢     {     BX     2   ,   1       }                        +            (       a     4   ,     1   ;   2     ,   2       -     Pr   ⁢     {     X     4   ,   1       }         )     ⁢   Pr   ⁢     {     BX     2   ,   2       }                    )                   =       ⁢         1   2     ⁢     (                  (     0.9   -   0.0312     )     ⁢   0.05267                    +            (     0.5   -   0.0312     )     ⁢   0.01                      +            (     0.5   -   0.0312     )     ⁢   0.02                  )       =   0.0299                 
Similarly,
 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       5 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       
                         
                           m 
                           5 
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             { 
                             
                               1 
                               , 
                               2 
                               , 
                               3 
                             
                             } 
                           
                         
                         
                             
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 S 
                                 kJ 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           
                               
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               g 
                               ∈ 
                               
                                 { 
                                 1 
                                 } 
                               
                             
                             
                                 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   X 
                                   ig 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     E 
                                     ⁡ 
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                   } 
                                 
                               
                                
                             
                             ⁢ 
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     H 
                                     kj 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         X 
                                         ig 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           H 
                           kj 
                         
                         ⁢ 
                         
                            
                           
                             E 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           } 
                         
                       
                        
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       3 
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         X 
                         
                           5 
                           , 
                           1 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                  
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     X 
                                     
                                       5 
                                       , 
                                       1 
                                     
                                   
                                 
                                 } 
                               
                               - 
                               
                                 Pr 
                                 ⁢ 
                                 
                                   { 
                                   
                                     BX 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                   
                                   } 
                                 
                                  
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   + 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         X 
                                         
                                           5 
                                           , 
                                           1 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                     
                                     } 
                                   
                                   ⁢ 
                                   
                                      
                                     + 
                                      
                                   
                                   ⁢ 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         2 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     X 
                                     
                                       5 
                                       , 
                                       1 
                                     
                                   
                                 
                               
                               } 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 Pr 
                               
                               ⁢ 
                               
                                 { 
                                 
                                   BX 
                                   
                                     2 
                                     , 
                                     2 
                                   
                                 
                                 } 
                               
                               ⁢ 
                               
                                  
                                 
                                   
                                     + 
                                     
                                        
                                       
                                         Pr 
                                         ⁢ 
                                         
                                           { 
                                           
                                             BX 
                                             
                                               3 
                                               , 
                                               1 
                                             
                                           
                                            
                                         
                                         ⁢ 
                                         
                                           X 
                                           
                                             5 
                                             , 
                                             1 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           3 
                                           , 
                                           1 
                                         
                                       
                                       } 
                                     
                                   
                                 
                                  
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.030331 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ρ 
                       6 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       
                         
                           m 
                           6 
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             { 
                             
                               2 
                               , 
                               3 
                             
                             } 
                           
                         
                         
                             
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 S 
                                 kJ 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           
                               
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               g 
                               ∈ 
                               
                                 { 
                                 
                                   1 
                                   , 
                                   2 
                                 
                                 } 
                               
                             
                             
                                 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   X 
                                   ig 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     E 
                                     ⁡ 
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                   } 
                                 
                               
                                
                             
                             ⁢ 
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     H 
                                     kj 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         X 
                                         ig 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           H 
                           kj 
                         
                         ⁢ 
                         
                            
                           
                             E 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           } 
                         
                       
                        
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         X 
                         
                           6 
                           , 
                           1 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         X 
                                         
                                           6 
                                           , 
                                           1 
                                         
                                       
                                     
                                     } 
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                       
                                       } 
                                     
                                     ⁢ 
                                     
                                        
                                       + 
                                        
                                     
                                     ⁢ 
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           2 
                                           , 
                                           2 
                                         
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       X 
                                       
                                         6 
                                         , 
                                         1 
                                       
                                     
                                   
                                 
                                 } 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         2 
                                       
                                     
                                     } 
                                   
                                 
                                  
                               
                               + 
                               
                                  
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         
                                           BX 
                                           
                                             3 
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                            
                                           
                                             X 
                                             
                                               6 
                                               , 
                                               1 
                                             
                                           
                                           } 
                                         
                                       
                                       - 
                                       
                                         Pr 
                                         ⁢ 
                                         
                                           { 
                                           
                                             BX 
                                             
                                               3 
                                               , 
                                               1 
                                             
                                           
                                           } 
                                         
                                       
                                     
                                      
                                   
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       + 
                       
                         1 
                         2 
                       
                     
                     ⁢ 
                     
                       Pr 
                       ( 
                       
                         X 
                         
                           6 
                           , 
                           2 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         X 
                                         
                                           6 
                                           , 
                                           2 
                                         
                                       
                                     
                                     } 
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                       
                                       } 
                                     
                                     ⁢ 
                                     
                                        
                                       + 
                                        
                                     
                                     ⁢ 
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           2 
                                           , 
                                           2 
                                         
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       X 
                                       
                                         6 
                                         , 
                                         2 
                                       
                                     
                                   
                                 
                                 } 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             Pr 
                                             ⁢ 
                                             
                                               { 
                                               
                                                 BX 
                                                 
                                                   2 
                                                   , 
                                                   2 
                                                 
                                               
                                               } 
                                             
                                           
                                            
                                         
                                         + 
                                       
                                        
                                     
                                     ⁢ 
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           3 
                                           , 
                                           1 
                                         
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       X 
                                       
                                         6 
                                         , 
                                         2 
                                       
                                     
                                   
                                   } 
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         3 
                                         , 
                                         1 
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.040694 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ρ 
                       7 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       
                         
                           m 
                           7 
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             { 
                             
                               1 
                               , 
                               2 
                             
                             } 
                           
                         
                         
                             
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 S 
                                 kJ 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           
                               
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               g 
                               ∈ 
                               
                                 { 
                                 1 
                                 } 
                               
                             
                             
                                 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   X 
                                   ig 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     E 
                                     ⁡ 
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                   } 
                                 
                               
                                
                             
                             ⁢ 
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     H 
                                     kj 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         X 
                                         ig 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           H 
                           kj 
                         
                         ⁢ 
                         
                            
                           
                             E 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           } 
                         
                       
                        
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         X 
                         
                           7 
                           , 
                           1 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                    
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       X 
                                       
                                         7 
                                         , 
                                         1 
                                       
                                     
                                   
                                   } 
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                         
                           
                             
                               
                                 
                                   + 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         X 
                                         
                                           7 
                                           , 
                                           1 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                         
                           
                             
                               
                                 
                                   + 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             2 
                                           
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         X 
                                         
                                           7 
                                           , 
                                           1 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         2 
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.020938 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ρ 
                       8 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       
                         
                           m 
                           8 
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             { 
                             
                               1 
                               , 
                               2 
                               , 
                               3 
                             
                             } 
                           
                         
                         
                             
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 S 
                                 kJ 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           
                               
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               g 
                               ∈ 
                               
                                 { 
                                 1 
                                 } 
                               
                             
                             
                                 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   X 
                                   ig 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     E 
                                     ⁡ 
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                   } 
                                 
                               
                                
                             
                             ⁢ 
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     H 
                                     kj 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         X 
                                         ig 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           H 
                           kj 
                         
                         ⁢ 
                         
                            
                           
                             E 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           } 
                         
                       
                        
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       3 
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         X 
                         
                           8 
                           , 
                           1 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                  
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     X 
                                     
                                       8 
                                       , 
                                       1 
                                     
                                   
                                 
                                 } 
                               
                               - 
                               
                                 Pr 
                                 ⁢ 
                                 
                                   { 
                                   
                                     BX 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                   
                                   } 
                                 
                                  
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   + 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         X 
                                         
                                           8 
                                           , 
                                           1 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                     
                                     } 
                                   
                                   ⁢ 
                                   
                                      
                                     + 
                                      
                                   
                                   ⁢ 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         2 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     X 
                                     
                                       8 
                                       , 
                                       1 
                                     
                                   
                                 
                               
                               } 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 Pr 
                               
                               ⁢ 
                               
                                 { 
                                 
                                   BX 
                                   
                                     2 
                                     , 
                                     2 
                                   
                                 
                                 } 
                               
                               ⁢ 
                               
                                  
                                 
                                   
                                     + 
                                     
                                        
                                       
                                         Pr 
                                         ⁢ 
                                         
                                           { 
                                           
                                             BX 
                                             
                                               3 
                                               , 
                                               1 
                                             
                                           
                                            
                                         
                                         ⁢ 
                                         
                                           X 
                                           
                                             8 
                                             , 
                                             1 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           3 
                                           , 
                                           1 
                                         
                                       
                                       } 
                                     
                                   
                                 
                                  
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.01785 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ρ 
                       9 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       
                         
                           m 
                           9 
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             { 
                             
                               1 
                               , 
                               2 
                             
                             } 
                           
                         
                         
                             
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 S 
                                 kJ 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           
                               
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               g 
                               ∈ 
                               
                                 { 
                                 1 
                                 } 
                               
                             
                             
                                 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   X 
                                   ig 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     E 
                                     ⁡ 
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                   } 
                                 
                               
                                
                             
                             ⁢ 
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     H 
                                     kj 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         X 
                                         ig 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           H 
                           kj 
                         
                         ⁢ 
                         
                            
                           
                             E 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           } 
                         
                       
                        
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         X 
                         
                           9 
                           , 
                           1 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                    
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       X 
                                       
                                         9 
                                         , 
                                         1 
                                       
                                     
                                   
                                   } 
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                         
                           
                             
                               
                                 
                                   + 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         X 
                                         
                                           9 
                                           , 
                                           1 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                         
                           
                             
                               
                                 
                                   + 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             2 
                                           
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         X 
                                         
                                           9 
                                           , 
                                           1 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         2 
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.020938 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ρ 
                       10 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       
                         
                           m 
                           10 
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             { 
                             
                               1 
                               , 
                               2 
                               , 
                               3 
                             
                             } 
                           
                         
                         
                             
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 S 
                                 kJ 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           
                               
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               g 
                               ∈ 
                               
                                 { 
                                 1 
                                 } 
                               
                             
                             
                                 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   X 
                                   ig 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     E 
                                     ⁡ 
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                   } 
                                 
                               
                                
                             
                             ⁢ 
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     H 
                                     kj 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         X 
                                         ig 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
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                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
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                           kj 
                         
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                               ( 
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                        
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       3 
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         X 
                         
                           10 
                           , 
                           1 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                  
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     X 
                                     
                                       10 
                                       , 
                                       1 
                                     
                                   
                                 
                                 } 
                               
                               - 
                               
                                 Pr 
                                 ⁢ 
                                 
                                   { 
                                   
                                     BX 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                   
                                   } 
                                 
                                  
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   + 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
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                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
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                                     } 
                                   
                                   ⁢ 
                                   
                                      
                                     + 
                                      
                                   
                                   ⁢ 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       BX 
                                       
                                         2 
                                         , 
                                         2 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     X 
                                     
                                       10 
                                       , 
                                       1 
                                     
                                   
                                 
                               
                               } 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 Pr 
                               
                               ⁢ 
                               
                                 { 
                                 
                                   BX 
                                   
                                     2 
                                     , 
                                     2 
                                   
                                 
                                 } 
                               
                               ⁢ 
                               
                                  
                                 
                                   
                                     + 
                                     
                                        
                                       
                                         Pr 
                                         ⁢ 
                                         
                                           { 
                                           
                                             BX 
                                             
                                               3 
                                               , 
                                               1 
                                             
                                           
                                            
                                         
                                         ⁢ 
                                         
                                           X 
                                           
                                             10 
                                             , 
                                             1 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           3 
                                           , 
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                                       } 
                                     
                                   
                                 
                                  
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.021232 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ρ 
                       11 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       
                         
                           m 
                           11 
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
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                           ∈ 
                           
                             { 
                             
                               2 
                               , 
                               3 
                             
                             } 
                           
                         
                         
                             
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 S 
                                 kJ 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
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                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               g 
                               ∈ 
                               
                                 { 
                                 1 
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                           ⁢ 
                           
                               
                           
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                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
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                             ⁢ 
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                             ⁢ 
                             
                               { 
                               
                                 
                                   
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                                 - 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
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                           kj 
                         
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                        
                     
                   
                 
               
             
             
               
                 
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                       1 
                       2 
                     
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                     Pr 
                     ⁢ 
                     
                       { 
                       
                         X 
                         
                           11 
                           , 
                           1 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   
                                      
                                     
                                       Pr 
                                       ⁢ 
                                       
                                         { 
                                         
                                           BX 
                                           
                                             2 
                                             , 
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                                           11 
                                           , 
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                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           2 
                                           , 
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                                       } 
                                     
                                     ⁢ 
                                     
                                        
                                       + 
                                        
                                     
                                     ⁢ 
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           2 
                                           , 
                                           2 
                                         
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       X 
                                       
                                         11 
                                         , 
                                         1 
                                       
                                     
                                   
                                 
                                 } 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           2 
                                           , 
                                           2 
                                         
                                       
                                       } 
                                     
                                     ⁢ 
                                     
                                        
                                       + 
                                        
                                     
                                     ⁢ 
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         BX 
                                         
                                           3 
                                           , 
                                           1 
                                         
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       X 
                                       
                                         11 
                                         , 
                                         1 
                                       
                                     
                                   
                                   } 
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
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                                
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.027049 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ρ 
                       12 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       
                         
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                           12 
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
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                         ∑ 
                         
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                           ∈ 
                           
                             { 
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                             } 
                           
                         
                         
                             
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
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                             j 
                             ∈ 
                             
                               { 
                               1 
                               } 
                             
                           
                           
                               
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               g 
                               ∈ 
                               
                                 { 
                                 1 
                                 } 
                               
                             
                             
                                 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   X 
                                   
                                     12 
                                     ⁢ 
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                                 ⁢ 
                                 
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                                     ⁡ 
                                     
                                       ( 
                                       0 
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                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     H 
                                     kj 
                                   
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                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           H 
                           kj 
                         
                         ⁢ 
                         
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                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
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                        
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         
                           1 
                           1 
                         
                         ⁢ 
                         Pr 
                         ⁢ 
                         
                           { 
                           
                             X 
                             
                               12 
                               , 
                               1 
                             
                           
                           } 
                         
                         ⁢ 
                         
                            
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 BX 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                                
                             
                             ⁢ 
                             
                               X 
                               
                                 12 
                                 , 
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                           } 
                         
                       
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                         ⁢ 
                         
                           { 
                           
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                           } 
                         
                       
                     
                   
                    
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.063 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ρ 
                       13 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
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                       ⁢ 
                       
                           
                       
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                             ∑ 
                             
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                                 { 
                                 1 
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                           ⁢ 
                           
                               
                           
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                             ⁢ 
                             
                               { 
                               
                                 
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                     ⁢ 
                     
                       { 
                       
                         
                           H 
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                               0 
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                        
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         
                           1 
                           1 
                         
                         ⁢ 
                         Pr 
                         ⁢ 
                         
                           { 
                           
                             X 
                             
                               13 
                               , 
                               1 
                             
                           
                           } 
                         
                         ⁢ 
                         
                            
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 BX 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                                
                             
                             ⁢ 
                             
                               X 
                               
                                 13 
                                 , 
                                 1 
                               
                             
                           
                           } 
                         
                       
                       - 
                       
                         Pr 
                         ⁢ 
                         
                           { 
                           
                             BX 
                             
                               1 
                               , 
                               1 
                             
                           
                           } 
                         
                       
                     
                   
                    
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.00792 
                 
               
             
           
         
       
     
     According to 7, take 
                   ρ   i     ⁡     (   y   )       =       max     l   ∈       S     X   ⁢   s       ⁡     (   y   )           ⁢     {       ρ   l     ⁡     (   y   )       }         ,         
one can get the following:
 
ρ 15 (0)=ρ 16 (0)=max{0.0299, 0.030331, 0.040694, 0.020938, 0.01785, 0.020838, 0.021232, 0.027049, 0.063, 0.00792}=0.063.
 
