Patent Publication Number: US-2010131863-A1

Title: Incremental collaborative filtering

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Patent Application 61/118,449, filed Nov. 27, 2008, which is incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to systems and methods for content distribution, and specifically to methods for individualized targeting of content. 
     BACKGROUND OF THE INVENTION 
     In network-based content distribution, promotional content, such as advertisements, is often pushed to the users. Typically, the advertisements contain hyperlinks, so that the user can respond, if interested, by clicking on the advertisement. The expected click rate is in the range of 1-2% when advertisements are sent randomly. A higher click rate is desirable in order to increase revenues from the advertising process. Methods of targeted advertising attempt to increase user response by matching advertisements to known or inferred user characteristics. 
     Recommendation systems apply knowledge discovery techniques in order to make personalized recommendations for delivery of certain content during an interaction with a given user. One technique that may be used for this purpose is collaborative filtering. This techniques works by building a database of preferences for items by users. A given user is matched against the database to discover “neighbors”—other users who have historically had similar taste to the given user. Items that the neighbors have liked are then recommended to the given user. Techniques of this sort are described, for example, by Sarwar et al., in “Item-Based Collaborative Filtering Recommendation Algorithms,”  Proceedings of the World Wide Web Conference WWW 10, pages 285-295 (Hong Kong, 2001), which is incorporated herein by reference. 
     Collaborative filtering (CF) requires expensive computations that grow polynomially with the number of users and items in the database. Papagelis et al. propose a method for addressing this scalability problem in “Incremental Collaborative Filtering for Highly-Scalable Recommendation Algorithms,” 15 th International Symposium on Methodologies for Intelligent Systems ISMIS  2005 (Saratoga Springs, N.Y., 2005), which is incorporated herein by reference. The method is based on incremental updates of user-to-user similarities and is said to provide recommendations orders of magnitude faster than classic CF. 
     SUMMARY OF THE INVENTION 
     Embodiments of the present invention that are described hereinbelow provide efficient methods and systems for collaborative filtering. These methods are described in the specific context of targeted advertising, but they may similarly be used in other interactive on-line applications. 
     There is therefore provided, in accordance with an embodiment of the present invention, a method for content delivery, including obtaining responses from a first plurality of users to a second plurality of items of content that were delivered to the users over a communication network. For each item, a respective vector is created corresponding to the responses of the users to the item. Distances are computed between the items responsively to the vectors. A first item is selected for delivery to a given user based on a previous response of the given user to a second item and to a distance computed between the first and second items responsively to the vectors. Following the delivery of the first item and receiving a response to the first item from the given user, the distances computed between the items are updated based on the response to the first item. The updated distances are applied in selecting a further item for delivery to another user. 
     In a disclosed embodiment, updating the distances includes computing an increment to a prior distance that was computed before the delivery of the first item to the given user, and applying the increment to the prior distance in order to find a new distance. Typically, computing the increment includes calculating changes in the vector corresponding to the first item without computation over the elements of the vector corresponding to the second item. 
     In some embodiments, selecting the first item includes selecting an advertisement for transmission over a wireless network to a mobile communication device operated by the given user, and receiving the response includes determining whether the user interacted with a hyperlink in the advertisement. 
     In disclosed embodiments, selecting the first item includes computing, based on the distances, a likelihood that the given user will return a positive response to the first item, and choosing the first item responsively to the likelihood. Computing the likelihood may include weighting the distances based on an age of the responses used in computing the distances. Additionally or alternatively, obtaining the responses may include delivering at least one of the items to at least some of the users multiple times, including at least first and second times, and creating the respective vector may include creating first and second vectors corresponding respectively to the responses of the users to the first and second times that the at least one of the items was presented to them. 
     There is also provided, in accordance with an embodiment of the present invention, apparatus for content delivery, including a memory, coupled to store responses from a first plurality of users to a second plurality of items of content that were delivered to the users over a communication network. A processor is coupled to the memory and is configured to create, for each item, a respective vector corresponding to the responses of the users to the item, to compute distances between the items responsively to the vectors, to select a first item for delivery to a given user based on a previous response of the given user to a second item and to a distance computed between the first and second items responsively to the vectors. The processor is configured, upon receiving a response to the first item from the given user following the delivery of the first item, to update the distances computed between the items based on the response to the first item, and to apply the updated distances in selecting a further item for delivery to another user. 
