Patent Application: US-68994410-A

Abstract:
methods for markov boundary discovery are important recent developments in pattern recognition and applied statistics , primarily because they offer a principled solution to the variable / feature selection problem and give insight about local causal structure . currently there exist two major local method families for identification of markov boundaries from data : methods that directly implement the definition of the markov boundary and newer compositional markov boundary methods that are more sample efficient and thus often more accurate in practical applications . however , in the datasets with hidden variables compositional markov boundary methods may miss some markov boundary members . the present invention circumvents this limitation of the compositional markov boundary methods and proposes a new method that can discover markov boundaries from the datasets with hidden variables and do so in a much more sample efficient manner than methods that directly implement the definition of the markov boundary . in general , the inventive method transforms a dataset with many variables into a minimal reduced dataset where all variables are needed for optimal prediction of some response variable . the power of the invention was empirically demonstrated with data generated by bayesian networks and with 13 real datasets from a diversity of application domains .

Description:
this specification teaches a novel method for discovery of a markov boundary of the response / target variable from datasets with hidden variables ( specifically , the method identifies a markov boundary of the response / target variable in the distribution over observed variables ). the novel method relies on the assumption that the distribution over all variables ( observed and unobserved ) involved in the underlying causal process is faithful to some dag ( spirtes et al ., 2000 ) ( whereas the distribution over a subset consisting of the observed variables may be unfaithful ). in general , the inventive method transforms a dataset with many variables into a minimal reduced dataset where all variables are needed for optimal prediction of some response variable . notation and key definitions are described in the appendix . the core method for finding a markov boundary of the response / target variable in the distributions where possibly not all variables have been observed is described in table 1 . several ways to apply this methodology are described herein . in particular , three generative methods cimb1 , cimb2 , cimb3 are described in tables 2 , 3 , 4 , respectively . the term “ generative method ” refers to a method that can be instantiated ( parameterized ) in a plurality of ways such that each instantiation provides a specific process to solve the problem of finding a markov boundary of t in the distributions where possibly not all variables have been observed such that the distribution over all ( observed and unobserved ) variables involved in the causal process is faithful . ( a ) the core method ( table 1 ). ( b ) the cimb1 , cimb2 , and cimb3 generative methods ( tables 2 - 4 ) being exemplars of the core method . ( c ) a plurality of instantiations of cimb1 , cimb2 , cimb3 demonstrating how these generative methods can be configured when reduced to practice ( e . g ., see table 5 ). ( d ) a method cimb * ( tables 6 - 8 ) that applies the core method while incorporating efficiency optimizations to speed up operation of the core method when implemented using a general - purpose digital computer . ( e ) variants of the cimb * method , termed cimb * 1 and cimb * 2 ( described below ). a pseudo - code to implement the method cimb1 is provided in table 5 . other implementations of the method cimb1 can be obtained by instantiating its steps as follows ( refer to table 2 for steps mentioned below ): step 2 : any strategy to iterate over variables z ∈ v \( tmb ( t )∪{ t }) can be employed . for example , one can use the strategy outlined in the pseudo - code that implements cimb1 ( table 5 ) or the more efficient strategy that is described in the cimb * method below ( tables 6 - 8 ). those who are skilled in the art can implement many additional known iteration strategies . step 3 : any backward elimination strategy can be used . those who are skilled in the art will recognize many suitable known methods such as the wrapper methods described in ( kohavi and john , 1997 ). step 1 of the sub - routine to determine whether x has a collider path to t : any available local or global method to learn a causal graph g to identify the existence of a collider path between x and t can be selected by those who are skilled in the art . for example , one can use the fci and pc methods implemented in tetrad software ( spirtes et al ., 2000 ). similarly , one can use the approach outlined in the cimb * method that is described below ( tables 6 - 8 ). implementations of the method cimb2 can be obtained by instantiating its steps as follows ( refer to table 3 for steps mentioned below ): step 2 : any method that learns a causal graph