And then:
 
                                             i   ρ i (0)                                                    4   0.0299           5   0.030331           6   0.040694           7   0.020938           8   0.01785           9   0.020938           10   0.021232           11   0.027049           12   0.063           13   0.00792           15   0.063           16   0.063                        
According to 6, let β i =1/α i  and assign their values as follows:
 
                                     i   α i     β i                                              4   100   0.01       5   2   0.5       6   50   0.02       7   1   1       8   2   0.5       9   2   0.5       10   2   0.5       11   1   1       12   1   1       13   1   1       15   1   1       16   1   1                    
According to 3, assume n=1, given m i (0), λ i (0) is calculated as follows:
 
                                     i   m i (0)   λ i (0)                                            4   2   1/2       5   3   1/3       6   2   1/2       7   1   1       8   3   1/3       9   1   1       10   1   1       11   1   1       12   1   1       13   1   1       15   1   1       16   1   1                    
In which, according to 7, let λ 15 (0)=λ 16 (0)=1.
 
According to 8, take
 
                   I   i     ⁡     (   y   )       =           λ   i     ⁡     (   y   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   y   )             Σ     i   ∈     S     X     (   y   )             ⁢       λ   i     ⁡     (   y   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   y   )             ,         
we can obtain the following results:
 
                 I   4     ⁡     (   0   )       =             λ   4     ⁡     (   0   )       ⁢     β   4     ⁢       ρ   4     ⁡     (   0   )             ∑     i   ∈     {     4   ,   …   ,   13   ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =           1   2     ×   0.01   ×   0.0299         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   0     ⁢   0   ⁢   0   ⁢   5   ⁢   44                         I   5     ⁡     (   0   )       =             λ   5     ⁡     (   0   )       ⁢     β   5     ⁢       ρ   5     ⁡     (   0   )             ∑     i   ∈     {     4   ,   …   ,   13   ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =           1   3     ×     0   .   5     ×     0   .   0     ⁢   3   ⁢   0   ⁢   3   ⁢   3   ⁢   1         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   0     ⁢   1   ⁢   8   ⁢   4   ⁢   1   ⁢   1                         I   6     ⁡     (   0   )       =             λ   6     ⁡     (   0   )       ⁢     β   6     ⁢       ρ   6     ⁡     (   0   )             ∑     i   ∈     {       4   ⁢     ,…,     ⁢   13     ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =           1   2     ×     0   .   0     ⁢   2   ×   0.040694         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   0     ⁢   01482                         I   7     ⁡     (   0   )       =             λ   7     ⁡     (   0   )       ⁢     β   7     ⁢       ρ   7     ⁡     (   0   )             ∑     i   ∈     {     4   ,       ...,     ⁢   13     ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =         1   ×   1   ×   0.020   ⁢   9   ⁢   3   ⁢   8         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   0     ⁢   7   ⁢   6   ⁢   2   ⁢   5   ⁢   5                         I   8     ⁡     (   0   )       =             λ   8     ⁡     (   0   )       ⁢     β   8     ⁢       ρ   8     ⁡     (   0   )             ∑     i   ∈     {     4   ,       ...,     ⁢   13     ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =           1   3     ×   0.5   ×   0.017   ⁢   8   ⁢   5         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   0     ⁢   10835                         I   9     ⁡     (   0   )       =             λ   9     ⁡     (   0   )       ⁢     β   9     ⁢       ρ   9     ⁡     (   0   )             ∑     i   ∈     {     4   ,       ...,     ⁢   13     ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =         1   ×   0.5   ×   0.020   ⁢   9   ⁢   3   ⁢   8       0   ⁢   2   ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   0     ⁢   38128                         I     1   ⁢   0       ⁡     (   0   )       =             λ     1   ⁢   0       ⁡     (   0   )       ⁢     β     1   ⁢   0       ⁢       ρ     1   ⁢   0       ⁡     (   0   )             ∑     i   ∈     {     4   ,       ...,     ⁢   13     ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =         1   ×   0.5   ×   0.021   ⁢   2   ⁢   3   ⁢   2       0.274   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   0     ⁢   38663                         I     1   ⁢   1       ⁡     (   0   )       =             λ     1   ⁢   1       ⁡     (   0   )       ⁢     β     1   ⁢   1       ⁢       ρ     1   ⁢   1       ⁡     (   0   )             ∑     i   ∈     {     4   ,       ...,1     ⁢   3     ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =         1   ×   1   ×   0.027   ⁢   0   ⁢   4   ⁢   9         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   0     ⁢   98511                         I     1   ⁢   2       ⁡     (   0   )       =             λ     1   ⁢   2       ⁡     (   0   )       ⁢     β     1   ⁢   2       ⁢       ρ     1   ⁢   2       ⁡     (   0   )             ∑     i   ∈     {     4   ,       ...,     ⁢   13     ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =         1   ×   1   ×   0.063         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   2     ⁢   29442                         I     1   ⁢   3       ⁡     (   0   )       =             λ     1   ⁢   3       ⁡     (   0   )       ⁢     β     1   ⁢   3       ⁢       ρ     1   ⁢   3       ⁡     (   0   )             ∑     i   ∈     {     4   ,       ...,     ⁢   13     ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =         1   ×   1   ×   0.007   ⁢   9   ⁢   2         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   0     ⁢   28844                         I   15     ⁡     (   0   )       =             λ   15     ⁡     (   0   )       ⁢     β   15     ⁢       ρ   15     ⁡     (   0   )             ∑     i   ∈     {     4   ,     ...     ,   13   ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =         1   ×   1   ×   0.063         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   2     ⁢   29442                         I     1   ⁢   6       ⁡     (   0   )       =             λ     1   ⁢   6       ⁡     (   0   )       ⁢     β     1   ⁢   6       ⁢       ρ     1   ⁢   6       ⁡     (   0   )             ∑     i   ∈     {     4   ,       ...,     ⁢   13     ,   15   ,   16     }         ⁢         λ   i     ⁡     (   0   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   0   )             =         1   ×   1   ×   0.063         0   .   2     ⁢   7   ⁢   4   ⁢   5   ⁢   7   ⁢   8   ⁢   6   ⁢   1       =       0   .   2     ⁢   2   ⁢   9   ⁢   4   ⁢   4   ⁢   2               
According to 9, the ranking results of X i  are:
 
     
       
         
           
               
               
               
             
               
                   
               
               
                 index 
                 i 
                 I i (0) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 1 
                 12 
                 0.229442 
               
               
                 2 
                 15 
                 0.229442 
               
               
                 3 
                 16 
                 0.229442 
               
               
                 4 
                 11 
                 0.098511 
               
               
                 5 
                 7 
                 0.076255 
               
               
                 6 
                 10 
                 0.038663 
               
               
                 7 
                 9 
                 0.038128 
               
               
                 8 
                 13 
                 0.028844 
               
               
                 9 
                 5 
                 0.018411 
               
               
                 10 
                 8 
                 0.010835 
               
               
                 11 
                 6 
                 0.001482 
               
               
                 12 
                 4 
                 0.000544 
               
               
                   
               
            
           
         
       
     
     Take the first 5 X-type variables into detection. Assuming the detection results are X 7,1 , X 11,0 , X 12,1 , X 15,0  and X 16,0 ,  FIG. 4  changes as  FIG. 8 , in which, X 15  and X 16  are cause-specific variables for BX 3,1  and all the detection results are negative (variables are all at state 0), thus we know that the state of BX 3  is BX 3,0 . 
     According to the parameters in Example 1,  FIG. 8  is simplified as  FIG. 9  by using the aforementioned simplification rules 2, 3 and 5, and E(1)=E + (0)E(0)=E + (0)=X 7,1 X 11,0 X 12,1 . Based on E(1) and similar to example 1,  FIG. 9  can be divided and simplified as  FIG. 10  and  FIG. 11  according to the DUCG algorithms in [8]. 
     According to the algorithms in [4]-[9], calculate the probabilities ζ i (y)=ζ i (1) of the sub-graph  FIG. 9  and  FIG. 10 : 
     Based on  FIG. 10 , 
                       ζ   1     ⁡     (   1   )       =       ⁢     Pr   ⁢     {     E   ⁡     (   1   )       }                   =       ⁢     Pr   ⁢     {       X     7   ,   1       ⁢     X       1   ⁢   1     ,   0       ⁢     X     12   ,   1         }                   =       ⁢     Pr   ⁢     {       X     7   ,   1       ⁢     X     12   ,   1         }                   =       ⁢     Pr   ⁢     {         r     7   ;   4         r   7       ⁢     A     7   ,     1   ;   4         ⁢       r     4   ;   1         r   4       ⁢     A     4   ;   1       ⁢   B   ⁢     X   1     ⁢     X     12   ,   1         }                   =       ⁢     Pr   ⁢     {       1   1     ⁢     A     7   ,     1   ;   4         ⁢     1   1     ⁢       A     4   ;   1       ⁡     (           r     1   ;   1         r   1       ⁢     A     1   ;   1       ⁢     B   1       +         r     1   ;   12         r   1       ⁢     A       1   ;   12     ,   1       ⁢     X     12   ,   1           )       ⁢     X     12   ,   1         }                   =       ⁢     Pr   ⁢     {       A     7   ,     1   ;   4         ⁢       A     4   ;   1       ⁡     (         1   2     ⁢     A     1   ;   1       ⁢     B   1       +       1   2     ⁢     A       1   ;   12     ,   1           )       ⁢     X     12   ,   1         }                   =       ⁢     Pr   ⁢     {       A     7   ,     1   ;   4         ⁢       A     4   ;   1       ⁡     (         1   2     ⁢     A     1   ;   1       ⁢     B   1       +       1   2     ⁢     A       1   ;   12     ,   1           )       ⁢     A       1   ⁢   2     ,     1   ;   D           }                   =       ⁢       a     7   ,     1   ;   4         ⁢       a     4   ;   1       ⁡     (         1   2     ⁢     a     1   ;   1       ⁢     b   1       +       1   2     ⁢     a       1   ;   12     ,   1           )       ⁢     a     12   ,     1   ;   D                       =       ⁢       (     -   0.7     )     ⁢     (         -       -           -       0.9         )     ⁢     (         1   2     ⁢           ⁢     (         -       -           -       1         )     ⁢     (         -           0.08         )       +       1   2     ⁢     (         -           0.7         )         )     ⁢   0.1                 =       ⁢     0.02   ⁢   457                 
Based on  FIG. 11 ,
 
                       ζ   2     ⁡     (   1   )       =       ⁢     Pr   ⁢     {     E   ⁡     (   1   )       }                   =       ⁢     Pr   ⁢     {       X     7   ,   1       ⁢     X     11   ,   0       ⁢     X     12   ,   1         }                   =       ⁢     Pr   ⁢     {       (         r     7   ;   4         r   7       ⁢     A     7   ,     1   ;   4         ⁢       r     4   ;   2         r   4       ⁢     A     4   ;   2       ⁢     BX   2       )     ⁢     (         r     11   ;   6         r   11       ⁢     A     11   ,     0   ;   6         ⁢       r     6   ;   2         r   6       ⁢     A     6   ;   2       ⁢     BX   2       )     ⁢     A     12   ,     1   ;   D           }                   =       ⁢     Pr   ⁢     {       (       1   1     ⁢     A     7   ,     1   ;   4         ⁢     1   1     ⁢     A     4   ;   2         )     *     (       1   1     ⁢     A     11   ,     0   ;   6         ⁢     1   1     ⁢     A     6   ;   2         )     ⁢     A     2   ;   2       ⁢     B   2     ⁢     A     12   ,     1   ;   D           }                   =       ⁢       (       a     7   ,     1   ;   4         ⁢     a     4   ;   2         )     *     (       a     11   ,     0   ;   6         ⁢     a     6   ;   2         )     ⁢     a     2   ;   2       ⁢     b   2     ⁢     a     12   ,     1   ;   D                       =       ⁢       (       (         -       0.7         )     ⁢     (         -       -       -           -       0.5       0.5         )       )     *     (     (         1         1   -   0.3           1   -   0.8           )                           ⁢     (         -       -       -           -       0.6       0.3           -       0.4       0.7         )     )     ⁢     (         -       -       -           -       1       -           -       -       1         )     ⁢     (         -           0.01           0.02         )     ⁢   0.1               =       ⁢   0.00042               
In which, the original parameter a 11,0;6 =( - - - ) is modified as a 11,0;6 =(1 1-0.3 1-0.8), because X 11,0  is a negative evidence, which means none of the abnormal states occurs and X 11,0 =1−X 11,1  according to [4].
 