     There is additionally provided, in accordance with an embodiment of the present invention, a computer software product, including a computer-readable medium in which program instructions are stored. The instructions, when read by a computer, cause the computer to store responses from a first plurality of users to a second plurality of items of content that were delivered to the users over a communication network, to create, for each item, a respective vector corresponding to the responses of the users to the item, to compute distances between the items responsively to the vectors, to select a first item for delivery to a given user based on a previous response of the given user to a second item and to a distance computed between the first and second items responsively to the vectors. The instructions cause the computer, upon receiving a response to the first item from the given user following the delivery of the first item, to update the distances computed between the items based on the response to the first item, and to apply the updated distances in selecting a further item for delivery to another user. 
     The present invention will be more fully understood from the following detailed description of the embodiments thereof, taken together with the drawings in which: 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram that schematically illustrates a system for mobile content distribution, in accordance with an embodiment of the present invention; and 
         FIG. 2  is a flow chart that schematically illustrates a method for distribution of advertisements based on incremental collaborative filtering, in accordance with an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
       FIG. 1  is a block diagram that schematically illustrates of a communication system  20 , in accordance with an embodiment of the present invention. Users  22  operate respective mobile communication devices  24 , such as digital cellular telephones, wireless-enabled personal digital assistants (PDAs), and other wireless-enabled computing devices. Mobile communication devices  24  are configured in hardware and/or software to receive and present items of content, such as images, video, audio, and/or text. Communication devices  24  are identified by respective unique identifiers, such as MSISDNs in GSM-based cellular networks, usernames or cookies in Web-based applications, or any other suitable subscriber identifier. Alternatively or additionally, although the pictured embodiment relates specifically to advertising via mobile communications, the methods described hereinbelow may similarly be applied in delivering targeted content of substantially any type to computing and multimedia devices over both wireless and wired networks. 
     Devices  24  communicate over a wireless network  26 , which comprises one or more base stations  28 . A mobile content server  32 , which comprises at least one processor and an interface, generates content, comprising images, video, audio, text, and/or other data, and sends the content to a switch  30 , which routes the content to devices  24  via network  26 . 
     At least some of the content that content server  32  generates contains intervals in which promotional content items, such as advertisements, may be inserted. An ad server  36  is invoked to supply these content items in the appropriate intervals. The intervals may be before, after or within the content provided by server  32  (in which case advertisements may also be inserted over some picture space in the content). The ad server receives an indication of the identity of the device to which the item is to be delivered. It uses this indication together with a recommendation provided by an incremental collaborative filtering (ICF) server  38  in order to select an item that is likely to engender a positive response from the user of the device. Some or all of the content items include user controls, such as hyperlinks, with which users  22  can interact in order to respond to the items. Such interaction (including activation and/or selection of the hyperlink) may take place by any suitable method that is known in the art, and is referred to herein for convenience as “clicking” on the item. 
     ICF server  38  derives its recommendation by processing previous responses of users  22 . For this purpose, when one of the users responds to a given content item, switch  30  routes the response to the ICF server, which performs ICF processing functions that are described in detail hereinbelow. ICF server  38  in the pictured embodiment comprises at least one processor  40  and a memory  42  for storing user response data. Processor  40  typically comprises a general-purpose computer, which is programmed in software to carry out the response processing functions that are described below. This software may be provided in electronic form, over a network, for example, or it may, alternatively or additionally, be stored on tangible storage media, such as optical, magnetic or electronic memory media. Alternatively, some or all of the functions of the ICF server  38  may be implemented using dedicated or programmable hardware circuits. Although ICF server  38  is shown and described, for the sake of clarity, as a separate, stand-alone unit, the functions of the ICF server may alternatively be integrated into other elements of system  20 , such as switch  30  or server  36 . 
     ICF server  38  processes the user response data in order to create a User-Item (UI) matrix in memory  42 , also referred to as a similarity matrix. Each of the columns of the matrix is a vector that tabulates the responses of the users to a particular item. For example, Table I below shows a simple UI matrix for five users and five items. Assuming the items offer users a binary choice (click or ignore), each UI cell can have three modes:
     1. Empty—The item was not introduced yet to that user.   2. 0—The item was presented to the user and the user did not respond (no click).   3. 1—The item was presented to the user and the user responded (clicked).   