g over v can be employed . those who are skilled in the art can recognize that the fci and pc methods implemented in tetrad software ( spirtes et al ., 2000 ) can be used . step 4 : any backward elimination strategy can be used . those who are skilled in the art will recognize many suitable known methods such as the wrapper methods described in ( kohavi and john , 1997 ). implementations of the method cimb3 can be obtained by instantiating its steps as follows ( refer to table 4 for steps mentioned below ): steps 2 and 3 : any forward selection and backward elimination strategies can be used . those who are skilled in the art will recognize many known suitable methods such as the wrapper methods described in ( kohavi and john , 1997 ). step 2 : apply the forward selection strategy by prioritizing variables for inclusion in tmb ( t ) according to : the strength of their association with t . the strength of their association with k where k is member of the current tmb ( t ). the membership of variables in gll - pc ( k ) where k is a member of the current tmb ( t ). the method cimb * described in table 6 is an instantiation of the core method and also can be seen as a variant of cimb1 . first , cimb * uses an efficient strategy to consider only potential members of the markov boundary . in other words , it does not iterate over all z ∈ v \( tmb ( t )∪{ t }), but it iterates only over a subset of v \( tmb ( t )∪{ t }). second , the approach used for identification of a collider path to t ( that is used in the sub - routine of cimb1 ) is based on recursive application of the gll - pc method ( to build regions of the network ) and subsequent application of the collider orientation rules that are described in the sub - routines find - spouses1 ( table 7 ) and find - spouses2 ( table 8 ) and in steps 19 - 29 of the cimb * method ( table 6 ). the examples provided below motivate the reasoning behind collider orientation rules that are described in steps 19 - 29 of the cimb * method ( and denoted as case a and b in the cimb * pseudo - code ): case a ( y and z are not adjacent ): consider two graphical structures shown in fig1 a and 2 a . assume that cimb * reached point of its operation when it identified the structures shown in fig1 b and 2 b . one wants to determine if z belongs to a mb ( t ). for both structures , w ={ r } is a sepset of y and z ( i . e ., y is independent of z given w ). since y is dependent on z given w ∪{ s }={ r , s }, z is mb ( t ) member . case b ( y and z are adjacent ): consider a graphical structure shown in fig3 a . assume that cimb * reached point of its operation when it identified the structure shown in fig3 b . one wants to determine if z belongs to mb ( t ). the sepset w of t and z is empty . since t is dependent on z given w ∪{ a 1 , a 2 , y , s }={ a 1 , a 2 , y , s }, z is mb ( t ) member . the following describes several ways to obtain variants of the method cimb * by modifying pseudo - code of the method : one variant of the cimb * method ( referred to as method cimb * 1 ) is the same as cimb * except that it does not consider case a and applies case b both when y and z are adjacent and when they are not adjacent . another heuristic variant of the cimb * method ( referred to as cimb * 2 ) improves upon cimb * 1 by conditioning not on all variables in the collider path but on subsets of limited size . e . g ., consider structure shown in fig4 . assume , one can condition on up to 3 variables . then if one of the following holds , z is a member of mb ( t ): i ( t , y | a 1 ), i ( t , y | a 1 , a 2 ), i ( t , y | a 1 , a 2 , a 3 ). here one hopes that there is a path without colliders between z and some a , that is located “ close ” to t . the same approach can be applied to make more sample efficient step 26 of the cimb * method ( case b ). as it was mentioned in this patent document , compositional markov boundary methods may miss some markov boundary members if the causal sufficiency assumption is violated ( spirtes et al ., 2000 ). the latter assumption implies that every common cause of any two or more variables is observed in the dataset . consider a graphical structure shown in fig2 a and assume that only variables shown in the figure are observed . clearly , data generated from this structure violate the causal sufficiency assumption ( e . g ., common causes of a 1 and a 2 are not observed ). now assume that the probability distribution over all variables ( i . e ., observed and unobserved ) is faithful to the graph and one can make correct inferences about independence relations from a given data sample from the underlying probability distribution . if one applies to the above data hiton - mb ( aliferis et al ., 2009a ; aliferis et al ., 2009b ), a state of the art compositional markov boundary method , the following markov boundary of t will be output by the method : { a 1 , a 2 }. notice however that this output set of variables does not satisfy the definition of the markov boundary ( pearl , 1988 ): variables y , s , and z will not be independent from t given { a 1 , a 2 }. on the other hand , the inventive method will correctly discover and output the markov boundary { a 1 , a 2 , y , s }. table s1 shows a list of bayesian networks used to simulate data . these bayesian networks were used in prior evaluation of markov boundary and causal discovery methods ( aliferis et al ., 2009a ; aliferis et al ., 2009c ; tsamardinos et al ., 2006a ) and were chosen on the basis of being representative of a wide range of problem domains ( emergency medicine , veterinary medicine , weather forecasting , financial modeling , molecular biology , and genomics ). for each of these bayesian networks , data was simulated using a logic sampling method ( russell and norvig , 2003 ). specifically , 5 datasets of 200 , 500 , 1000 , 2000 , and 5000 samples were simulated . notice that all these datasets do not contain hidden variables and thus cannot be used in the original form to demonstrate benefits of the invention . that is why the method stated in table s3 was applied to generate experiment tuples of the form & lt ; t , s , mb s ( t )& gt ;, where each tuple instructs first to run the invention and baseline comparison method on a target variable t after removing from the dataset variables s and then to compare the output variable set with the correct answer mb s ( t ). the following markov boundary methods were applied to those datasets with g 2 test of statistical independence ( agresti , 2002 ): cimb *, iamb ( tsamardinos and aliferis , 2003 ; tsamardinos et al ., 2003b ), blcd - mb ( mani and cooper , 2004 ), fast - iamb ( yaramakala and margaritis , 2005 ), hiton - pc ( aliferis et al ., 2009a ; aliferis et al ., 2009b ), and hiton - mb ( aliferis et al ., 2009a ; aliferis et al ., 2009b ). in addition , iamb ( tsamardinos and aliferis , 2003 ; tsamardinos et al ., 2003b ) with mutual information ( cover et al ., 1991 ) ( this method is denoted as “ iamb - mi ”) was applied . the results for sensitivity , specificity , and error of markov boundary discovery are shown in tables 9 , 10 , 11 , respectively . the results for sensitivity and error of markov boundary discovery are also plotted in fig5 and 6 , respectively . as can be seen , cimb * yields larger sensitivity ( table 9 , fig5 ) and similar specificity ( table 10 ) compared to other methods , which results in smaller error of markov boundary discovery ( table 11 , fig6 ). these results demonstrate the advantages of the invention in terms of accurate detection of the markov boundary . table s2 shows a list of real datasets used in experiments . the datasets were used in prior evaluation of markov boundary methods ( aliferis et al ., 2009a ; aliferis et al ., 2009c ) and were chosen on the basis of being representative of a wide range of problem domains ( biology , medicine , economics , ecology , digit recognition , text categorization , and computational biology ) in which markov boundary induction and feature selection are essential . these datasets are challenging since they have a large number of features with small - to - large sample sizes . several datasets used in prior feature selection and classification challenges were included . all datasets have a single binary response variable . it is also likely to assume that these datasets have hidden variables ( because these are real - life data from domains where only a subset of variables are observed with respect to all known observables in each domain ) and the causal sufficiency assumption is violated with certainty . thus these datasets can be used to demonstrate the benefits of the inventive method . the following markov boundary methods were applied to those datasets with g 2 test of statistical independence ( agresti , 2002 ): cimb *, iamb ( tsamardinos and aliferis , 2003 ; tsamardinos et al ., 2003b ), blcd - mb ( mani and cooper , 2004 ), fast - iamb ( yaramakala and margaritis , 2005 ), hiton - pc ( aliferis et al ., 2009a ; aliferis et al ., 2009b ), and hiton - mb ( aliferis et al ., 2009a ; aliferis et al ., 2009b ). in addition , iamb ( tsamardinos and aliferis , 2003 ; tsamardinos et al ., 2003b ) with mutual information ( cover et al ., 1991 ) ( this method is denoted as “ iamb - mi ”) was applied , and likewise the set of all variables in the dataset ( denoted as “ all ”) was also included in the comparison . once features were selected , svm classifiers were trained and tested on selected features according to the cross - validation protocol stated in table s2 ( vapnik , 1998 ). the results are shown in table 12 ( classification performance , measured by area under roc curve ) and table 13 ( proportion of selected features ). as can be seen from the row “ median ” of table 12 , cimb * yields larger median classification performance than other methods , including using all variables in the dataset . specifically , cimb * achieves the largest classification performance in acpletiology , gisette , sylva , and hiva datasets . in terms of mean classification performance , its results are comparable to the best baseline comparison method ( hiton - mb ) ( table 12 , row “ mean ”). at the same time according