     According to the algorithms in [8], the weighting coefficients 
                 ξ   i     ⁡     (   y   )       =         ζ   i     ⁡     (   y   )           ∑   j     ⁢       ζ   j     ⁡     (   y   )                 
of the sub-graphs are:
 
                 ξ   1     ⁡     (   1   )       =           ζ   1     ⁡     (   1   )             ζ   1     ⁡     (   1   )       +       ζ   2     ⁡     (   1   )           =           0   .   0     ⁢   2   ⁢   4   ⁢   5   ⁢   7           0   .   0     ⁢   2   ⁢   4   ⁢   5   ⁢   7     +       0   .   0     ⁢   0   ⁢   0   ⁢   4   ⁢   2         =       0   .   9     ⁢   832                         ξ   2     ⁡     (   1   )       =           ζ   2     ⁡     (   1   )             ζ   1     ⁡     (   1   )       +       ζ   2     ⁡     (   1   )           =           0   .   0     ⁢   0042           0   .   0     ⁢   2   ⁢   4   ⁢   5   ⁢   7     +     0.000   ⁢   4   ⁢   2         =       0   .   0     ⁢   1   ⁢   6   ⁢   8   ⁢   1               
According to 1 and based on the DUCG algorithms in [4]-[9], when y=0+1=1, the state probability h kj   s (y)=h kj   s (1) of H kj  is:
 
                       h     1   ,   1     s     ⁡     (   1   )       =       ⁢         ξ   1     ⁡     (   1   )       ⁢   Pr   ⁢     {       BX     1   ,   1       ❘     E   ⁡     (   1   )         }                   =       ⁢         ξ   1     ⁡     (   1   )       ⁢   Pr   ⁢     {       BX     1   ,   1       ❘       X     7   ,   1       ⁢     X     11   ,   0       ⁢     X     12   ,   1           }                   =       ⁢         ξ   1     ⁡     (   1   )       ⁢       Pr   ⁢     {       BX     1   ,   1       ⁢     X     7   ,   1       ⁢     X     11   ,   0       ⁢     X     12   ,   1         }         Pr   ⁢     {       X     7   ,   1       ⁢     X     11   ,   0       ⁢     X     12   ,   1         }                       =       ⁢     0.983193   ×     0.00022932   0.0003574                   =       ⁢     0.983193   ×   0.64163                 =       ⁢   0.63085               
In which
 
     
       
         
           
             
               Pr 
               ⁢ 
               
                 { 
                 
                   
                     X 
                     
                       7 
                       , 
                       1 
                     
                   
                   ⁢ 
                   
                     X 
                     
                       11 
                       , 
                       0 
                     
                   
                   ⁢ 
                   
                     X 
                     
                       12 
                       , 
                       1 
                     
                   
                 
                 } 
               
             
             = 
             
               
                 Pr 
                 ⁢ 
                 
                   { 
                   
                     
                       
                         r 
                         
                           7 
                           ; 
                           4 
                         
                       
                       
                         r 
                         7 
                       
                     
                     ⁢ 
                     
                       
                         A 
                         
                           7 
                           , 
                           
                             1 
                             ; 
                             4 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 r 
                                 
                                   4 
                                   ; 
                                   1 
                                 
                               
                               
                                 r 
                                 4 
                               
                             
                             ⁢ 
                             
                               A 
                               
                                 4 
                                 ; 
                                 1 
                               
                             
                             ⁢ 
                             
                               BX 
                               1 
                             
                           
                           + 
                           
                             
                               
                                 r 
                                 
                                   4 
                                   ; 
                                   2 
                                 
                               
                               
                                 r 
                                 4 
                               
                             
                             ⁢ 
                             
                               A 
                               
                                 4 
                                 ; 
                                 2 
                               
                             
                             ⁢ 
                             
                               BX 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         r 
                         
                           11 
                           ; 
                           6 
                         
                       
                       
                         r 
                         11 
                       
                     
                     ⁢ 
                     
                       A 
                       
                         11 
                         , 
                         
                           0 
                           ; 
                           6 
                         
                       
                     
                     ⁢ 
                     
                       
                         r 
                         
                           6 
                           ; 
                           2 
                         
                       
                       
                         r 
                         6 
                       
                     
                     ⁢ 
                     
                       A 
                       
                         6 
                         ; 
                         2 
                       
                     
                     ⁢ 
                     
                       BX 
                       2 
                     
                     ⁢ 
                     
                       X 
                       
                         12 
                         , 
                         1 
                       
                     
                   
                   } 
                 
               
               = 
               
                 
 
               
               ⁢ 
               
                 
                   Pr 
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 r 
                                 
                                   7 
                                   ; 
                                   4 
                                 
                               
                               
                                 r 
                                 7 
                               
                             
                             ⁢ 
                             
                               A 
                               
                                 7 
                                 , 
                                 
                                   1 
                                   ; 
                                   4 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 r 
                                 
                                   4 
                                   ; 
                                   1 
                                 
                               
                               
                                 r 
                                 4 
                               
                             
                             ⁢ 
                             
                               A 
                               
                                 4 
                                 ; 
                                 1 
                               
                             
                             ⁢ 
                             
                               BX 
                               1 
                             
                             ⁢ 
                             
                               
                                 r 
                                 
                                   11 
                                   ; 
                                   6 
                                 
                               
                               
                                 r 
                                 11 
                               
                             
                             ⁢ 
                             
                               A 
                               
                                 11 
                                 , 
                                 
                                   0 
                                   ; 
                                   6 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 r 
                                 
                                   6 
                                   ; 
                                   2 
                                 
                               
                               
                                 r 
                                 6 
                               
                             
                             ⁢ 
                             
                               A 
                               
                                 6 
                                 ; 
                                 2 
                               
                             
                             ⁢ 
                             
                               BX 
                               2 
                             
                             ⁢ 
                             
                               X 
                               
                                 12 
                                 , 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               + 
                               
                                 
                                   BX 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           7 
                                           ; 
                                           4 
                                         
                                       
                                       
                                         r 
                                         7 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         7 
                                         , 
                                         
                                           1 
                                           ; 
                                           4 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           4 
                                           ; 
                                           2 
                                         
                                       
                                       
                                         r 
                                         4 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         4 
                                         ; 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             * 
                             
                               ( 
                               
                                 
                                   
                                     r 
                                     
                                       11 
                                       ; 
                                       6 
                                     
                                   
                                   
                                     r 
                                     11 
                                   
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     11 
                                     , 
                                     
                                       0 
                                       ; 
                                       6 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     r 
                                     
                                       6 
                                       ; 
                                       2 
                                     
                                   
                                   
                                     r 
                                     6 
                                   
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     6 
                                     ; 
                                     2 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               BX 
                               2 
                             
                             ⁢ 
                             
                               X 
                               
                                 12 
                                 , 
                                 1 
                               
                             
                           
                         
                       
                     
                     } 
                   
                 
                 = 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       r 
                                       
                                         7 
                                         ; 
                                         4 
                                       
                                     
                                     
                                       r 
                                       7 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       7 
                                       , 
                                       
                                         1 
                                         ; 
                                         4 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         4 
                                         ; 
                                         1 
                                       
                                     
                                     
                                       r 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       A 
                                       
                                         4 
                                         ; 
                                         1 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   r 
                                                   
                                                     1 
                                                     ; 
                                                     1 
                                                   
                                                 
                                                 
                                                   r 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 A 
                                                 
                                                   1 
                                                   ; 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 B 
                                                 1 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 + 
                                                 
                                                   
                                                     r 
                                                     
                                                       1 
                                                       ; 
                                                       12 
                                                     
                                                   
                                                   
                                                     r 
                                                     1 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 A 
                                                 
                                                   1 
                                                   ; 
                                                   12 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         11 
                                         ; 
                                         6 
                                       
                                     
                                     
                                       r 
                                       11 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       11 
                                       , 
                                       
                                         0 
                                         ; 
                                         6 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                     
                                       r 
                                       6 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       6 
                                       ; 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     + 
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             
                                               7 
                                               ; 
                                               4 
                                             
                                           
                                           
                                             r 
                                             7 
                                           
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             7 
                                             , 
                                             
                                               1 
                                               ; 
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               4 
                                               ; 
                                               2 
                                             
                                           
                                           
                                             r 
                                             4 
                                           
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             4 
                                             ; 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   * 
                                   
                                     ( 
                                     
                                       
                                         
                                           r 
                                           
                                             11 
                                             ; 
                                             6 
                                           
                                         
                                         
                                           r 
                                           11 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           11 
                                           , 
                                           
                                             0 
                                             ; 
                                             6 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           r 
                                           
                                             6 
                                             ; 
                                             2 
                                           
                                         
                                         
                                           r 
                                           6 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           6 
                                           ; 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             r 
                             
                               2 
                               ; 
                               2 
                             
                           
                           
                             r 
                             2 
                           
                         
                         ⁢ 
                         
                           A 
                           
                             2 
                             ; 
                             2 
                           
                         
                         ⁢ 
                         
                           B 
                           2 
                         
                         ⁢ 
                         
                           X 
                           
                             12 
                             , 
                             1 
                           
                         
                       
                       } 
                     
                   
                   = 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           7 
                                           ; 
                                           4 
                                         
                                       
                                       
                                         r 
                                         7 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         7 
                                         , 
                                         
                                           1 
                                           ; 
                                           4 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           4 
                                           ; 
                                           1 
                                         
                                       
                                       
                                         r 
                                         4 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         a 
                                         
                                           4 
                                           ; 
                                           1 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     r 
                                                     
                                                       1 
                                                       ; 
                                                       1 
                                                     
                                                   
                                                   
                                                     r 
                                                     1 
                                                   
                                                 
                                                 ⁢ 
                                                 
                                                   a 
                                                   
                                                     1 
                                                     ; 
                                                     1 
                                                   
                                                 
                                                 ⁢ 
                                                 
                                                   b 
                                                   
                                                     1 
                                                     , 
                                                     1 
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   + 
                                                   
                                                     
                                                       r 
                                                       
                                                         1 
                                                         ; 
                                                         12 
                                                       
                                                     
                                                     
                                                       r 
                                                       1 
                                                     
                                                   
                                                 
                                                 ⁢ 
                                                 
                                                   a 
                                                   
                                                     1 
                                                     , 
                                                     
                                                       ; 
                                                       12 
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           11 
                                           ; 
                                           6 
                                         
                                       
                                       
                                         r 
                                         11 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         11 
                                         , 
                                         
                                           0 
                                           ; 
                                           6 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           6 
                                           ; 
                                           2 
                                         
                                       
                                       
                                         r 
                                         6 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   + 
                                   
                                     ( 
                                     
                                       
                                         
                                           r 
                                           
                                             7 
                                             ; 
                                             4 
                                           
                                         
                                         
                                           r 
                                           7 
                                         
                                       
                                       ⁢ 
                                       
                                         a 
                                         
                                           7 
                                           , 
                                           
                                             1 
                                             ; 
                                             4 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           r 
                                           
                                             4 
                                             ; 
                                             2 
                                           
                                         
                                         
                                           r 
                                           4 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           4 
                                           ; 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           11 
                                           ; 
                                           6 
                                         
                                       
                                       
                                         r 
                                         11 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         11 
                                         , 
                                         
                                           0 
                                           ; 
                                           6 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           6 
                                           ; 
                                           2 
                                         
                                       
                                       
                                         r 
                                         6 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         a 
                         
                           2 
                           ; 
                           2 
                         
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                       ⁢ 
                       
                         a 
                         
                           
                             12 
                             , 
                             
                               1 
                               ; 
                               D 
                             
                           
                           ⁢ 
                           
                             
 
                           
                         
                       
                     
                     = 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               0.7 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         1 
                                         2 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.9 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               1 
                                               2 
                                             
                                             ⁢ 
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     - 
                                                   
                                                   
                                                     - 
                                                   
                                                 
                                                 
                                                   
                                                     - 
                                                   
                                                   
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                             ⁢ 
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     - 
                                                   
                                                 
                                                 
                                                   
                                                     0.08 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           
                                             
                                               1 
                                               2 
                                             
                                             ⁢ 
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     - 
                                                   
                                                 
                                                 
                                                   
                                                     0.08 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                   · 
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               1 
                                             
                                             
                                               
                                                 1 
                                                 - 
                                                 0.3 
                                               
                                             
                                             
                                               
                                                 1 
                                                 - 
                                                 0.8 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.6 
                                             
                                             
                                               0.3 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.4 
                                             
                                             
                                               0.7 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     + 
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 - 
                                               
                                               
                                                 0.7 
                                               
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           1 
                                           2 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               
                                                 - 
                                               
                                               
                                                 - 
                                               
                                               
                                                 - 
                                               
                                             
                                             
                                               
                                                 - 
                                               
                                               
                                                 0.5 
                                               
                                               
                                                 0.5 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                   * 
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               1 
                                             
                                             
                                               
                                                 1 
                                                 - 
                                                 0.3 
                                               
                                             
                                             
                                               
                                                 1 
                                                 - 
                                                 0.8 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.6 
                                             
                                             
                                               0.3 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.4 
                                             
                                             
                                               0.7 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 - 
                               
                               
                                 - 
                               
                               
                                 - 
                               
                             
                             
                               
                                 - 
                               
                               
                                 1 
                               
                               
                                 - 
                               
                             
                             
                               
                                 - 
                               
                               
                                 - 
                               
                               
                                 1 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 - 
                               
                             
                             
                               
                                 0.01 
                               
                             
                             
                               
                                 0.02 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         0.1 
                       
                       ⁢ 
                       
                           
                       
                       = 
                       
                         
 
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 0.12285 
                                 × 
                                 
                                   ( 
                                   
                                     
                                       
                                         - 
                                       
                                       
                                         0.5 
                                       
                                       
                                         0.35 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 0.39 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         - 
                                       
                                       
                                         0.175 
                                       
                                       
                                         0.175 
                                       
                                     
                                   
                                   ) 
                                 
                                 * 
                                 
                                   ( 
                                   
                                     
                                       
                                         - 
                                       
                                       
                                         0.5 
                                       
                                       
                                         0.35 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   - 
                                 
                               
                               
                                 
                                   0.01 
                                 
                               
                               
                                 
                                   0.02 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           0.1 
                         
                         ⁢ 
                         
                             
                         
                         = 
                         
                           
 
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     - 
                                   
                                   
                                     0.148925 
                                   
                                   
                                     0.104248 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     - 
                                   
                                 
                                 
                                   
                                     0.01 
                                   
                                 
                                 
                                   
                                     0.02 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             0.1 
                           
                           ⁢ 
                           
                               
                           
                           = 
                           
                               
                           
                           ⁢ 
                           0.0003574 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               Pr 
               ⁢ 
               
                 { 
                 
                   
                     BX 
                     
                       1 
                       , 
                       1 
                     
                   
                   ⁢ 
                   
                     X 
                     
                       7 
                       , 
                       1 
                     
                   
                   ⁢ 
                   
                     X 
                     
                       11 
                       , 
                       0 
                     
                   
                   ⁢ 
                   
                     X 
                     
                       12 
                       , 
                       1 
                     
                   
                 
                 } 
               
             
             = 
             
               
 
             
             ⁢ 
             
               
                 Pr 
                 ⁢ 
                 
                   { 
                   
                     
                       BX 
                       
                         1 
                         , 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         r 
                         
                           7 
                           ; 
                           4 
                         
                       
                       
                         r 
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                     ⁢ 
                     
                       
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                         ( 
                         
                           
                             
                               
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                             ⁢ 
                             
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                         ) 
                       
                     
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                         r 
                         
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                         r 
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                         r 
                         
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                         r 
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                     ⁢ 
                     
                       A 
                       
                         6 
                         ; 
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                     ⁢ 
                     
                       BX 
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                         12 
                         , 
                         
                           1 
                           ; 
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                   } 
                 
               
               ⁢ 
               
                   
               
               = 
               
                 
 
               
               ⁢ 
               
                 
                   Pr 
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 r 
                                 
                                   7 
                                   ; 
                                   4 
                                 
                               
                               
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                                   BX 
                                   
                                     1 
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                                   ( 
                                   
                                     
                                       
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                 ⁢ 
                 
                     
                 
                 = 
                 
                   
 
                 
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                       { 
                       
                         
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                             12 
                             , 
                             
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                               ; 
                               D 
                             
                           
                         
                       
                       } 
                     
                   
                   ⁢ 
                   
                       
                   
                   = 
                   
                     
 
                   
                   ⁢ 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           7 
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                                         r 
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                                         r 
                                         
                                           4 
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                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     r 
                                                     
