Patent Application: US-201414215820-A

Abstract:
discovery of causal networks is essential for understanding and manipulating complex systems in numerous data analysis application domains . several methods have been proposed in the last two decades for solving this problem . the inventive method uses local causal discovery methods for global causal network learning in a divide - and - conquer fashion . the usefulness of the invention is demonstrated in data capturing characteristics of several domains . the inventive method outputs more accurate networks compared to other discovery approaches .

Description:
the present invention describes use of local causal discovery methods for global causal network learning in a divide - and - conquer fashion . we remind that a major motivation for pursuing local causal learning methods is scaling up causal discovery and causal feature selection as explained in [ 13 ]. the key points of the following description are that ( a ) the local - to - global framework can be configured in several ways with excellent empirical results ; ( b ) an important previously unnoticed factor is the variable order in which to execute local learning , and ( c ) trying to use non - causal feature selection in order to facilitate global learning ( instead of causal local learning ) is not as a promising strategy as previously thought . given powerful methods for finding local neighborhoods , for example those provided by the gll framework , one can circumvent the need for uniform connectivity ( as well as user knowledge of that connectivity ) and avoid the application of inefficient heuristics employed in sca [ 6 ] thus improving on quality and speed of execution . fig1 provides the general method termed lgl ( for local - to - global learning ). steps # 1 - 3 can be configured in numerous ways . an “ admissible configuration ” of the general lgl method and system is one that guarantees correct output for the configured method . for example , by configuring lgl to use gll - pc with symmetry correction as the underlying sub - process for identifying local causes and effects for each variable ( gll - pc defined in [ 13 ]) then the following constitute admissible configuration criteria : i . configure gll - pc with admissible configurations of steps # 1 , and step # 2 [ 13 ]; ii . employ a sound orientation scheme in step # 3 , of lgl . the above 2 configuration criteria render the total configuration sound under the assumptions of correctness of gll - pc [ 13 ] and comprise the admissibility requirements ( criteria ) for the admissible configuration of the lgl method . stated more generally : statement of operating characteristics . under the following sufficient conditions lgl outputs correctly oriented causal graphs : a . configure gll - pc in step # 1 of lgl using any admissible configuration of gll - pc [ 13 ]. b . connect local neighborhoods discovered by gll - pc in step # 2 of lgl so that an edge between a and b is established if and only if a is a member of the output of gll - pc run on b and a is a member of the output of gll - pc run on b , c . orient the skeleton produced at step # 2 of lgl using any sound orientation method ( e . g ., pc [ 10 , 13 ]). d . employ lgl in data that have the property that there is a causal bayesian network faithful to the data distribution p [ 13 ]; e . employ statistically correct procedures for the determination of variable dependences and independence from the sample data d [ 13 ]; f . employ lgl in data that have the property that causal sufficiency in v [ 13 ]. to demonstrate the generality and robustness of the lgl general method we describe here as a configuration of lgl , a global learning method termed hhc ( fig2 ), and compare it empirically to the state - of - the - art mmhc method [ 7 ]. we also show that the two methods are not identical in edge quality or computational efficiency , with the new method being at least as good on average as mmhc . fig6 presents results for missing / extra edges in undirected skeleton , number of statistical tests for construction of skeleton , structural hamming distance ( shd ), bayesian score , and execution time on 9 of the largest datasets used for the evaluation of mmhc [ 7 ]. hhc used gll - pc without symmetry correction . since the datasets were simulated from known networks , the method output can be compared to the true structure . as can be seen , in all 9 datasets , hhc performs equally well with mmhc in terms of shd and bayesian score . in 8 out of 9 datasets it performs from 10 % to 50 % fewer tests , and in one dataset ( link ) it performs & gt ; 10 times the tests performed by mmhc resulting in running 35 % slower in terms of execution time . because mmhc was found to be superior to a number of other methods for the datasets tested , hhc &# 39 ; s better performance over mmhc in 8 out of 9 datasets ( in terms of number of statistical tests for skeleton construction ) and similar performance in 9 out of 9 datasets ( in terms of quality metrics ) translates also to excellent performance of hhc relative to greedy search , ges , or , pc , tpda , and sca [ 7 ]. an important parameter of local - to - global learning previously unnoticed in methods such as sca and mmhc is the ordering of variables when executing the local causal discovery variable - by - variable ( i . e ., not in parallel ). we will assume that results are shared among local learning runs of gll - pc , i . e . when we start learning pc ( x ) by gll - pc rather than starting with an empty tpc ( x ) set , we start with all variables y : x ∈ pc ( y ). this constitutes a sound configuration of the gll - pc method template as explained in [ 13 ]. fig1 gives two extreme examples where the right order can “ make - or - break ” an lgl method . in fig1 ( a ) it follows that an order of local learning & lt ; x 1 , x 2 , . . . , x 100 , y & gt ; without symmetry correction requires a quadratic number of conditional independence tests ( cits ) for the unoriented graph to be correctly learned . however , the order of local learning & lt ; y , x 1 , x 2 , . . . , x 100 & gt ; requires up to an exponential number of cits as the max - k parameter of gll - pc and sample are allowed to grow without bounds . even with modest max - k values , the number of cits is higher - order polynomial and thus intractable . even when y is not in the beginning but as long as a non - trivial number of x &# 39 ; s are after it in the ordering , the method will be intractable or at least very slow . the latter setting occurs in the majority of runs of the method with random orderings . in fig7 we provide data from a simulation experiment showing the above in concrete terms and exploring the effects of limited sample and connectivity at the same time . as