Patent Application: US-201414215782-A

Abstract:
methods for discovery of a markov boundary from data constitute one of the most important recent developments in pattern recognition and applied data analysis and modeling , primarily because they offer a principled solution to the variable / feature selection problem and give insight about local causal structure . even though there is always a single markov boundary of the response variable in faithful distributions , distributions with violations of the intersection property of probability theory may have multiple markov boundaries . such distributions are abundant in practical data - analytic applications , and there are several reasons why it is important to discover and extract all markov boundaries from such data as a critical step of data analysis . the present invention is a novel fast generative method that can discover all markov boundaries from a sample drawn from a distribution . the new method has been tested with simulated data and then applied to discover markov boundaries in datasets from several application domains including but not limited to : biology , medicine , economics , ecology , image recognition , text processing , and computational biology .

Description:
the inventive methods itie * and generalized - itie * are shown in fig1 and 2 , respectively . itie * is a configuration of the method generalized - itie * which has the following inclusion sub - process in step 2 : a ) sort in descending order the variables in the priority queue according to their pairwise association with the response variable t ; b ) remove from the priority queue variables with zero association with the response variable t ; c ) insert in the seed markov boundary m the highest - priority variable a in the priority queue and remove it from the priority queue ; a ) if the priority queue is empty , then remove every member a of the seed markov boundary m that is probabilistically independent of the response variable t given a subset b of the remaining variables in m ; b ) else if the priority queue is not empty , then if the last variable a that entered the seed markov boundary m is probabilistically independent from the response variable t given a subset b of the rest of the variables in m , then remove a from m ; c ) if a variable a that has been removed from m in the above step b ) because a is probabilistically independent of the response variable t given an subset b of m , record in the equivalency catalogue θ that a and b contain equivalent information ( with respect to t ) if b is probabilistically independent of the response variable t given a ; its interleaving sub - process consists of repeating inclusion and elimination / analysis strategies until the priority queue is empty . consider running the itie * method on data d generated from the example causal bayesian network shown in fig3 . the response variable t is directly caused by c , d , f . the underlying distribution is such that variables a and c contain equivalent information about t ; likewise variables b and d contain equivalent information about t . itie *, when applied to data d would output correctly 4 markov boundaries of t : ( a , b , f ), ( a , d , f ), ( c , b , f ), and ( c , d , f ). in what follows , we present an empirical evaluation of methods for extraction of multiple markov boundaries and variable sets . the evaluated methods and their parameterizations are shown in fig4 . these methods were chosen for our evaluation as they are the current state - of - the - art techniques for discovery of multiple markov boundaries and variable sets . all experiments involving assessment of classification performance were executed by holdout validation or cross - validation ( see below ), whereby markov boundaries and variable sets are discovered in a training subset of data samples ( training set ), classification models based on the above variables are also developed in the training set , and the reported performance of classification models is estimated in an independent testing set . assessment of classification performance of the extracted markov boundaries and variable sets was done in the presented validation using support vector machines ( svms ) [ 28 ]. we chose to use svms due to their excellent empirical performance across a wide range of application domains ( especially with high - dimensional data and relatively small sample sizes ), regularization capabilities , ability to learn both simple and complex classification functions , and tractable computational time [ 28 - 31 ]. when the response variable was multiclass , we applied svms in one - versus - rest fashion [ 30 ]. we used libsvm v . 2 . 9 . 1 ( http :// www . csie . ntu . edu . tw /˜ cjlin / libsvm /) implementation of svms in all experiments [ 32 ]. polynomial kernels were used in svms as they have shown good classification performance across the data domains considered in this study . the degree d of the polynomial kernel and the penalty parameter c of svm were optimized by cross - validation on the training data . each variable in a dataset was scaled to [ 0 , 1 ] range to facilitate svm training . the scaling constants were computed on the training set of samples and then applied to the entire dataset . below we present an evaluation of methods for extraction of multiple markov boundaries and variable sets in simulated data . simulated data allows us to evaluate methods in a controlled setting where the underlying causal process and all markov boundaries of the response variable t are known exactly . two datasets were used in this evaluation . one of these datasets , referred to as tied , was previously used in an international causality challenge [ 33 ]. tied contains 30 variables , including the response variable t . the underlying causal graph and its parameterization are given in [ 33 ] there are 72 distinct markov boundaries of t . each markov boundary contains 5 variables : variable x 10 and one variable from each of the four subsets ( x 1 , x 2 , x 3 , x 11 ), ( x 5 , x 9 ), ( x 12 , x 13 , x 14 ) and ( x 19 , x 20 , x 21 ). another simulated dataset , referred to as tied1000 , contains 1 , 000 variables in total and was generated by the causal process of tied augmented with an additional 970 variables that have no association with t . tied1000 has the same set of markov boundaries of t as tied . tied1000 allows us to study the behavior of different methods for learning multiple markov boundaries and variable sets in an environment where the fraction of variables carrying relevant information about t is small . for each of the two datasets , 750 observations were used for discovery of markov boundaries / variable sets and training of the svm classification models of the response variable t ( with the goal to predict its values from the inferred markov boundary variables ), and an independent testing set of 3 , 000 observations was used for evaluation of the models &# 39 ; classification performance . all methods for extracting multiple markov boundaries and variable sets were assessed based on the following six performance criteria : i . the number of distinct markov boundaries / variable sets output by the method . ii . the average size of an output markov boundary / variable set ( number of variables ). iii . the number of true markov boundaries identified exactly , i . e ., without false positives and false negatives . iv . the average proportion of false positives ( pfp ) in the output markov boundaries / variable sets . v . the average false negative rate ( fnr ) in the output markov boundaries / variable sets . vi . the average svm classification performance ( weighted accuracy ) over all output markov boundaries / variable sets . we also compared the average classification performance of the svm models with the maximum a posteriori classifier in the true bayesian network ( denoted as map - bn ) using the same data sample . as can be seen in fig5 - 7 , itie * identified exactly all and only true markov boundaries of t in both simulated datasets , and their classification performance with the svm classifier was statistically comparable to performance of the map - bn classifier . none of the comparator methods , regardless of the number of markov boundaries / variable sets output , were able to identify exactly any of the 72 true markov boundaries , except for resampling + rfe ( without statistical comparison ) and ir - hiton - pc that identified exactly 1 - 2 out of 72 true markov boundaries , depending on the dataset . overall prior methods had either large proportion of false positives or large false negative rate , and often their classification performance was significantly worse that the performance of the map - bn classifier . for evaluation of methods for learning multiple markov boundaries and variable sets in real data , we used 13 datasets that cover a broad range of application domains ( clinical outcome prediction , gene expression , proteomics , drug discovery , text categorization , digit recognition , ecology and finance ), dimensionalities ( from 86 to over 100 , 000 ), and sample sizes ( from hundreds to thousands ) that are representative of those appearing in practical applications . these datasets have recently been used in a broad benchmark [ 8 ] of the current state - of - the - art single markov boundary induction and feature selection methods , which is another reason why we chose to use the same data in this study . the datasets are described in detail in fig8 . the datasets were preprocessed ( imputed , discretized , etc .) as described in [ 8 ]. in datasets with relatively large sample sizes (& gt ; 600 ), classification performance of the output markov boundaries and variable sets was estimated by holdout validation with 75 % of samples used for markov boundary / variable set induction and svm classifier training , and the remaining 25 % of samples used for estimation of classification performance . in small - sample datasets , 10 - fold cross - validation was used instead . markov boundary / variable set induction and classifier training were both performed on the training sets from the 10 - fold cross - validation design , with classification performance being subsequently estimated on the respective testing sets . evaluation of markov boundary / variable selection methods in real data is challenging due to the lack of knowledge of the true markov boundaries . in practical applications , however , the interest typically lies in the most compact subsets of variables that give the highest classification performance for reasonable and widely used classifiers [ 6 ]. this consideration motivated the following two primary evaluation criteria ( with the averages taken over all markov boundaries / variable sets output by each method ): i . the average proportion of variables ( pv ) in the output markov boundaries / variable sets . ii . the average classification performance ( auc ) of the output markov boundaries / variable sets . in addition to the above two primary criteria , in some problems we are also interested in extracting as many of the maximally compact and predictive variable sets ( i . e ., optimal solutions to the variable selection problem ) as possible . detailed results of itie * are shown in fig9 and comparison with other methods is given in fig1 . as can be seen , itie * extracted multiple compact markov boundaries with high classification performance and surpassed all other methods on the combined ( pv , auc ) criterion . egs - cmim — ensemble gene selection with conditional mutual information maximization criterion ( method for selecting multiple variable sets ) egs - ncmigs — ensemble gene selection with normalized conditional mutual information gene selection ( method for selecting multiple variable sets ) egsg — ensemble gene selection by grouping ( method for selecting multiple variable sets ) generalized - itie *— generalized individual target information equivalency ( method for discovery of multiple markov boundaries ) ir - hiton - pc — iterative removal with hiton - pc ( method for discovery of multiple markov boundaries ) ir - splr — iterative removal with sparse logistic regression ( method for selecting multiple variable sets ) itie *— individual target information equivalency ( method for discovery of multiple markov boundaries ) map - bn — maximum a posteriori classification method in the true / data generating bayesian network pv — proportion of variables relative to the number of variables in the original dataset before variable selection resampling + rfe — resampling followed by application of support vector machines - based recursive feature elimination ( method for selecting multiple variable sets ) resampling + uaf — resampling followed by application of univariate attribute filtering ( method for selecting multiple variable sets ) the relationships , correlations , and significance ( thereof ) discovered by application of the method of this invention may be output as graphic displays ( multidimensional as required ), probability plots , linkage / pathway maps , data tables , and other methods as are well known to those skilled in the art . for instance , the structured data stream of the method &# 39 ; s output can be routed to a number of presentation , data / format conversion , data storage , and analysis devices including but not limited to the following : ( a ) electronic graphical displays such as crt , led , plasma , and lcd screens capable of displaying text and images ; ( b ) printed graphs , maps , plots , and reports produced by printer devices and printer control software ; ( c ) electronic data files stored and manipulated in a general purpose digital computer or other device with data storage and / or processing capabilities ; ( d ) digital or analog network connections capable of transmitting data ; ( e ) electronic databases and file systems . the data output is transmitted or stored after data conversion and formatting steps appropriate for the receiving device have been executed . 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 general purpose digital computer with suitable software programming ( i . e ., hardware instruction set ) ( fig1 describes the architecture of modern digital computer systems ). such computer systems are 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 including , but not limited to , c , java , and python . in addition , where applicable , appropriate commercially available software programs or routines may be incorporated . 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 . to implement parts of the software code , the inventors have used mathworks matlab ® and a personal computer with an intel xeon cpu 2 . 4 ghz with 24 gb of ram and 2 tb hard disk . 1 . davenport t h , harris j g : competing on analytics : the new science of winning : harvard business press ; 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