                                                       1 
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                                   ) 
                                 
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                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           11 
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                                         6 
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                                                 r 
                                                 
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                                                 r 
                                                 1 
                                               
                                             
                                             ⁢ 
                                             
                                               a 
                                               
                                                 1 
                                                 , 
                                                 
                                                   1 
                                                   ; 
                                                   1 
                                                 
                                                 , 
                                                 1 
                                               
                                             
                                             ⁢ 
                                             
                                               b 
                                               
                                                 1 
                                                 , 
                                                 1 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               + 
                                               
                                                 
                                                   r 
                                                   
                                                     1 
                                                     ; 
                                                     12 
                                                   
                                                 
                                                 
                                                   r 
                                                   1 
                                                 
                                               
                                             
                                             ⁢ 
                                             
                                               a 
                                               
                                                 1 
                                                 , 
                                                 
                                                   1 
                                                   ; 
                                                   12 
                                                 
                                                 , 
                                                 1 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           7 
                                           ; 
                                           4 
                                         
                                       
                                       
                                         r 
                                         7 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         7 
                                         , 
                                         
                                           1 
                                           ; 
                                           4 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           4 
                                           ; 
                                           2 
                                         
                                       
                                       
                                         r 
                                         4 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         4 
                                         ; 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 * 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           11 
                                           ; 
                                           6 
                                         
                                       
                                       
                                         r 
                                         11 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         11 
                                         , 
                                         
                                           0 
                                           ; 
                                           6 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           6 
                                           ; 
                                           2 
                                         
                                       
                                       
                                         r 
                                         6 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         a 
                         
                           2 
                           ; 
                           2 
                         
                       
                       ⁢ 
                       
                         b 
                         2 
                       
                       ⁢ 
                       
                         a 
                         
                           
                             12 
                             , 
                             
                               1 
                               ; 
                               D 
                             
                           
                           ⁢ 
                           
                             
 
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     = 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               0.7 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         1 
                                         2 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.9 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               1 
                                               2 
                                             
                                             × 
                                             0.08 
                                           
                                           + 
                                           
                                             
                                               1 
                                               2 
                                             
                                             × 
                                             0.7 
                                           
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               1 
                                             
                                             
                                               
                                                 1 
                                                 - 
                                                 0.3 
                                               
                                             
                                             
                                               
                                                 1 
                                                 - 
                                                 0.8 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.6 
                                             
                                             
                                               0.3 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.4 
                                             
                                             
                                               0.7 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     + 
                                     
                                       ( 
                                       
                                         
                                           
                                             1 
                                             2 
                                           
                                           × 
                                           0.08 
                                         
                                         + 
                                         
                                           
                                             1 
                                             2 
                                           
                                           × 
                                           0.7 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               0.7 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         1 
                                         2 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.5 
                                             
                                             
                                               0.5 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                   * 
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               1 
                                             
                                             
                                               
                                                 1 
                                                 - 
                                                 0.3 
                                               
                                             
                                             
                                               
                                                 1 
                                                 - 
                                                 0.8 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                             
                                               - 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.6 
                                             
                                             
                                               0.3 
                                             
                                           
                                           
                                             
                                               - 
                                             
                                             
                                               0.4 
                                             
                                             
                                               0.7 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 - 
                               
                             
                             
                               
                                 0.01 
                               
                             
                             
                               
                                 0.02 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         0.1 
                       
                       ⁢ 
                       
                           
                       
                       = 
                       
                         
 
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 0.12285 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         - 
                                       
                                       
                                         0.5 
                                       
                                       
                                         0.35 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 0.39 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         - 
                                       
                                       
                                         0.175 
                                       
                                       
                                         0.175 
                                       
                                     
                                   
                                   ) 
                                 
                                 * 
                                 
                                   ( 
                                   
                                     
                                       
                                         - 
                                       
                                       
                                         0.5 
                                       
                                       
                                         0.35 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   - 
                                 
                               
                               
                                 
                                   0.01 
                                 
                               
                               
                                 
                                   0.02 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           0.1 
                         
                         ⁢ 
                         
                             
                         
                         = 
                         
                             
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     - 
                                   
                                   
                                     0.09555 
                                   
                                   
                                     0.066885 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     - 
                                   
                                 
                                 
                                   
                                     0.01 
                                   
                                 
                                 
                                   
                                     0.02 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             0.1 
                           
                           ⁢ 
                           
                               
                           
                           = 
                           
                               
                           
                           ⁢ 
                           0.00022932 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 In 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 the 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 same 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 way 
               
               , 
               
                 
 
               
               ⁢ 
               
                 
                   
                     
                       
                         
                           h 
                           
                             2 
                             , 
                             1 
                           
                           s 
                         
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             ξ 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             1 
                             ) 
                           
                         
                         ⁢ 
                         Pr 
                         ⁢ 
                         
                           { 
                           
                             
                               BX 
                               
                                 2 
                                 , 
                                 1 
                               
                             
                             ❘ 
                             
                               E 
                               ⁡ 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             ξ 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             1 
                             ) 
                           
                         
                         ⁢ 
                         Pr 
                         ⁢ 
                         
                           { 
                           
                             
                               BX 
                               
                                 2 
                                 , 
                                 1 
                               
                             
                             ❘ 
                             
                               
                                 X 
                                 
                                   7 
                                   , 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 X 
                                 
                                   11 
                                   , 
                                   0 
                                 
                               
                               ⁢ 
                               
                                 X 
                                 
                                   12 
                                   , 
                                   1 
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       0.007 
                     
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       h 
                       
                         2 
                         , 
                         2 
                       
                       s 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ξ 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           BX 
                           
                             2 
                             , 
                             2 
                           
                         
                         ❘ 
                         
                           E 
                           ⁡ 
                           
                             ( 
                             1 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ξ 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           BX 
                           
                             2 
                             , 
                             2 
                           
                         
                         ❘ 
                         
                           
                             X 
                             
                               7 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               11 
                               , 
                               0 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               12 
                               , 
                               1 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.009781 
                 
               
             
           
         
       
     
     Based on 
                   h   kj   r     ⁡     (   1   )       =         h   kj   s     ⁡     (   1   )           ∑       H   kj     ∈       S   H     ⁡     (   1   )           ⁢       h   kj   s     ⁡     (   1   )             ,         
the rank probabilities are:
 
                                     index   H kj     h kj   r  (1)                  1   BX 1, 1     0.974089       2   BX 2, 2     0.015103       3   BX 2, 1     0.010809                    
Compared to the rank probabilities h kj   r (0) before detections, we see (1) BX 3,1  which ranks first is eliminated and the possible result is a reduced state space S H (1), and (2) the rank probability of BX 1,1  is far bigger than the rest two, which means that the cause of abnormality in the real target system can be verified as BX 1,1  with only five X-type variables being detected.
 
     Example 2: y=1 
     According to 2, when no evidence (y=0), based on  FIG. 4 , S H (0)={H 1,1 , H 2,1 , H 2,2 , H 3,1 }={BX 1,1 , BX 2,1 , BX 2,2 , BX 3,1 }, and the state pending X-type variables are {X 4 , X 5 , X 6 , X 7 , X 8 , X 9 , X 10 , X 11 , X 12 , X 13 , X 15 , X 16 }. All these states are unknown and single-connect to H k  in S H (0) without state-known blocking variables, so that S X (0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16}. Since X 15  and X 16  are cause-specific variable for BX 3,1 , they are eliminated from S X (0), that means S Xs (0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. Accordingly, S 4G (0)=S 5G (0)=S 7G (0)=S 8G (0)=S 9G (0)=S 10G (0)=S 11G (0)=S 12G (0)=S 13G (0)={1} and S 6G (0)={1, 2}. 
     As example 1 and based on  FIG. 9 , E(1)=X 7,1 X 11,0 X 12,1 , S H (1)={H 1,1 , H 2,1 , H 2,2 }={BX 1,1 , BX 2,1 , BX 2,2 } and the testing X variables are {X 4 , X 5 , X 6 , X 8 , X 9 , X 10 }. All of them are states unknown and single-connect to like H k ∈{BX 1 , BX 2 } without state-known blocking variables. According to 9, X 4  is the only upstream variable of X 7,1  and can be eliminated from the ranking, thus S X (1)={5, 6, 8, 9, 10}. Since there is no cause-specific variable, then S Xs (1)=S X (1). As the same as y=0, one can get S 4G (1)=S 5G (1)=S 8G (1)=S 9G (1)=S 10G (1)={1} and S 6G (1)={1, 2}. 
     According to 3 and based on  FIG. 9 , one can get the following equations: S 13K (1)={1}, S 4K (1)=S 5K (1)=S 8K (1)=S 9K (1)=S 10K (1)={1,2}, and S 6K (1)={2}; and accordingly m 6 (1)=1, m 4 (1)=m 5 (1)=m 8 (1)=m 10 (1) m 9 (1)=2, S 1J (1)={1}, and S 2J (1)={2}. Still same as y=0, ω kj =1 (k∈{1,2}). 
     According to 5, take the following formula for calculation as same as in example 1: 
                 ρ   i     ⁡     (   y   )       =       1       m   i     ⁡     (   y   )         ⁢       ∑     k   ∈       S   iK     ⁡     (   y   )           ⁢       ∑     j   ∈       S   kJ     ⁡     (   y   )           ⁢       ω   kj     ⁢       ∑     g   ∈       S   iG     ⁡     (   y   )           ⁢     Pr   ⁢     {       X   ig     |     E   ⁡     (   y   )         }     ⁢            Pr   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   y   )           }       -     Pr   ⁢     {       H   kj     |     E   ⁡     (   y   )         }                                
where i∈S Xs (0)={4, 5, 6, 8, 9, 10}. Note m′ i (y)=m i (y) in this example.
 
     According to  FIG. 9 , since ω kj =1 and k∈{1, 2}, we have 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       4 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       
                         
                           m 
                           4 
                         
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             { 
                             
                               1 
                               , 
                               2 
                             
                             } 
                           
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 S 
                                 kJ 
                               
                               ⁡ 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               g 
                               ∈ 
                               
                                 { 
                                 1 
                                 } 
                               
                             
                           
                           ⁢ 
                           
                             Pr 
                             ⁢ 
                             
                               { 
                               
                                 
                                   X 
                                   ig 
                                 
                                 | 
                                 
                                   E 
                                   ⁡ 
                                   
                                     ( 
                                     1 
                                     ) 
                                   
                                 
                               
                               } 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         H 
                                         kj 
                                       
                                       | 
                                       
                                         
                                           X 
                                           ig 
                                         
                                         ⁢ 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         H 
                                         kj 
                                       
                                       | 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           X 
                           
                             4 
                             , 
                             1 
                           
                         
                         | 
                         
                           E 
                           ⁡ 
                           
                             ( 
                             1 
                             ) 
                           
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                                
                               
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         H 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                       | 
                                       
                                         
                                           X 
                                           
                                             4 
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         H 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                       | 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                         
                           
                             
                               + 
                               
                                  
                                 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           H 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                         | 
                                         
                                           
                                             X 
                                             
                                               4 
                                               , 
                                               1 
                                             
                                           
                                           ⁢ 
                                           
                                             E 
                                             ⁡ 
                                             
                                               ( 
                                               1 
                                               ) 
                                             
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           H 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                         | 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                  
                               
                             
                           
                         
                         
                           
                             
                               + 
                               
                                  
                                 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           H 
                                           
                                             2 
                                             , 
                                             2 
                                           
                                         
                                         | 
                                         
                                           
                                             X 
                                             
                                               4 
                                               , 
                                               1 
                                             
                                           
                                           ⁢ 
                                           
                                             E 
                                             ⁡ 
                                             
                                               ( 
                                               1 
                                               ) 
                                             
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           H 
                                           
                                             2 
                                             , 
                                             2 
                                           
                                         
                                         | 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                  
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           X 
                           
                             4 
                             , 
                             1 
                           
                         
                         | 
                         
                           E 
                           ⁡ 
                           
                             ( 
                             1 
                             ) 
                           
                         
                       
                       } 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                                
                               
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         BX 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                       | 
                                       
                                         
                                           X 
                                           
                                             4 
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         BX 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                       | 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                         
                           
                             
                               + 
                               
                                  
                                 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                         | 
                                         
                                           
                                             X 
                                             
                                               4 
                                               , 
                                               1 
                                             
                                           
                                           ⁢ 
                                           
                                             E 
                                             ⁡ 
                                             
                                               ( 
                                               1 
                                               ) 
                                             
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                         | 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                  
                               
                             
                           
                         
                         
                           
                             
                               + 
                               
                                  
                                 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             2 
                                           
                                         
                                         | 
                                         
                                           
                                             X 
                                             
                                               4 
                                               , 
                                               1 
                                             
                                           
                                           ⁢ 
                                           
                                             E 
                                             ⁡ 
                                             
                                               ( 
                                               1 
                                               ) 
                                             
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           BX 
                                           
                                             2 
                                             , 
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                                         | 
                                         
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                       ( 
                       
                         
                           
                             
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                                     { 
                                     
                                       
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                                         E 
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                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
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                                   ⁢ 
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                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         B 
                                         
                                           1 
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                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                         
                           
                             
                               + 
                               
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                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           BX 
                                           
                                             2 
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                                           E 
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                                   - 
                                   
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                                       { 
                                       
                                         
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                                     ⁢ 
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
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                                             2 
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                                     Pr 
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                                       { 
                                       
                                         
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                                   - 
                                   
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                                       { 
                                       
                                         
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                                     ⁢ 
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     In which 
     
       
         
           
             
               
                 
                   
                     Pr 
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                       { 
                       
                         
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                   = 
                     
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                                           0.9 
                                         
                                       
                                     
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                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           1 
                                         
                                         
                                           
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                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             - 
                                           
                                           
                                             0.5 
                                           
                                           
                                             0.5 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           1 
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.3 
                                           
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.8 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
                                           - 
                                         
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           - 
                                         
                                         
                                           0.6 
                                         
                                         
                                           0.3 
                                         
                                       
                                       
                                         
                                           - 
                                         
                                         
                                           0.4 
                                         
                                         
                                           0.7 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             - 
                           
                         
                         
                           
                             0.01 
                           
                         
                         
                           
                             0.02 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     0.1 
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     0.00035742 
                   
                   ⁢ 
                   
                     
 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           
                             BX 
                             
                               1 
                               , 
                               1 
                             
                           
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                             X 
                             
                               4 
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                             E 
                             ⁡ 
                             
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                               1 
                               ) 
                             
                           
                         
                         } 
                       
                     
                     = 
                       
                     ⁢ 
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           
                             BX 
                             
                               1 
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                             X 
                             
                               4 
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                             X 
                             
                               7 
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                                 2 
                               
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                   = 
                     
                   ⁢ 
                   
                     
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                     = 
                     
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                     Pr 
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                       { 
                       
                         
                           
                             
                               
                                 
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                   = 
                     
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                                     1 
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                                   0.9 
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                                     ( 
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                                 ) 
                               
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                                           1 
                                         