                     TABLE I                  UI MATRIX EXAMPLE                                         Item 1   Item 2   Item 3   Item 4   Item 5                                                 User 1   1   1   0               User 2   0   0   1   0       User 3   1   1   0   1   1       User 4   0   0   1   0       User 5   ?   1   0                    
In this example, ICF server  38  is to select an item to deliver to User 5 by predicting the item or items to which this user is likely to give a positive response (click).
 
     The prediction process is based on extracting similarities between the columns (item vectors) of the UI matrix. These similarities can be expressed in terms of distances between the item vectors, wherein the smaller the distance, the greater the similarity. Two items are considered similar when many users co-rate both items. For mobile advertising applications, such as in system  20 , there are typically many users and a relatively small number of items (such as advertisements). Therefore the UI matrix is composed of a relatively small number of long columns. 
     In the example shown in Table I, server  38  is called upon to predict whether user 5 will like item 1. Because item 1 and item 2 have similar item vectors, and user 5 has clicked on item 2, it can be predicted will probably click on item 1, as well. On the other hand, the item vector of item 3 is inverse to that of item 1. 
     Therefore, if user 5 did not like item 3, he will probably like item 1 (although this sort of prediction, based on large distance between vectors, is typically less reliable than prediction based on small distance, such as between items 1 and 2). There is also some similarity between item 4 and item 1, but the response of user 5 to item 4 is not yet known. Similarly, since there is only one cell in common between the item vectors of item 1 and item 5, there is not yet enough information to measure the similarity between these items. 
     Although distances can be inferred visually in the simple example of Table I, in practice processor  40  computes a quantitative distance measure in order to assess the similarity between item vectors. Small distance corresponds to close similarity, and vice versa. These distances are then used in predicting user responses to new items. In some embodiments, the processor computes a prediction value {circumflex over (R)} u     a     i     a    (indicating the likelihood that user u a  will respond positively to item i a ) based on the Pearson distance Sim i     x     i     y    between the item vectors. These factors are defined as follows: 
     
       
         
           
             
               
                 
                   
                     
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     In the above equations, R u     a     i     a    is the actual response that user u a  gave to item i a  (1 or 0 in the present binary example), while  R   i     a    is the average response for item i a . In equation (2), the distance calculation Sim i     x     i     y    between items i x  and i y  includes only users who rated both items (and the sums are over N u  such users, as explained further in the Appendix). N i  is the number of items used in the prediction calculation (either all the available items or a certain subset). Alternatively, other distance measures (such as cosine distance) and prediction formulas may be used. When there are several candidate items for delivery to user u a , server  38  typically selects the item with the highest prediction value. 
     Prediction calculations over a large user base are computation-intensive, particularly if the distance (similarity) values must be updated each time. For example, assuming that N i ˜200 and N u ˜10 6 , a direct calculation of a single prediction requires about 6*10 8  multiplications and 1.4*10 9  additions. This computational burden limits the number of predictions that server  38  can perform per second, which thus limits the number of targeted advertisements that can be delivered in system  20 . 
     In order to overcome this problem, server  38  uses an incremental method to update the distance measures each time new data is added to the UI matrix. In other words, the server applies item-based ICF to update each Sim i     x     i     y    value by incrementing its previous value, without completely recomputing equation (2) over the entire updated UI matrix. The server then uses the updated distance measures in selecting the subsequent content items for delivery. 
       FIG. 2  is a flow chart that schematically illustrates a method for distribution of content items based on ICF, in accordance with an embodiment of the present invention. The method is described hereinbelow, for the sake of clarity and convenience, with reference to the mobile advertising system that is shown in  FIG. 1 , but the principles of this method may likewise be applied in targeted distribution of items of other types, in both wireless and wired network environments. 
     ICF server  38  collects user responses to content items delivered by ad server  36  to devices  24 , at a click collection step  50 . The ICF server records both positive responses (“clicks”) and negative responses (item displayed with no user click) in memory  42 . Processor  40  arranges the responses in a UI matrix, as defined above, in a UI building step  52 . Once a sufficient number of responses has accumulated, processor  40  computes initial distance values between the item vectors (i.e., between the columns of the UI matrix), at a distance calculation step  54 . For this purpose, the processor may use the distance measure defined above in equation (2), or another suitable distance measure. Typically, a distance measure between two given items is considered to be valid only if at least a certain minimum number of users (for example, 100 users) have responded, positively or negatively, to both items. 
     Ad server  36  receives new requests to deliver a content item to a given user  22  via the respective device  24 , at an ad request step  56 . The ad server asks ICF server  38  to provide a recommendation in response to each request, or at least some of the requests. In order to make the recommendation, processor  40  computes prediction values {circumflex over (R)} u     a     i     a    for the given user with respect to one or more of the content items that have not yet been delivered to the user, using equation (1) and the current distance values Sim i     x     i     y   . Typically, the prediction will be considered valid only if at least a certain minimum number of the component distance values (for example, five distance values) are available. 
     ICF server  38  then recommends to ad server  36  a content item with a high prediction value—either the highest value among the available items or at least a value above a certain threshold. For example, processor may check whether {circumflex over (R)} u     a     i     a   &gt;  R   i     a   , or equivalently may check whether the deviation from the mean component Δ u     a     i     a    is positive: 
     