to table 13 , the proportion of features selected by cimb * is only a few percent larger than for other markov boundary methods . due to large numbers of data elements in the datasets , which the present invention is designed to analyze , the invention is best practiced by means of a computational device . for example , a general purpose digital computer with suitable software program ( i . e ., hardware instruction set ) is needed to handle the large datasets and to practice the method in realistic time frames . based on the complete disclosure of the method in this patent document , software code to implement the invention may be written by those reasonably skilled in the software programming arts in any one of several standard programming languages . the software program may be stored on a computer readable medium and implemented on a single computer system or across a network of parallel or distributed computers linked to work as one . the inventors have used mathworks matlab ® and a personal computer with an intel xeon cpu 2 . 4 ghz with 4 gb of ram and 160 gb hard disk . in the most basic form , the invention receives on input a dataset and a response variable index corresponding to this dataset , and outputs a markov boundary ( described by indices of variables in this dataset ) which can be either stored in a data file , or stored in computer memory , or displayed on the computer screen . likewise , the invention can transform an input dataset into a minimal reduced dataset that contains only variables that are needed for optimal prediction of the response variable ( i . e ., markov boundary ). agresti , a . 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( 2005 ) speculative markov blanket discovery for optimal feature selection . proceedings of the fifth ieee international conference on data mining , 809 - 812 . in this specification capital letters in italics denote variables ( e . g ., a , b , c ) and bold letters denote variable sets ( e . g ., x , y , z ). the following standard notation of statistical independence relations is adopted : i ( t , a ) means that t is independent of variable set a . similarly , if t is independent of variable set a given ( conditioned on ) variable set b , this denoted as i ( t , a | b ). if i ( )” is used instead of “ i ( ), this means dependence instead of independence . if a graph contains an edge x →& gt ; y , then x is a parent of y and y is a child of x . the edge x y means that x and y are confounded by hidden variable ( s ) ( i . e ., they share at least one unobserved common cause ). the edge x o → y denotes either x → y or x y . finally , the edge x o - o y denotes either x → y , or x y , or x ← y . the set of all variables involved in the causal process is denoted by a = v ∪ h , where v is the set of observed variables ( including the response / target variable t ) and h is the set of unobserved ( hidden ) variables . definition of bayesian network & lt ; v , g , j & gt ;: let v be a set of variables and j be a joint probability distribution over all possible instantiations of v . let g be a directed acyclic graph ( dag ) such that all nodes of g correspond one - to - one to members of v . it is required that for every node a ∈ v , a is probabilistically independent of all non - descendants of a , given the parents of a ( i . e . markov condition holds ). then the triplet & lt ; v , g , j & gt ; is called a bayesian network ( abbreviated as “ bn ”), or equivalently a belief network or probabilistic network ( neapolitan , 1990 ). definition of markov blanket : a markov blanket m of the response / target variable t ∈ v in the joint probability distribution p over variables v is a set of variables conditioned on which all other variables are independent of t , i . e . for every x ∈( v \ m \{ t }), i ( t , x | m ). definition of market boundary : if m is a markov blanket of t in the joint probability distribution p over variables v and no proper subset of m satisfies the definition of markov blanket of t , then m is called a markov boundary of t . the markov boundary of t is denoted as mb ( t ). definition of the set of parents and children : x belongs to the set of parents and children of t ( denoted as pc ( t )) if and only if x is adjacent with t in the underlying causal graph g over variables v . definition of putative parent : x is a putative parent of y if x is a parent of y or x and y are confounded by hidden variable ( s ), i . e . x → y or x y . this can be also denoted as x o → y . definition of putative child : x is a putative child of y if x is a child of y or x and y are confounded by hidden variable ( s ), i . e . x ← y or x y . this can be also denoted as x ← o y . definition of collider path : x is connected to y via a collider path p if the length of p is at least two edges and every variable on the path p is a collider . here are a few examples of collider paths between x and y : definition of bidirectional path : x is connected to y via a bidirectional path p if every edge on the path is here are a few examples of bidirectional paths between x and y :