                                         
                                           
                                             1 
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                                             1 
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                                     ) 
                                   
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                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
                                           - 
                                         
                                         
                                           - 
                                         
                                       
                                       
                                         
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                                           0.6 
                                         
                                         
                                           0.3 
                                         
                                       
                                       
                                         
                                           - 
                                         
                                         
                                           0.4 
                                         
                                         
                                           0.7 
                                         
                                       
                                     
                                     ) 
                                   
                                 
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                                       1 
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                                             - 
                                           
                                           
                                             0.5 
                                           
                                           
                                             0.5 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           1 
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.3 
                                           
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.8 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
                                           - 
                                         
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           - 
                                         
                                         
                                           0.6 
                                         
                                         
                                           0.3 
                                         
                                       
                                       
                                         
                                           - 
                                         
                                         
                                           0.4 
                                         
                                         
                                           0.7 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 0.39 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             - 
                           
                         
                         
                           
                             0.01 
                           
                         
                         
                           
                             0.02 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     0.1 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.0002293 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           
                             BX 
                             
                               2 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               4 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             E 
                             ⁡ 
                             
                               ( 
                               1 
                               ) 
                             
                           
                         
                         } 
                       
                     
                     = 
                       
                     ⁢ 
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           
                             BX 
                             
                               2 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               4 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               7 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               11 
                               , 
                               0 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               12 
                               , 
                               1 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               r 
                               
                                 7 
                                 ; 
                                 4 
                               
                             
                             
                               r 
                               7 
                             
                           
                           ⁢ 
                           
                             A 
                             
                               7 
                               , 
                               
                                 1 
                                 ; 
                                 4 
                               
                             
                           
                           ⁢ 
                           
                             X 
                             4 
                           
                           ⁢ 
                           
                             X 
                             
                               4 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               11 
                               , 
                               0 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               12 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             BX 
                             
                               2 
                               , 
                               1 
                             
                           
                         
                         } 
                       
                     
                     = 
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               r 
                               
                                 7 
                                 ; 
                                 4 
                               
                             
                             
                               r 
                               7 
                             
                           
                           ⁢ 
                           
                             A 
                             
                               7 
                               , 
                               
                                 1 
                                 ; 
                                 4 
                               
                             
                           
                           ⁢ 
                           
                             X 
                             
                               4 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               11 
                               , 
                               0 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               12 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             BX 
                             
                               2 
                               , 
                               1 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             r 
                             
                               7 
                               ; 
                               4 
                             
                           
                           
                             r 
                             7 
                           
                         
                         ⁢ 
                         
                           
                             A 
                             
                               7 
                               , 
                               
                                 1 
                                 ; 
                                 4 
                               
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     r 
                                     
                                       4 
                                       ; 
                                       1 
                                     
                                   
                                   
                                     r 
                                     4 
                                   
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     4 
                                     , 
                                     
                                       1 
                                       ; 
                                       1 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   BX 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   
                                     r 
                                     
                                       4 
                                       ; 
                                       2 
                                     
                                   
                                   
                                     r 
                                     4 
                                   
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     4 
                                     , 
                                     
                                       1 
                                       ; 
                                       2 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   BX 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             r 
                             
                               11 
                               ; 
                               6 
                             
                           
                           
                             r 
                             11 
                           
                         
                         ⁢ 
                         
                           A 
                           
                             11 
                             , 
                             
                               0 
                               ; 
                               6 
                             
                           
                         
                         ⁢ 
                         
                           
                             r 
                             
                               6 
                               ; 
                               2 
                             
                           
                           
                             r 
                             6 
                           
                         
                         ⁢ 
                         
                           A 
                           
                             6 
                             ; 
                             2 
                           
                         
                         ⁢ 
                         
                           BX 
                           2 
                         
                         ⁢ 
                         
                           A 
                           
                             12 
                             , 
                             
                               1 
                               ; 
                               D 
                             
                           
                         
                         ⁢ 
                         
                           BX 
                           
                             2 
                             , 
                             1 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   r 
                                   
                                     7 
                                     ; 
                                     4 
                                   
                                 
                                 
                                   r 
                                   7 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   7 
                                   , 
                                   
                                     1 
                                     ; 
                                     4 
                                   
                                   , 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   r 
                                   
                                     4 
                                     ; 
                                     1 
                                   
                                 
                                 
                                   r 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   4 
                                   , 
                                   
                                     1 
                                     ; 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 BX 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   r 
                                   
                                     11 
                                     ; 
                                     6 
                                   
                                 
                                 
                                   r 
                                   11 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   11 
                                   , 
                                   
                                     0 
                                     ; 
                                     6 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   r 
                                   
                                     6 
                                     ; 
                                     2 
                                   
                                 
                                 
                                   r 
                                   6 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   
                                     6 
                                     ; 
                                     2 
                                   
                                   , 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 BX 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   12 
                                   , 
                                   
                                     1 
                                     ; 
                                     D 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 + 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           7 
                                           ; 
                                           4 
                                         
                                       
                                       
                                         r 
                                         7 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         7 
                                         , 
                                         
                                           1 
                                           ; 
                                           4 
                                         
                                         , 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           4 
                                           ; 
                                           2 
                                         
                                       
                                       
                                         r 
                                         4 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         4 
                                         , 
                                         
                                           1 
                                           ; 
                                           2 
                                         
                                         , 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         11 
                                         ; 
                                         6 
                                       
                                     
                                     
                                       r 
                                       11 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       11 
                                       , 
                                       
                                         0 
                                         ; 
                                         6 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                     
                                       r 
                                       6 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                       , 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 BX 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   112 
                                   , 
                                   
                                     1 
                                     ; 
                                     D 
                                   
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     P 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     r 
                     ⁢ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       r 
                                       
                                         7 
                                         ; 
                                         4 
                                       
                                     
                                     
                                       r 
                                       7 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       7 
                                       , 
                                       
                                         1 
                                         ; 
                                         4 
                                       
                                       , 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         4 
                                         ; 
                                         1 
                                       
                                     
                                     
                                       r 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       A 
                                       
                                         4 
                                         , 
                                         
                                           1 
                                           ; 
                                           1 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   r 
                                                   
                                                     1 
                                                     ; 
                                                     1 
                                                   
                                                 
                                                 
                                                   r 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 A 
                                                 
                                                   1 
                                                   ; 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 B 
                                                 1 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 + 
                                                 
                                                   
                                                     r 
                                                     
                                                       1 
                                                       ; 
                                                       12 
                                                     
                                                   
                                                   
                                                     r 
                                                     1 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 A 
                                                 
                                                   1 
                                                   ; 
                                                   12 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         11 
                                         ; 
                                         6 
                                       
                                     
                                     
                                       r 
                                       11 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       11 
                                       , 
                                       
                                         0 
                                         ; 
                                         6 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                     
                                       r 
                                       6 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                       , 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     + 
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             
                                               7 
                                               ; 
                                               4 
                                             
                                           
                                           
                                             r 
                                             7 
                                           
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             7 
                                             , 
                                             
                                               1 
                                               ; 
                                               4 
                                             
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               4 
                                               ; 
                                               2 
                                             
                                           
                                           
                                             r 
                                             4 
                                           
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             4 
                                             , 
                                             
                                               1 
                                               ; 
                                               2 
                                             
                                             , 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           r 
                                           
                                             11 
                                             ; 
                                             6 
                                           
                                         
                                         
                                           r 
                                           11 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           11 
                                           , 
                                           
                                             0 
                                             ; 
                                             6 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           r 
                                           
                                             6 
                                             ; 
                                             2 
                                           
                                         
                                         
                                           r 
                                           6 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           
                                             6 
                                             ; 
                                             2 
                                           
                                           , 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             r 
                             
                               2 
                               ; 
                               2 
                             
                           
                           
                             r 
                             2 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           A 
                           
                             2 
                             , 
                             
                               1 
                               ; 
                               2 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           2 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           A 
                           
                             12 
                             , 
                             
                               1 
                               ; 
                               D 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         7 
                                         ; 
                                         4 
                                       
                                     
                                     
                                       r 
                                       7 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       7 
                                       , 
                                       
                                         1 
                                         ; 
                                         4 
                                       
                                       , 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         4 
                                         ; 
                                         1 
                                       
                                     
                                     
                                       r 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       a 
                                       
                                         4 
                                         , 
                                         
                                           1 
                                           ; 
                                           1 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   r 
                                                   
                                                     1 
                                                     ; 
                                                     1 
                                                   
                                                 
                                                 
                                                   r 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 a 
                                                 
                                                   1 
                                                   ; 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 b 
                                                 1 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 + 
                                                 
                                                   
                                                     r 
                                                     
                                                       1 
                                                       ; 
                                                       12 
                                                     
                                                   
                                                   
                                                     r 
                                                     1 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 a 
                                                 
                                                   1 
                                                   ; 
                                                   12 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         11 
                                         ; 
                                         6 
                                       
                                     
                                     
                                       r 
                                       11 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       11 
                                       , 
                                       
                                         0 
                                         ; 
                                         6 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                     
                                       r 
                                       6 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                       , 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 + 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           7 
                                           ; 
                                           4 
                                         
                                       
                                       
                                         r 
                                         7 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         7 
                                         , 
                                         
                                           1 
                                           ; 
                                           4 
                                         
                                         , 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           4 
                                           ; 
                                           2 
                                         
                                       
                                       
                                         r 
                                         4 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         4 
                                         , 
                                         
                                           1 
                                           ; 
                                           2 
                                         
                                         , 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         11 
                                         ; 
                                         6 
                                       
                                     
                                     
                                       r 
                                       11 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       11 
                                       , 
                                       
                                         0 
                                         ; 
                                         6 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                     
                                       r 
                                       6 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                       , 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       a 
                       
                         2 
                         , 
                         
                           1 
                           ; 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       b 
                       2 
                     
                     ⁢ 
                     
                       a 
                       
                         12 
                         , 
                         
                           1 
                           ; 
                           D 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   0.7 
                                   ⁢ 
                                   
                                     1 
                                     2 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
                                           0.9 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           0.39 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           1 
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.3 
                                           
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.8 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           0.6 
                                         
                                       
                                       
                                         
                                           0.4 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 + 
                                 
                                   ( 
                                   
                                     0.7 
                                     ⁢ 
                                     
                                       1 
                                       2 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       0.5 
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           1 
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.3 
                                           
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.8 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           0.6 
                                         
                                       
                                       
                                         
                                           0.4 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       0.01 
                       ) 
                     
                     ⁢ 
                     0.1 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0.0001489 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           BX 
                           
                             2 
                             , 
                             2 
                           
                         
                         ⁢ 
                         
                           X 
                           
                             4 
                             , 
                             1 
                           
                         
                         ⁢ 
                         
                           E 
                           ⁡ 
                           
                             ( 
                             1 
                             ) 
                           
                         
                       
                       } 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           BX 
                           
                             2 
                             , 
                             2 
                           
                         
                         ⁢ 
                         
                           X 
                           
                             4 
                             , 
                             1 
                           
                         
                         ⁢ 
                         
                           X 
                           
                             7 
                             , 
                             1 
                           
                         
                         ⁢ 
                         
                           X 
                           
                             11 
                             , 
                             0 
                           
                         
                         ⁢ 
                         
                           X 
                           
                             12 
                             , 
                             1 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               r 
                               
                                 7 
                                 ; 
                                 4 
                               
                             
                             
                               r 
                               7 
                             
                           
                           ⁢ 
                           
                             A 
                             
                               7 
                               , 
                               
                                 1 
                                 ; 
                                 4 
                               
                             
                           
                           ⁢ 
                           
                             X 
                             4 
                           
                           ⁢ 
                           
                             X 
                             
                               4 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               11 
                               , 
                               0 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               12 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             BX 
                             
                               2 
                               , 
                               1 
                             
                           
                         
                         } 
                       
                     
                     = 
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               r 
                               
                                 7 
                                 ; 
                                 4 
                               
                             
                             
                               r 
                               7 
                             
                           
                           ⁢ 
                           
                             A 
                             
                               7 
                               , 
                               
                                 1 
                                 ; 
                                 4 
                               
                             
                           
                           ⁢ 
                           
                             X 
                             
                               4 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               11 
                               , 
                               0 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               12 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             BX 
                             
                               2 
                               , 
                               2 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             r 
                             
                               7 
                               ; 
                               4 
                             
                           
                           
                             r 
                             7 
                           
                         
                         ⁢ 
                         
                           
                             A 
                             
                               7 
                               , 
                               
                                 1 
                                 ; 
                                 4 
                               
                               , 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     r 
                                     
                                       4 
                                       ; 
                                       1 
                                     
                                   
                                   
                                     r 
                                     4 
                                   
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     4 
                                     , 
                                     
                                       1 
                                       ; 
                                       1 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   BX 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   
                                     r 
                                     
                                       4 
                                       ; 
                                       2 
                                     
                                   
                                   
                                     r 
                                     4 
                                   
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     4 
                                     , 
                                     
                                       1 
                                       ; 
                                       2 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   BX 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             r 
                             
                               11 
                               ; 
                               6 
                             
                           
                           
                             r 
                             11 
                           
                         
                         ⁢ 
                         
                           A 
                           
                             11 
                             , 
                             
                               0 
                               ; 
                               6 
                             
                           
                         
                         ⁢ 
                         
                           
                             r 
                             
                               6 
                               ; 
                               2 
                             
                           
                           
                             r 
                             6 
                           
                         
                         ⁢ 
                         
                           A 
                           
                             6 
                             ; 
                             2 
                           
                         
                         ⁢ 
                         
                           BX 
                           2 
                         
                         ⁢ 
                         
                           A 
                           
                             12 
                             , 
                             
                               1 
                               ; 
                               D 
                             
                           
                         
                         ⁢ 
                         
                           BX 
                           
                             2 
                             , 
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   r 
                                   
                                     7 
                                     ; 
                                     4 
                                   
                                 
                                 
                                   r 
                                   7 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   7 
                                   , 
                                   
                                     1 
                                     ; 
                                     4 
                                   
                                   , 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   r 
                                   
                                     4 
                                     ; 
                                     1 
                                   
                                 
                                 
                                   r 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   4 
                                   , 
                                   
                                     1 
                                     ; 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 BX 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   r 
                                   
                                     11 
                                     ; 
                                     6 
                                   
                                 
                                 
                                   r 
                                   11 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   11 
                                   , 
                                   
                                     0 
                                     ; 
                                     6 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   r 
                                   
                                     6 
                                     ; 
                                     2 
                                   
                                 
                                 
                                   r 
                                   6 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   
                                     6 
                                     ; 
                                     2 
                                   
                                   , 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 BX 
                                 
                                   2 
                                   , 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   12 
                                   , 
                                   
                                     1 
                                     ; 
                                     D 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 + 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           7 
                                           ; 
                                           4 
                                         
                                       
                                       
                                         r 
                                         7 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         7 
                                         , 
                                         
                                           1 
                                           ; 
                                           4 
                                         
                                         , 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           4 
                                           ; 
                                           2 
                                         