       
         
           
             
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     When there are a number of items to choose from, processor  40  may produce a sorted listed of candidates according to the value of Δ u     a     i     a   , from most favorite to less favorite. Usually the ICF server will recommend a content item that has not yet been delivered to the user in question, although in some cases content items may be repeated. Ad server  36  then delivers the recommended item to the user, at an ad delivery step  58 . 
     After the content item has been delivered to the user, switch  30  reports the user response to ICF server  38 , at a status recording step  60 . The server updates the UI matrix with the response data, at a UI update step  62 . For example, the server may enter a “1” or “0” in the corresponding cell of the matrix, depending on whether or not the user clicked on the link in the advertisement that was presented to him or her. The server may, from time to time, use the data in the UI matrix to completely recalculate the column distance values, as it did at step  54 , in order to eliminate accumulated errors due to the incremental calculations. 
     In general, however, processor  40  updates the column distances incrementally, at a distance update step  64 . The incremental computations take into account the new response value and the change it causes in the corresponding column average, while avoiding recomputation of sums and products over the entire UI matrix. For this purpose, each of the initially-calculated distance values can be framed parametrically as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     The distance between any pair of columns (items) x and y can be derived from the corresponding elements of the matrices B, C and D. Each time a user response is received, the affected elements of the matrices are updated incrementally: 
     
       
      
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     The values of the increments to the elements of e, f and g depend on the specific circumstances that apply to each matrix element (for example, which items have already been delivered to and responded to—positively or negatively—by the user in question). Detailed formulas are presented below in an Appendix. 
     ICF server  38  uses the updated distances computed at step  64 , based on equations (3) and (4), in deciding what items to recommend for delivery to other users in subsequent passes through steps  56 - 64 . In an experiment conducted on a standard Pentium-based computer, the inventors found that they were able to generate approximately 50,000 recommendations (prediction operations) per second in this manner. 
     Although the formulas presented above give equal weights to all user responses, server  38  may alternatively take into account temporal factors and, for example, give greater weight to more recent responses. Responses to items delivered in the distant past may be considered obsolete and eliminated. For this purpose, the corresponding columns may be removed from the UI matrix and the distances recalculated over the remaining items (according to the procedure at step  54 ). 
     For the remaining items, processor  40  may apply an aging factor, which will tend to give a higher weight to similarity with recently-introduced items. For this purpose, equation (1) may be reformulated as follows: 
     
       
         
           
             
               
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     In this formula, w k =e −λ·T     k   , λ is a constant aging factor, and T k  is the elapsed time (in months, for example) since item number k was introduced. Thus, if λ=0.1, the weight will be reduced to about 0.5 for items that are six months old. 
     The methods presented above are designed to predict user response to newly-introduced items on the assumption that any given item is presented to a user only once. In some cases, however, an item may be delivered several times to the same user, during an advertising campaign, for example. This situation may be handled simply by modifying the above methods in a number of ways, for example:
         Only the first response counts, and subsequent responses to the same item are ignored.   Only the last response counts, and earlier responses are ignored.       

     Alternatively, for more accurate prediction, each repeat delivery of a given item can be treated as a new “pseudo-item,” with its own column in the UI matrix. The matrix may be processed in various ways in order to predict user responses to a given item, including:
         The processor may use the entire UI matrix, including all columns, in the calculation.   The processor may consider only the subset of columns corresponding to the given item (for initial and repeat deliveries).       

     It will be appreciated that the embodiments described above are cited by way of example, and that the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention includes both combinations and subcombinations of the various features described hereinabove, as well as variations and modifications thereof which would occur to persons skilled in the art upon reading the foregoing description and which are not disclosed in the prior art. 
     Appendix—Calculation OF ICF Increments 
     The section that follows presents the formulas for calculation of the distance increments e, f and g, as defined above in equation (4). The formulas vary depending on circumstances pertaining to the user in question and the different items in the UI matrix. In each case, the increments are computed based on the changes that have occurred in the specific columns (vectors) corresponding to the items in question without computation over all elements in the columns as required by equation (2). 
     Case 1—In this case, an increment for updating Sim i     a     i     y    is to be calculated after a user u a  has responded to item i a  for the first time, without having yet responded to item i y . Here the increments in equation (4) result from dR i     a   , i .e . , the change in  R   i     a   . In the formulas below, N denotes the number of users who have responded to both of items i a  and i y , and R S1 , R S2  denote the sums of the UI matrix entries in the columns for items i a  and i y , respectively. (Thus, the sums are taken only over users who responded to both items. The symbol “h” is used to index members of this subset of users.) 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           B 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           B 
                           - 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           C 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           - 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     a 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             dR 
                             
                               i 
                               a 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           C 
                           - 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     D 
                     ′ 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           h 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         
                           ( 
                           
                             
                               R 
                               
                                 
                                   u 
                                   h 
                                 
                                  
                                 
                                   i 
                                   y 
                                 
                               
                             
                             - 
                             
                               
                                 R 
                                 _ 
                               
                               
                                 i 
                                 y 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     = 
                     D 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     The incremental steps are then given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         e 
                         = 
                           
                          
                         
                           - 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                           
                           · 
                           
                             ( 
                             
                               
                                 R 
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                               + 
                               
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                                 · 
                                 N 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         f 
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                   · 
                                   N 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                             · 
                             N 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   g 
                   = 
                   0 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Since only the entries in the column belonging to i a  have changed, calculating the increments to the distance values requires only that processor  40  update sums and differences over this column, followed by a few multiplications. The same simplification occurs in the other cases listed below, as well. 
     Case 2—In this case, the increment for updating Sim i     a     i     y    is to be calculated after user u a  has responded to item i a  for the first time, after having previously responded to item i y , as well. Here the increment to Sim i     a     i     y    results both from the change in  R   i     a    and from an additional element in the sums due to R u     a     i   a . 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           B 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                   - 
                                   
                                     dR 
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ( 
                             
                               
                                 R 
                                 
                                   
                                     u 
                                     a 
                                   
                                    
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               - 
                               
                                 
                                   R 
                                   _ 
                                 
                                 
                                   i 
                                   a 
                                 
                               
                               - 
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               
                                 R 
                                 
                                   
                                     u 
                                     a 
                                   
                                    
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               - 
                               
                                 
                                   R 
                                   _ 
                                 
                                 
                                   i 
                                   y 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           B 
                           - 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           C 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                   - 
                                   
                                     dR 
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           - 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           C 
                           - 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ( 
                             
                               
                                 R 
                                 
                                   
                                     u 
                                     a 
                                   
                                    
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               - 
                               
                                 
                                   R 
                                   _ 
                                 
                                 
                                   i 
                                   a 
                                 
                               
                               - 
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           D 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         a 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           D 
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The incremental steps are given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         e 
                         = 
                           
                          
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       R 
                                       
                                         
                                           u 
                                           h 
                                         
                                          
                                         
                                           i 
                                           y 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         R 
                                         _ 
                                       
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                   · 
                                   N 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         f 
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                   · 
                                   N 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                             · 
                             N 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   g 
                   = 
                   
                     
                       ( 
                       
                         
                           R 
                           
                             
                               u 
                               a 
                             
                              
                             
                               i 
                               y 
                             
                           
                         
                         - 
                         
                           
                             R 
                             _ 
                           
                           
                             i 
                             y 
                           
                         
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Case 3—User u a  has updated a previous response to item i a , but has not yet entered any response to item i y . Here the difference results from the change in  R   i     a   . 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           B 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           B 
                           - 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           C 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           - 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             dR 
                             
                               i 
                               a 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           C 
                           - 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           D 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         D 
                       
                     
                   
                 
               
             
           
         
       
     
     The incremental steps are given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         e 
                         = 
                           
                          
                         
                           - 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                           
                           · 
                           
                             ( 
                             
                               
                                 R 
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                               + 
                               
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                                 · 
                                 N 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         f 
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                   · 
                                   N 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                             · 
                             N 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   g 
                   = 
                   0 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Case 4–User u a  has updated his previous response to item i a , and user u a  has also responded to item i y . Here the difference results both from the change in  R   i     a    and from the change in the response, dR u     a     i     a   . 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           B 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                   - 
                                   