                                       
                                       
                                         r 
                                         4 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         4 
                                         , 
                                         
                                           1 
                                           ; 
                                           2 
                                         
                                         , 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         11 
                                         ; 
                                         6 
                                       
                                     
                                     
                                       r 
                                       11 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       11 
                                       , 
                                       
                                         0 
                                         ; 
                                         6 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                     
                                       r 
                                       6 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                       , 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 BX 
                                 
                                   2 
                                   , 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   12 
                                   , 
                                   
                                     1 
                                     ; 
                                     D 
                                   
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     P 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     r 
                     ⁢ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       r 
                                       
                                         7 
                                         ; 
                                         4 
                                       
                                     
                                     
                                       r 
                                       7 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       7 
                                       , 
                                       
                                         1 
                                         ; 
                                         4 
                                       
                                       , 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         4 
                                         ; 
                                         1 
                                       
                                     
                                     
                                       r 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       A 
                                       
                                         4 
                                         , 
                                         
                                           1 
                                           ; 
                                           1 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   r 
                                                   
                                                     1 
                                                     ; 
                                                     1 
                                                   
                                                 
                                                 
                                                   r 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 A 
                                                 
                                                   1 
                                                   ; 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 B 
                                                 1 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 + 
                                                 
                                                   
                                                     r 
                                                     
                                                       1 
                                                       ; 
                                                       12 
                                                     
                                                   
                                                   
                                                     r 
                                                     1 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 A 
                                                 
                                                   
                                                     1 
                                                     ; 
                                                     12 
                                                   
                                                   , 
                                                   1 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         11 
                                         ; 
                                         6 
                                       
                                     
                                     
                                       r 
                                       11 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       11 
                                       , 
                                       
                                         0 
                                         ; 
                                         6 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                     
                                       r 
                                       6 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                       , 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     + 
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             
                                               7 
                                               ; 
                                               4 
                                             
                                           
                                           
                                             r 
                                             7 
                                           
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             7 
                                             , 
                                             
                                               1 
                                               ; 
                                               4 
                                             
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               4 
                                               ; 
                                               2 
                                             
                                           
                                           
                                             r 
                                             4 
                                           
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             4 
                                             , 
                                             
                                               1 
                                               ; 
                                               2 
                                             
                                             , 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   * 
                                   
                                     ( 
                                     
                                       
                                         
                                           r 
                                           
                                             11 
                                             ; 
                                             6 
                                           
                                         
                                         
                                           r 
                                           11 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           11 
                                           , 
                                           
                                             0 
                                             ; 
                                             6 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           r 
                                           
                                             6 
                                             ; 
                                             2 
                                           
                                         
                                         
                                           r 
                                           6 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           
                                             6 
                                             ; 
                                             2 
                                           
                                           , 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             r 
                             
                               2 
                               ; 
                               2 
                             
                           
                           
                             r 
                             2 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           A 
                           
                             2 
                             , 
                             
                               2 
                               ; 
                               2 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           2 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           A 
                           
                             12 
                             , 
                             
                               1 
                               ; 
                               D 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         7 
                                         ; 
                                         4 
                                       
                                     
                                     
                                       r 
                                       7 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       7 
                                       , 
                                       
                                         1 
                                         ; 
                                         4 
                                       
                                       , 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         4 
                                         ; 
                                         1 
                                       
                                     
                                     
                                       r 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       a 
                                       
                                         4 
                                         , 
                                         
                                           1 
                                           ; 
                                           1 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   r 
                                                   
                                                     1 
                                                     ; 
                                                     1 
                                                   
                                                 
                                                 
                                                   r 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 a 
                                                 
                                                   1 
                                                   ; 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 b 
                                                 1 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 + 
                                                 
                                                   
                                                     r 
                                                     
                                                       1 
                                                       ; 
                                                       12 
                                                     
                                                   
                                                   
                                                     r 
                                                     1 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 a 
                                                 
                                                   
                                                     1 
                                                     ; 
                                                     12 
                                                   
                                                   , 
                                                   1 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         11 
                                         ; 
                                         6 
                                       
                                     
                                     
                                       r 
                                       11 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       11 
                                       , 
                                       
                                         0 
                                         ; 
                                         6 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                     
                                       r 
                                       6 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                       , 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 + 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           7 
                                           ; 
                                           4 
                                         
                                       
                                       
                                         r 
                                         7 
                                       
                                     
                                     ⁢ 
                                     
                                       a 
                                       
                                         7 
                                         , 
                                         
                                           1 
                                           ; 
                                           4 
                                         
                                         , 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           4 
                                           ; 
                                           2 
                                         
                                       
                                       
                                         r 
                                         4 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         4 
                                         , 
                                         
                                           1 
                                           ; 
                                           2 
                                         
                                         , 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         11 
                                         ; 
                                         6 
                                       
                                     
                                     
                                       r 
                                       11 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       11 
                                       , 
                                       
                                         0 
                                         ; 
                                         6 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                     
                                     
                                       r 
                                       6 
                                     
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       
                                         6 
                                         ; 
                                         2 
                                       
                                       , 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       a 
                       
                         2 
                         , 
                         
                           2 
                           ; 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       b 
                       2 
                     
                     ⁢ 
                     
                       a 
                       
                         12 
                         , 
                         
                           1 
                           ; 
                           D 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   0.7 
                                   ⁢ 
                                   
                                     1 
                                     2 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
                                           0.9 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           0.39 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           1 
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.3 
                                           
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.8 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           0.3 
                                         
                                       
                                       
                                         
                                           0.7 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 + 
                                 
                                   ( 
                                   
                                     0.7 
                                     ⁢ 
                                     
                                       1 
                                       2 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       0.5 
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           1 
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.3 
                                           
                                         
                                         
                                           
                                             1 
                                             - 
                                             0.8 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           0.3 
                                         
                                       
                                       
                                         
                                           0.7 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       0.02 
                       ) 
                     
                     ⁢ 
                     0.1 
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     0.000208 
                   
                   ⁢ 
                   
                     
 
                   
                 
               
             
           
         
       
     
     Thus 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       4 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       1 
                       
                         Pr 
                         ⁢ 
                         
                           { 
                           
                             E 
                             ⁡ 
                             
                               ( 
                               1 
                               ) 
                             
                           
                           } 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                                
                               
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         BX 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         X 
                                         
                                           4 
                                           , 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 - 
                                 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         X 
                                         
                                           4 
                                           , 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                   ⁢ 
                                   Pr 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         BX 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                       ❘ 
                                       
                                         E 
                                         ⁡ 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                                
                             
                           
                         
                         
                           
                             
                               + 
                               
                                  
                                 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           X 
                                           
                                             4 
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           X 
                                           
                                             4 
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                     ⁢ 
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                         ❘ 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                  
                               
                             
                           
                         
                         
                           
                             
                               + 
                               
                                  
                                 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             2 
                                           
                                         
                                         ⁢ 
                                         
                                           X 
                                           
                                             4 
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   - 
                                   
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           X 
                                           
                                             4 
                                             , 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                     ⁢ 
                                     Pr 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           BX 
                                           
                                             2 
                                             , 
                                             2 
                                           
                                         
                                         ❘ 
                                         
                                           E 
                                           ⁡ 
                                           
                                             ( 
                                             1 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                  
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       1 
                       0.0003574 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                                
                               
                                 0.0002293 
                                 - 
                                 
                                   0.00035742 
                                   × 
                                   0.64163 
                                 
                               
                                
                             
                           
                         
                         
                           
                             
                               + 
                               
                                  
                                 
                                   0.0001489 
                                   - 
                                   
                                     0.00035742 
                                     × 
                                     0.41662 
                                   
                                 
                                  
                               
                             
                           
                         
                         
                           
                             
                               + 
                               
                                  
                                 
                                   0.000208 
                                   - 
                                   
                                     0.00035742 
                                     × 
                                     0.58286 
                                   
                                 
                                  
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   0 
                 
               
             
           
         
       
     
     Similarly, 
                       ρ   5     ⁡     (   1   )       =       ⁢       1       m   5     ⁡     (   1   )         ⁢       ∑     k   ∈       {     1   ,   2     }     ⁢   j         ⁢       ∑     ∈       S   kJ     ⁡     (   1   )           ⁢       ∑     g   ∈     {   1   }         ⁢     P   ⁢   r   ⁢     {       X   ig     |     E   ⁡     (   1   )         }     ⁢            Pr   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   1   )           }       -     P   ⁢   r   ⁢     {       H   kj     |     E   ⁡     (   1   )         }                                    =       ⁢       1   2     ⁢   P   ⁢   r   ⁢     {       X     5   ,   1       |     E   ⁡     (   1   )         }     ⁢     (                  Pr   ⁢     {       BX     1   ,   1       ⁢     X     5   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     1   ,   1       ❘     E   ⁡     (   1   )         }                        +            Pr   ⁢     {       BX     2   ,   1       ⁢     X     5   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   1       ❘     E   ⁡     (   1   )         }                          +            Pr   ⁢     {       BX     2   ,   2       ⁢     X     5   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   2       ❘     E   ⁡     (   1   )         }                      )                   =       ⁢   0.092519                               ρ   6     ⁡     (   1   )       =       ⁢       1       m   6     ⁡     (   1   )         ⁢       ∑     k   ∈       {   2   }     ⁢   j         ⁢       ∑     ∈       S   kJ     ⁡     (   1   )           ⁢       ∑     g   ∈     {     1   ,   2     }         ⁢     P   ⁢   r   ⁢     {       X   ig     |     E   ⁡     (   1   )         }     ⁢            Pr   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   1   )           }       -     P   ⁢   r   ⁢     {       H   kj     |     E   ⁡     (   1   )         }                                    =       ⁢       1   1     ⁢   P   ⁢   r   ⁢     {       X     6   ,   1       |     E   ⁡     (   1   )         }     ⁢     (              Pr   ⁢     {       BX     2   ,   1       ⁢     X     6   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   1       ❘     E   ⁡     (   1   )         }              +            Pr   ⁢     {       BX     2   ,   2       ⁢     X     6   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   2       ❘     E   ⁡     (   1   )         }                )                       ⁢       +     1   1       ⁢   P   ⁢   r   ⁢     {       X     6   ,   2       |     E   ⁡     (   1   )         }     ⁢     (              Pr   ⁢     {       BX     2   ,   1       ⁢     X     6   ,   2       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   1       ❘     E   ⁡     (   1   )         }              +            Pr   ⁢     {       BX     2   ,   2       ⁢     X     6   ,   2       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   2       ❘     E   ⁡     (   1   )         }                )                   =       ⁢   0.233261                               ρ   8     ⁡     (   1   )       =       ⁢       1       m   8     ⁡     (   1   )         ⁢       ∑     k   ∈       {     1   ,   2     }     ⁢   j         ⁢       ∑     ∈       S   kJ     ⁡     (   1   )           ⁢       ∑     g   ∈     {   1   }         ⁢     P   ⁢   r   ⁢     {       X   ig     |     E   ⁡     (   1   )         }     ⁢            Pr   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   1   )           }       -     P   ⁢   r   ⁢     {       H   kj     |     E   ⁡     (   1   )         }                                    =       ⁢       1   2     ⁢   P   ⁢   r   ⁢     {       X     8   ,   1       |     E   ⁡     (   1   )         }     ⁢     (                  Pr   ⁢     {       BX     1   ,   1       ⁢     X     8   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     1   ,   1       ❘     E   ⁡     (   1   )         }                        +            Pr   ⁢     {       BX     2   ,   1       ⁢     X     8   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   1       ❘     E   ⁡     (   1   )         }                          +            Pr   ⁢     {       BX     2   ,   2       ⁢     X     8   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   2       ❘     E   ⁡     (   1   )         }                      )                   =       ⁢   0.065047                               ρ   9     ⁡     (   1   )       =       ⁢       1       m   9     ⁡     (   1   )         ⁢       ∑     k   ∈       {     1   ,   2     }     ⁢   j         ⁢       ∑     ∈       S   kJ     ⁡     (   1   )           ⁢       ∑     g   ∈     {   1   }         ⁢     P   ⁢   r   ⁢     {       X   ig     |     E   ⁡     (   1   )         }     ⁢            Pr   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   1   )           }       -     P   ⁢   r   ⁢     {       H   kj     |     E   ⁡     (   1   )         }                                    =       ⁢       1   2     ⁢   P   ⁢   r   ⁢     {       X     9   ,   1       |     E   ⁡     (   1   )         }     ⁢     (                  Pr   ⁢     {       BX     1   ,   1       ⁢     X     9   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     1   ,   1       ❘     E   ⁡     (   1   )         }                        +            Pr   ⁢     {       BX     2   ,   1       ⁢     X     9   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   1       ❘     E   ⁡     (   1   )         }                          +            Pr   ⁢     {       BX     2   ,   2       ⁢     X     9   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   2       ❘     E   ⁡     (   1   )         }                      )                   =       ⁢   0.000199                               ρ   10     ⁡     (   1   )       =       ⁢       1       m   10     ⁡     (   1   )         ⁢       ∑     k   ∈       {     1   ,   2     }     ⁢   j         ⁢       ∑     ∈       S   kJ     ⁡     (   1   )           ⁢       ∑     g   ∈     {   1   }         ⁢     P   ⁢   r   ⁢     {       X   ig     |     E   ⁡     (   1   )         }     ⁢            Pr   ⁢     {       H   kj     |       X   ig     ⁢     E   ⁡     (   1   )           }       -     P   ⁢   r   ⁢     {       H   kj     |     E   ⁡     (   1   )         }                                    =       ⁢       1   2     ⁢       Pr   ⁢     {       X     10   ,   1       |     E   ⁢     (   1   )         }         Pr   ⁢     {     E   ⁢     (   1   )       }         ⁢     (                  Pr   ⁢     {       BX     1   ,   1       ⁢     X     10   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     1   ,   1       ❘     E   ⁡     (   1   )         }                        +            Pr   ⁢     {       BX     2   ,   1       ⁢     X     10   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   1       ❘     E   ⁡     (   1   )         }                          +            Pr   ⁢     {       BX     2   ,   2       ⁢     X     10   ,   1       ⁢     E   ⁡     (   1   )         }       -     Pr   ⁢     {       BX     2   ,   2       ❘     E   ⁡     (   1   )         }                      )                   =       ⁢   0.06229               
The calculation results are listed in the following:
 
                                             i   ρ i (1)                                                    4   0           5   0.092519           6   0.233261           8   0.065047           9   0.000199           10   0.06299                        
According to 6, the same as example 1, assume β i =1/α i  and the values are follows:
 
     
       
         
           
               
               
               
             
               
                   
               
               
                 i 
                 α i   
                 β i   
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 4 
                 100 
                 0.01 
               
               
                 5 
                 2 
                 0.5 
               
               
                 6 
                 50 
                 0.02 
               
               
                 8 
                 2 
                 0.5 
               
               
                 9 
                 2 
                 0.5 
               
               
                 10 
                 2 
                 0.5 
               
               
                   
               
            
           
         
       
     
     Given m i (1), λ i (1) is calculated as follows: 
     
       
         
           
               
               
               
             
               
                   
               
               
                 i 
                 m i (1) 
                 λ i (1) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 4 
                 2 
                 1/2 
               
               
                 5 
                 2 
                 1/2 
               
               
                 6 
                 1 
                 1/1 
               
               
                 8 
                 3 
                 1/3 
               
               
                 9 
                 1 
                 1 
               
               
                 10 
                 1 
                 1 
               
               
                   
               
            
           
         
       
     
     According to 8, take 
                   I   i     ⁡     (   y   )       =           λ   i     ⁡     (   y   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   y   )             ∑     i   ∈     S     X   ⁡     (   y   )             ⁢         λ   i     ⁡     (   y   )       ⁢     β   i     ⁢       ρ   i     ⁡     (   y   )               ,         
we can obtain the following results:
 
     
       
         
           
             
               
                 I 
                 4 
               
               ⁡ 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       λ 
                       4 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     β 
                     4 
                   
                   ⁢ 
                   
                     
                       ρ 
                       4 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   
                     ∑ 
                     
                       i 
                       ∈ 
                       
                         { 
                         
                           4 
                           , 
                           ⋯ 
                           ⁢ 
                           
                               
                           
                           , 
                           9 
                           , 
                           10 
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         λ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     ⁢ 
                     
                       β 
                       i 
                     
                     ⁢ 
                     
                       
                         ρ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     
                       1 
                       2 
                     
                     × 
                     
                       0 
                       . 
                       0 
                     
                     ⁢ 
                     1 
                     × 
                     0 
                   
                   
                     
                       0 
                       . 
                       0 
                     
                     ⁢ 
                     7 
                     ⁢ 
                     0 
                     ⁢ 
                     2 
                     ⁢ 
                     3 
                     ⁢ 
                     1 
                   
                 
                 = 
                 0 
               
             
           
         
       
       
         
           
             
               
                 I 
                 5 
               
               ⁡ 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       λ 
                       5 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     β 
                     5 
                   
                   ⁢ 
                   
                     
                       ρ 
                       5 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   
                     ∑ 
                     
                       i 
                       ∈ 
                       
                         { 
                         
                           4 
                           , 
                           ⋯ 
                           ⁢ 
                           
                               
                           
                           , 
                           9 
                           , 
                           10 
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         λ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     ⁢ 
                     
                       β 
                       i 
                     
                     ⁢ 
                     
                       
                         ρ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     
                       1 
                       2 
                     
                     × 
                     0.5 
                     × 
                     0.0231 
                     ⁢ 
                     2 
                     ⁢ 
                     9 
                     ⁢ 
                     8 
                   
                   
                     0.070 
                     ⁢ 
                     2 
                     ⁢ 
                     3 
                     ⁢ 
                     1 
                   
                 
                 = 
                 
                   
                     0 
                     . 
                     3 
                   
                   ⁢ 
                   2 
                   ⁢ 
                   9 
                   ⁢ 
                   3 
                   ⁢ 
                   3 
                   ⁢ 
                   8 
                 
               
             
           
         
       
       
         
           
             
               
                 I 
                 6 
               
               ⁡ 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       λ 
                       6 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     β 
                     6 
                   
                   ⁢ 
                   
                     
                       ρ 
                       6 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   
                     ∑ 
                     
                       i 
                       ∈ 
                       
                         { 
                         
                           4 
                           , 
                           ⋯ 
                           ⁢ 
                           
                               
                           
                           , 
                           9 
                           , 
                           10 
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         λ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     ⁢ 
                     
                       β 
                       i 
                     
                     ⁢ 
                     
                       
                         ρ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     1 
                     × 
                     
                       0 
                       . 
                       0 
                     
                     ⁢ 
                     2 
                     × 
                     0.004 
                     ⁢ 
                     6 
                     ⁢ 
                     6 
                     ⁢ 
                     5 
                     ⁢ 
                     2 
                   
                   
                     
                       0 
                       . 
                       0 
                     
                     ⁢ 
                     7 
                     ⁢ 
                     0 
                     ⁢ 
                     2 
                     ⁢ 
                     3 
                     ⁢ 
                     1 
                   
                 
                 = 
                 
                   
                     0 
                     . 
                     0 
                   
                   ⁢ 
                   6 
                   ⁢ 
                   6 
                   ⁢ 
                   4 
                   ⁢ 
                   2 
                   ⁢ 
                   7 
                 
               
             
           
         
       
       
         
           
             
               
                 I 
                 8 
               
               ⁡ 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       λ 
                       8 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     β 
                     8 
                   
                   ⁢ 
                   
                     
                       ρ 
                       8 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   
                     ∑ 
                     
                       i 
                       ∈ 
                       
                         { 
                         
                           4 
                           , 
                           ⋯ 
                           ⁢ 
                           
                               
                           
                           , 
                           9 
                           , 
                           10 
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         λ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     ⁢ 
                     
                       β 
                       i 
                     
                     ⁢ 
                     
                       
                         ρ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     
                       1 
                       3 
                     
                     × 
                     0.5 
                     × 
                     
                       0 
                       . 
                       0 
                     
                     ⁢ 
                     1 
                     ⁢ 
                     0 
                     ⁢ 
                     8 
                     ⁢ 
                     4 
                     ⁢ 
                     1 
                   
                   
                     
                       0 
                       . 
                       0 
                     
                     ⁢ 
                     7 
                     ⁢ 
                     0 
                     ⁢ 
                     2 
                     ⁢ 
                     3 
                     ⁢ 
                     1 
                   
                 
                 = 
                 
                   
                     0 
                     . 
                     1 
                   
                   ⁢ 
                   5 
                   ⁢ 
                   4 
                   ⁢ 
                   3 
                   ⁢ 
                   6 
                   ⁢ 
                   3 
                 
               
             
           
         
       
       
         
           
             
               
                 I 
                 9 
               
               ⁡ 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       λ 
                       9 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     β 
                     9 
                   
                   ⁢ 
                   
                     
                       ρ 
                       9 
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   
                     ∑ 
                     
                       i 
                       ∈ 
                       
                         { 
                         
                           4 
                           , 
                           ⋯ 
                           ⁢ 
                           
                               
                           
                           , 
                           9 
                           , 
                           10 
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         λ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     ⁢ 
                     
                       β 
                       i 
                     
                     ⁢ 
                     
                       
                         ρ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     1 
                     × 
                     
                       0 
                       . 
                       5 
                     
                     × 
                     0.000 
                     ⁢ 
                     0 
                     ⁢ 
                     9 
                     ⁢ 
                     9 
                     ⁢ 
                     5 
                   
                   
                     
                       0 
                       . 
                       0 
                     
                     ⁢ 
                     7 
                     ⁢ 
                     0 
                     ⁢ 
                     2 
                     ⁢ 
                     3 
                     ⁢ 
                     1 
                   
                 
                 = 
                 
                   
                     0 
                     . 
                     0 
                   
                   ⁢ 
                   0 
                   ⁢ 
                   1 
                   ⁢ 
                   4 
                   ⁢ 
                   1 
                   ⁢ 
                   7 
                 
               
             
           
         
       
       
         
           
             
               
                 I 
                 
                   1 
                   ⁢ 
                   0 
                 
               
               ⁡ 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       λ 
                       
                         1 
                         ⁢ 
                         0 
                       
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     β 
                     
                       1 
                       ⁢ 
                       0 
                     
                   
                   ⁢ 
                   
                     
                       ρ 
                       
                         1 
                         ⁢ 
                         0 
                       
                     
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   
                     ∑ 
                     
                       i 
                       ∈ 
                       
                         { 
                         
                           4 
                           , 
                           ⋯ 
                           ⁢ 
                           
                               
                           
                           , 
                           9 
                           , 
                           10 
                         
                         } 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         λ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     ⁢ 
                     
                       β 
                       i 
                     
                     ⁢ 
                     
                       
                         ρ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     1 
                     × 
                     0.5 
                     × 
                     0.031 
                     ⁢ 
                     4 
                     ⁢ 
                     9 
                     ⁢ 
                     5 
                   
                   
                     
                       0 
                       . 
                       0 
                     
                     ⁢ 
                     7 
                     ⁢ 
                     0 
                     ⁢ 
                     2 
                     ⁢ 
                     3 
                     ⁢ 
                     1 
                   
                 
                 = 
                 
                   
                     0 
                     . 
                     4 
                   
                   ⁢ 
                   4 
                   ⁢ 
                   8 
                   ⁢ 
                   4 
                   ⁢ 
                   4 
                   ⁢ 
                   9 
                 
               
             
           
         
       
     
     The ranking results are: 
     
       
         
           
               
               
               
             
               
                   
               
               
                 index 
                 i 
                 I i (1) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 1 
                 10 
                 0.448449 
               
               
                 2 
                 5 
                 0.329338 
               
               
                 3 
                 8 
                 0.154363 
               
               
                 4 
                 6 
                 0.066427 
               
               
                 5 
                 9 
                 0.001417 
               
               
                 6 
                 4 
                 0 
               
               
                   
               
            
           
         
       
     
     According to 9, since X 5  is the only upstream variable of X 10 , X 5  can be eliminated from the ranking. Since the rank probability of X 4  is equal to zero and there is no meaning to detect X 4 , thus X 4  can be eliminated from the ranking. So the above ranking becomes: 
     
       
         
           
               
               
               
             
               
                   
               
               
                 index 
                 i 
                 I i (1) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 1 
                 10 
                 0.448449 
               
               
                 2 
                 8 
                 0.154363 
               
               
                 3 
                 6 
                 0.066427 
               
               
                 4 
                 9 
                 0.001417 
               
               
                   
               
            
           
         
       
     
     Take the first three X-type variables into detection. Assume the results are X 8,1 , and X 6,0  respectively. Then  FIG. 9  becomes  FIG. 12 . Since α 8,1;6,0 =a 11,0;6,0 =“-”, which means no corresponding causal relationship,  FIG. 12  is simplified as  FIG. 13  based on the simplification rules, where E(2)=E + (1)E(1)=X 10,1 X 8,1 X 6,0 X 7,1 X 8,1 X 10,1 X 12,1 . To calculate h 1,1   r (2), h 2,1   r (2), and h 2,2   r (2),  FIG. 13  can be divided and simplified as  FIG. 14  and  FIG. 15 . According to the DUCG algorithms in [8], we can obtain that: 
     Based on  FIG. 14 , 
     
       
         
           
             
               
                 
                   
                     
                       ζ 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       2 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           E 
                           ⁡ 
                           
                             ( 
                             2 
                             ) 
                           
                         
                         } 
                       
                     
                     = 
                     
                       P 
                       ⁢ 
                       r 
                       ⁢ 
                       
                         { 
                         
                           
                             X 
                             
                               7 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               8 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               10 
                               , 
                               1 
                             
                           
                           ⁢ 
                           
                             X 
                             
                               12 
                               , 
                               1 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 
                                   7 
                                   ; 
                                   4 
                                 
                               
                               
                                 r 
                                 7 
                               
                             
                             ⁢ 
                             
                               A 
                               
                                 7 
                                 , 
                                 
                                   1 
                                   ; 
                                   4 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 r 
                                 
                                   4 
                                   ; 
                                   1 
                                 
                               
                               
                                 r 
                                 4 
                               
                             
                             ⁢ 
                             
                               A 
                               
                                 4 
                                 ; 
                                 1 
                               
                             
                             ⁢ 
                             B 
                             ⁢ 
                             
                               X 
                               1 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       r 
                                       
                                         8 
                                         ; 
                                         4 
                                       
                                     
                                     
                                       r 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
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                                       8 
                                       , 
                                       
                                         1 
                                         ; 
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                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         4 
                                         ; 
                                         1 
                                       
                                     
                                     
                                       r 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       4 
                                       ; 
                                       1 
                                     
                                   
                                   ⁢ 
                                   B 
                                   ⁢ 
                                   
                                     X 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     + 
                                     
                                       
                                         r 
                                         
                                           8 
                                           ; 
                                           5 
                                         
                                       
                                       
                                         r 
                                         8 
                                       
                                     
                                   
                                   ⁢ 
                                   
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                                       8 
                                       , 
                                       
                                         1 
                                         ; 
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                                       r 
                                       
                                         5 
                                         ; 
                                         1 
                                       
                                     
                                     
                                       r 
                                       5 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       5 
                                       ; 
                                       1 
                                     
                                   
                                   ⁢ 
                                   B 
                                   ⁢ 
                                   
                                     X 
                                     1 
                                   
                                 
                               
                             
                           
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                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 r 
                                 
                                   10 
                                   ; 
                                   5 
                                 
                               
                               
                                 r 
                                 
                                   1 
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                             ⁢ 
                             
                               A 
                               
                                 10 
                                 , 
                                 
                                   1 
                                   ; 
                                   5 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 r 
                                 
                                   5 
                                   ; 
                                   1 
                                 
                               
                               
                                 r 
                                 5 
                               
                             
                             ⁢ 
                             
                               A 
                               
                                 5 
                                 ; 
                                 1 
                               
                             
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                             B 
                             ⁢ 
                             
                               X 
                               1 
                             
                           
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                         ⁢ 
                         
                           X 
                           
                             12 
                             , 
                             1 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         7 
                                         ; 
                                         4 
                                       
                                     
                                     
                                       r 
                                       7 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       7 
                                       , 
                                       
                                         1 
                                         ; 
                                         4 
                                       
                                     
                                   
                                   * 
                                   
                                     
                                       r 
                                       
                                         8 
                                         ; 
                                         4 
                                       
                                     
                                     
                                       r 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       8 
                                       , 
                                       
                                         1 
                                         ; 
                                         4 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         4 
                                         ; 
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                                       r 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       4 
                                       ; 
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                               * 
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         10 
                                         ; 
                                         5 
                                       
                                     
                                     
                                       r 
                                       
                                         1 
                                         ⁢ 
                                         0 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       10 
                                       , 
                                       
                                         1 
                                         ; 
                                         5 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       
                                         5 
                                         ; 
                                         1 
                                       
                                     
                                     
                                       r 
                                       5 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       5 
                                       ; 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 BX 
                                 1 
                               
                               ⁢ 
                               
                                 X 
                                 
                                   12 
                                   , 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 + 
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         
                                           7 
                                           ; 
                                           4 
                                         
                                       
                                       
                                         r 
                                         7 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         7 
                                         , 
                                         
                                           1 
                                           ; 
                                           4 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         r 
                                         
                                           4 
                                           ; 
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                                         r 
                                         4 
                                       
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         4 
                                         ; 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       
                                         8 
                                         ; 
                                         5 
                                       
                                     
                                     
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                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       8 
                                       , 
                                       
                                         1 
                                         ; 
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                                   * 
                                   
                                     
                                       r 
                                       
                                         10 
                                         ; 
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                                       r 
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                                     A 
                                     
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                                       , 
                                       
                                         1 
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                                       r 
                                       
                                         5 
                                         ; 
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                                       r 
                                       5 
                                     