                                     dR 
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 + 
                                 
                                   dR 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ( 
                             
                               
                                 R 
                                 
                                   
                                     u 
                                     a 
                                   
                                    
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               - 
                               
                                 
                                   R 
                                   _ 
                                 
                                 
                                   i 
                                   a 
                                 
                               
                               - 
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               
                                 R 
                                 
                                   
                                     u 
                                     a 
                                   
                                    
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               - 
                               
                                 
                                   R 
                                   _ 
                                 
                                 
                                   i 
                                   y 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               dR 
                               
                                 
                                   u 
                                   a 
                                 
                                  
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           B 
                           - 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               dR 
                               
                                 
                                   u 
                                   a 
                                 
                                  
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           C 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                   - 
                                   
                                     dR 
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                                 + 
                                 
                                   dR 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ( 
                             
                               
                                 R 
                                 
                                   
                                     u 
                                     a 
                                   
                                    
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               - 
                               
                                 
                                   R 
                                   _ 
                                 
                                 
                                   i 
                                   a 
                                 
                               
                               - 
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           - 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                           + 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 
                                   u 
                                   a 
                                 
                                  
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     a 
                                   
                                 
                                 - 
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             dR 
                             
                               
                                 u 
                                 a 
                               
                                
                               
                                 i 
                                 a 
                               
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           C 
                           - 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                           + 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 
                                   u 
                                   a 
                                 
                                  
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ( 
                             
                               
                                 R 
                                 
                                   
                                     u 
                                     a 
                                   
                                    
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               - 
                               
                                 
                                   R 
                                   _ 
                                 
                                 
                                   i 
                                   a 
                                 
                               
                               - 
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             dR 
                             
                               
                                 u 
                                 a 
                               
                                
                               
                                 i 
                                 a 
                               
                             
                             2 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           D 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               h 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       h 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         D 
                       
                     
                   
                 
               
             
           
         
       
     
     The incremental steps are given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         e 
                         = 
                           
                          
                         
                           
                             - 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 
                                   dR 
                                   
                                     i 
                                     a 
                                   
                                 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       R 
                                       
                                         
                                           u 
                                           h 
                                         
                                          
                                         
                                           i 
                                           y 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         R 
                                         _ 
                                       
                                       
                                         i 
                                         y 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           
                             
                               dR 
                               
                                 
                                   u 
                                   a 
                                 
                                  
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       y 
                                     
                                   
                                   · 
                                   N 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               dR 
                               
                                 
                                   u 
                                   a 
                                 
                                  
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     
                                       u 
                                       a 
                                     
                                      
                                     
                                       i 
                                       y 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     _ 
                                   
                                   
                                     i 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         f 
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   h 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     
                                       
                                         u 
                                         h 
                                       
                                        
                                       
                                         i 
                                         a 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 h 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                           
                           + 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 
                                   u 
                                   a 
                                 
                                  
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ( 
                             
                               
                                 R 
                                 
                                   
                                     u 
                                     a 
                                   
                                    
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               - 
                               
                                 
                                   R 
                                   _ 
                                 
                                 
                                   i 
                                   a 
                                 
                               
                               - 
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             dR 
                             
                               
                                 u 
                                 a 
                               
                                
                               
                                 i 
                                 a 
                               
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                             · 
                             
                               dR 
                               
                                 i 
                                 a 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       R 
                                       _ 
                                     
                                     
                                       i 
                                       a 
                                     
                                   
                                   · 
                                   N 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               dR 
                               
                                 i 
                                 a 
                               
                               2 
                             
                             · 
                             N 
                           
                           + 
                           
                             2 
                             · 
                             
                               dR 
                               
                                 
                                   u 
                                   a 
                                 
                                  
                                 
                                   i 
                                   a 
                                 
                               
                             
                             · 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ( 
                             
                               
                                 R 
                                 
                                   
                                     u 
                                     a 
                                   
                                    
                                   
                                     i 
                                     a 
                                   
                                 
                               
                               - 
                               
                                 
                                   R 
                                   _ 
                                 
                                 
                                   i 
                                   a 
                                 
                               
                               - 
                               
                                 dR 
                                 
                                   i 
                                   a 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             dR 
                             
                               
                                 u 
                                 a 
                               
                                
                               
                                 i 
                                 a 
                               
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   g 
                   = 
                   0