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       5 
                                       ; 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 BX 
                                 1 
                               
                               ⁢ 
                               
                                 X 
                                 
                                   12 
                                   , 
                                   1 
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           r 
                                           
                                             7 
                                             ; 
                                             4 
                                           
                                         
                                         
                                           r 
                                           7 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           7 
                                           , 
                                           
                                             1 
                                             ; 
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                                       * 
                                       
                                         
                                           r 
                                           
                                             8 
                                             ; 
                                             4 
                                           
                                         
                                         
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                                           8 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           8 
                                           , 
                                           
                                             1 
                                             ; 
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                                       ⁢ 
                                       
                                         
                                           r 
                                           
                                             4 
                                             ; 
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                                       ⁢ 
                                       
                                         A 
                                         
                                           4 
                                           ; 
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                                   * 
                                   
                                     ( 
                                     
                                       
                                         
                                           r 
                                           
                                             10 
                                             ; 
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                                             1 
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                                           , 
                                           
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                                             ; 
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                                       ⁢ 
                                       
                                         
                                           r 
                                           
                                             5 
                                             ; 
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                                           5 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           5 
                                           ; 
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                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     + 
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             
                                               7 
                                               ; 
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                                             r 
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                                         ⁢ 
                                         
                                           A 
                                           
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                                               ; 
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                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               4 
                                               ; 
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                                             r 
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                                         ⁢ 
                                         
                                           A 
                                           
                                             4 
                                             ; 
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                                     ( 
                                     
                                       
                                         
                                           r 
                                           
                                             8 
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                                       ⁢ 
                                       
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                                           r 
                                           
                                             10 
                                             ; 
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                                           , 
                                           
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                                             ; 
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                           ) 
                         
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                           BX 
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                         ⁢ 
                         
                           X 
                           
                             12 
                             , 
                             1 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         1 
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           7 
                                           , 
                                           
                                             1 
                                             ; 
                                             4 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         1 
                                         2 
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           8 
                                           , 
                                           
                                             1 
                                             ; 
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                                       ⁢ 
                                       
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                                       ⁢ 
                                       
                                         A 
                                         
                                           4 
                                           ; 
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                                     ) 
                                   
                                   * 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         1 
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           10 
                                           , 
                                           
                                             1 
                                             ; 
                                             5 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         1 
                                         1 
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           5 
                                           ; 
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                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
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                                       ( 
                                       
                                         
                                           1 
                                           1 
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             7 
                                             , 
                                             
                                               1 
                                               ; 
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           1 
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                                         ⁢ 
                                         
                                           A 
                                           
                                             4 
                                             ; 
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                                       ) 
                                     
                                   
                                   * 
                                   
                                     ( 
                                     
                                       
                                         1 
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                                       ⁢ 
                                       
                                         A 
                                         
                                           8 
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                                             1 
                                             ; 
                                             5 
                                           
                                         
                                       
                                       * 
                                       
                                         1 
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                                       ⁢ 
                                       
                                         A 
                                         
                                           10 
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                                             1 
                                             ; 
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                                       ⁢ 
                                       
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                                       ⁢ 
                                       
                                         A 
                                         
                                           5 
                                           ; 
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                                   r 
                                   
                                     1 
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                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   ; 
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                                   r 
                                   
                                     1 
                                     ; 
                                     12 
                                   
                                 
                                 
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                               ⁢ 
                               
                                 A 
                                 
                                   
                                     1 
                                     ; 
                                     12 
                                   
                                   , 
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                               ⁢ 
                               
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                                   12 
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                           ) 
                         
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                           X 
                           
                             12 
                             , 
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                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         A 
                                         
                                           7 
                                           , 
                                           
                                             1 
                                             ; 
                                             4 
                                           
                                         
                                       
                                       * 
                                       
                                         1 
                                         2 
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           8 
                                           , 
                                           
                                             1 
                                             ; 
                                             4 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           4 
                                           ; 
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                                     ) 
                                   
                                   * 
                                   
                                     ( 
                                     
                                       
                                         A 
                                         
                                           10 
                                           , 
                                           
                                             1 
                                             ; 
                                             5 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           5 
                                           ; 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     + 
                                     
                                       ( 
                                       
                                         
                                           A 
                                           
                                             7 
                                             , 
                                             
                                               1 
                                               ; 
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             4 
                                             ; 
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                                       ) 
                                     
                                   
                                   * 
                                   
                                     ( 
                                     
                                       
                                         1 
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                                         A 
                                         
                                           8 
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                                             1 
                                             ; 
                                             5 
                                           
                                         
                                       
                                       * 
                                       
                                         A 
                                         
                                           10 
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                                             1 
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                                             5 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           5 
                                           ; 
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                                     ) 
                                   
                                 
                               
                             
                           
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                                 1 
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                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   ; 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 B 
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                                 1 
                                 2 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   
                                     1 
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                                   , 
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                         ⁢ 
                         
                           X 
                           
                             12 
                             , 
                             1 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     Pr 
                     ⁢ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         A 
                                         
                                           7 
                                           , 
                                           
                                             1 
                                             ; 
                                             4 
                                           
                                         
                                       
                                       * 
                                       
                                         1 
                                         2 
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           8 
                                           , 
                                           
                                             1 
                                             ; 
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                                       ⁢ 
                                       
                                         A 
                                         
                                           4 
                                           ; 
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                                   * 
                                   
                                     ( 
                                     
                                       
                                         A 
                                         
                                           10 
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                                             1 
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                                         A 
                                         
                                           5 
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                                           A 
                                           
                                             7 
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                                   * 
                                   
                                     ( 
                                     
                                       
                                         1 
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                                         A 
                                         
                                           8 
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                                             1 
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                                             5 
                                           
                                         
                                       
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                                         A 
                                         
                                           5 
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                                           10 
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                                 1 
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                                 A 
                                 
                                   1 
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                                   1 
                                 
                               
                               ⁢ 
                               
                                 B 
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                                 1 
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                                 A 
                                 
                                   
                                     1 
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                       } 
                     
                   
                 
               
             
             
               
                 
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                       ) 
                     
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                       ) 
                     
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                         12 
                         , 
                         
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                   = 
                     
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                       ( 
                       
                         
                           
                             
                               
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                                     ) 
                                   
                                   * 
                                   
                                     1 
                                     2 
                                   
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                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
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                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           - 
                                         
                                         
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                                     ) 
                                   
                                 
                                 ) 
                               
                               * 
                               
                                 ( 
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             - 
                                           
                                           
                                             0.7 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
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                                             - 
                                           
                                         
                                         
                                           
                                             - 
                                           
                                           
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                                       ) 
                                     
                                   
                                   ) 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 + 
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             - 
                                           
                                           
                                             0.7 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             - 
                                           
                                           
                                             - 
                                           
                                         
                                         
                                           
                                             - 
                                           
                                           
                                             0.9 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 ( 
                                 
                                   
                                     1 
                                     2 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
                                           0.7 
                                         
                                       
                                     
                                     ) 
                                   
                                   * 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
                                           0.7 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                         
                                         
                                           - 
                                         
                                       
                                       
                                         
                                           - 
                                         
                                         
                                           0.7 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   - 
                                 
                                 
                                   - 
                                 
                               
                               
                                 
                                   - 
                                 
                                 
                                   1 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   - 
                                 
                               
                               
                                 
                                   0.08 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   - 
                                 
                               
                               
                                 
                                   0.7 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     0.1 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               - 
                               
                                   
                               
                               ⁢ 
                               
                                 0 
                                 . 
                                 1 
                               
                             
                             ⁢ 
                             2 
                             ⁢ 
                             3 
                             ⁢ 
                             4 
                             ⁢ 
                             8 
                           
                           ) 
                         
                         + 
                         
                           ( 
                           
                             
                               - 
                               
                                   
                               
                               ⁢ 
                               
                                 0 
                                 . 
                                 1 
                               
                             
                             ⁢ 
                             0 
                             ⁢ 
                             8 
                             ⁢ 
                             0 
                             ⁢ 
                             4 
                             ⁢ 
                             5 
                           
                           ) 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             - 
                           
                         
                         
                           
                             0.39 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     0.1 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     0.00 
                     ⁢ 
                     9 
                     ⁢ 
                     029 
                   
                 
               
             
           
         
       
     
     Based on  FIG. 15 , 
                       ζ   2     ⁡     (   2   )       =       ⁢       Pr   ⁢     {     E   ⁡     (   2   )       }       =     P   ⁢   r   ⁢     {       X     6   ,   0       ⁢     X     7   ,   1       ⁢     X     8   ,   1       ⁢     X     10   ,   1       ⁢     X     12   ,   1         }                     =       ⁢     Pr   ⁢     {         r     6   ;   2         r   6       ⁢     A     6   ,     0   ;   2         ⁢   B   ⁢     X   2     ⁢       r     7   ,   4         r   7       ⁢     A     7   ,     1   ;   4         ⁢       r     4   ;   2         r   4       ⁢     A     4   ;   2       ⁢   B   ⁢       X   2     ⁡     (               r     8   ;   4         r   8       ⁢     A     8   ,     1   ;   4         ⁢       r     4   ;   2         r   4       ⁢     A     4   ;   2       ⁢   B   ⁢     X   2                   +       r     8   ;   5         r   8         ⁢     A     8   ,     1   ;   5         ⁢       r     5   ;   2         r   5       ⁢     A     5   ;   2       ⁢     BX   2             )       ⁢       r     10   ;   5         r   5       ⁢     A     10   ,     1   ;   5         ⁢       r     5   ;   2         r   5       ⁢     A     5   ;   2       ⁢   B   ⁢     X   2     ⁢     A     12   ,     1   ;   D           }                   =       ⁢     Pr   ⁢     {               r     6   ;   2         r   6       ⁢     A     6   ,     0   ;   2         *     (         r     7   ,   4         r   7       ⁢     A     7   ,     1   ;   4         *       r     8   ;   4         r   8       ⁢     A     8   ,     1   ;   4         ⁢       r     4   ;   2         r   4       ⁢     A     4   ;   2         )     *     (         r     10   ;   5         r   5       ⁢     A     10   ,     1   ;   5         ⁢       r     5   ;   2         r   5       ⁢     A     5   ;   2         )     ⁢     BX   2     ⁢     A     12   ,     1   ;   D                       +       r     6   ;   2         r   6         ⁢     A     6   ,     0   ;   2         *       r     7   ,   4         r   7       ⁢     A     7   ,     1   ;   4         ⁢       r     4   ;   2         r   4       ⁢     A     4   ;   2       *     (         r     8   ;   5         r   8       ⁢     A     8   ,     1   ;   5         *     A     10   ,     1   ;   5         ⁢       r     5   ;   2         r   5       ⁢     A     5   ;   2         )     ⁢     BX   2     ⁢     A     12   ,     1   ;   D                 }                   =       ⁢     Pr   ⁢     {       (           (       1   1     ⁢     A     6   ,     0   ;   2         *     (       1   1     ⁢     A     7   ,     1   ;   4         *     1   2     ⁢     A     8   ,     1   ;   4         ⁢     1   1     ⁢     A     4   ;   2         )     *     (       1   1     ⁢     A     10   ,     1   ;   5         ⁢     1   1     ⁢     A     5   ;   2         )                     +     1   1       ⁢     A     6   ,     0   ;   2         *     1   1     ⁢     A     7   ,     1   ;   4         ⁢     1   1     ⁢     A     4   ;   2       *     (       1   2     ⁢     A     8   ,     1   ;   5         *     1   1     ⁢     A     10   ,     1   ;   5         ⁢     1   1     ⁢     A     5   ;   2         )             )     ⁢     BX   2     ⁢     A     12   ,     1   ;   D           }                   =       ⁢     Pr   ⁢     {       A     6   ,     0   ;   2         *     (             (       A     7   ,     1   ;   4         *     1   2     ⁢     A     8   ,     1   ;   4         ⁢     A     4   ;   2         )     *     (       A     10   ,     1   ;   5         ⁢     A     5   ;   2         )                   +     A     7   ,     1   ;   4           ⁢     A     4   ;   2       *     (       1   2     ⁢     A     8   ,     1   ;   5         *     A     10   ,     1   ;   5         ⁢     A     5   ;   2         )             )     ⁢     A     2   ;   2       ⁢     B   2     ⁢     A     12   ,     1   ;   D           }                   =       ⁢       a     6   ,     0   ;   2         *     (             (       a     7   ,     1   ;   4         *     1   2     ⁢     a     8   ,     1   ;   4         ⁢     a     4   ;   2         )     *     (       a     10   ,     1   ;   5         ⁢     a     5   ;   2         )                   +     a     7   ,     1   ;   4           ⁢     a     4   ;   2       *     (       1   2     ⁢     a     8   ,     1   ;   5         *     a     10   ,     1   ;   5         ⁢     a     5   ;   2         )             )     ⁢     a     2   ;   2       ⁢     b   2     ⁢     a     12   ,     1   ;   D                       =       ⁢       (             1         1   -   0.6   -   0.4           ⁢   1     -   0.3   -   0.7     )     *     (           (       (         -       0.7         )     *     1   2     ⁢     (         -       0.8         )     ⁢     (         -       -       -           -       0.5       0.5         )       )               *     (       (         -       0.7         )     ⁢     (         -       -       -           -       0.4       0.8         )       )                   +     (       (         -       0.7         )     ⁢     (         -       -       -           -       0.5       0.5         )       )       *               (       1   2     ⁢     (         -       0.7         )     *     (         -       0.7         )     ⁢     (         -       -       -           -       0.4       0.8         )       )           )     ⁢     (         -       -       -           -       1       -           -       -       1         )     ⁢     (         -           0.01           0.02         )     ⁢   0.1                 =       ⁢   0               
According to the algorithms in [8], the weighting coefficients
 
                 ξ   i     ⁡     (   y   )       =         ζ   i     ⁡     (   y   )           ∑   j     ⁢       ζ   j     ⁡     (   y   )                 
of the sub-graphs are:
 
                   ξ   1     ⁡     (   2   )       =           ζ   1     ⁡     (   2   )             ζ   1     ⁡     (   2   )       +       ζ   2     ⁡     (   2   )           =           0   .   0     ⁢   0   ⁢   9   ⁢   0   ⁢   2   ⁢   9           0   .   0     ⁢   0   ⁢   9   ⁢   0   ⁢   2   ⁢   9     +   0       =   1         ⁢     
     ⁢         ξ   2     ⁡     (   2   )       =           ζ   2     ⁡     (   2   )             ζ   1     ⁡     (   2   )       +       ζ   2     ⁡     (   2   )           =       0         0   .   0     ⁢   0   ⁢   9   ⁢   0   ⁢   2   ⁢   9     +   0       =   0               
Since ζ 2 (0)=0 or ξ 2 (0)=0,  FIG. 15  does not hold and should be eliminated.  FIG. 14  is the only valid sub-graph, which means, BX 1,1  is the only event in S H (2). According to 10, the ranking procedure ends. The cause of abnormality of the current system is verified as BX 1,1 .