can be seen , under fixed sample , running hhc with order from larger to smaller connectivity , as long as the sample is enough for the number of parents to be learned ( i . e ., number of parents is ≦ 20 ), increases run time by more than 100 - fold . however because sample is fixed , as the number of parents grows the number of conditional independence tests equalizes between the two strategies because cits that have too large conditioning sets for the fixed sample size are not executed . although the number of cits is self - limiting under these conditions , quality ( in terms of number of missing edges , i . e ., number of undiscovered parents of t ) drops very fast as number of parents increases . the random ordering strategy trades off quality for execution time with the wrong ( larger - to - smaller connectivity ) ordering , however in all instances the right ordering offers better quality and 2 to 100 - fold faster execution that random ordering . a more dramatic difference exists for the structure in fig1 ( b ) where y is parent of all x &# 39 ; s . here the number of tests required to find the parent ( y ) of each x i is quadratic to the number of variables with the right ordering ( low - to - high connectivity ) whereas an exponential number is needed with the wrong ordering ( large - to - small connectivity ). because the sample requirements are constant to the number of children of y , quality is affected very little and there is no self - restricting effect of the number of cits , opposite to what holds for causal structure in fig1 ( a ). hence the number of cits grows exponentially larger for the large - to - small connectivity ordering versus the opposite ordering and a similar trend is also present for the average random ordering in full concordance with our theoretical expectations . see fig8 for results of related simulation experiments . these results show that in some cases , it is possible to transform an intractable local learning problem into a tractable one by employing a global learning strategy ( i . e ., by exploiting asymmetries in connectivity ). thus the variable order in local - to - global learning may have promise for substantial speedup and improved quality in real - life datasets ( assuming the order of connectivity is known or can be estimated ). however the optimal order is a priori unknown for many domains . can we use local variable connectivity as a proxy to optimal order in real data ? the next experiment assumes the existence of an oracle that gives the true local connectivity for each variable . the experiment examines empirically the effect of three orders ( low - to - high connectivity , lexicographical ( random ) order , and high - to - low connectivity order ) on the quality of learning and number of cits in the mmhc evaluation datasets . it also compares the sensitivity of hhc and mmhc to order . as can be seen in fig1 , the order does have an effect on computational efficiency however not nearly as dramatic in the majority of these more realistic datasets compared to the simpler structures of fig1 . an exception is the link dataset in which low - to - high connectivity allows hhc to run 17 times faster than lexicographical ( random ) order and 27 times faster than high - to - low connectivity order . for the majority of cases , running these methods with lexicographical ( i . e . random ) order is very robust and does not affect quality adversely but affects run time and number of cits to a small degree . thus , while connectivity affects which variable order is optimal in lgl methods , ranking by local connectivity does not exactly correspond to the optimal order . fig1 shows the number of cits plotted against true local connectivity in each one of the 9 datasets used in this section . related to the above , fig1 also shows the distribution of true local connectivity in each dataset . consistent trends indicating the shape of the distributions by which the degree of local connectivity may determine an advantage of orderings low - to - high to high - to - low connectivity are not apparent in these datasets . we hypothesized that more robust criteria for the effect of variable ordering in lgl learning can be devised . for example , the number or total cost of cits required to locally learn the neighborhood of each variable . such criteria are also more likely to be available or to be approximated well during practical execution of a method than true connectivity . a variant of lgl , method lgl - oo ( standing for “ lgl with optimal order ”) orders variables dynamically according to heuristic approximations of the total number of cits for each variable ( fig4 ). fig5 shows the general method hhc - oo . we also anticipate that in many cases the strategy for piecing together the local learning results strongly interacts with the local variable ordering to determine the tradeoff between the quality and efficiency of the lgl methods . in recent years several researchers have proposed that because modern feature selection methods can deal with large dimensionality / small sample datasets , they could also be used to speed up or approximate large scale causal discovery ( e . g ., kohane et al . use univariate feature selection to build so - called “ relevance networks ” [ 14 ], or hybrid methods can be employed that use feature selection as a pre - processing to build a skeleton and then an orientation method like greedy search in the spirit of mmhc and lgl [ 15 ]. the results of [ 13 ] contradict this postulate because they show that non - causal feature selection does not give locally correct results . however it is still conceivable that orientation - and - repair post - processing methods ( e . g ., with bayesian search - and - score ) can still provide a high quality final causal graph . we test this hypothesis by examining several such hybrid methods using respectively rfe , lars - en and uaf post - processed by greedy tabu bayesian search - and - score . we use simulated datasets from 5 out of 9 bayesian networks employed earlier in the present section . this is because the other 4 networks cannot be used for reliable training and testing of the underlying classifier since they have several variables with very unbalanced distributions . as shown in fig9 , the hypothesis is not corroborated by the experimental results . in particular , greedy search with feature selection - based skeleton , exhibits substantial drops in quality of the returned networks ( measured by structural hamming distance [ 7 ]) and typically more than one order of magnitude longer running times compared to hhc with lexicographical ( random ) variable ordering . on the basis of these findings , which are consistent with the results in [ 13 ], we find strong evidence in favor of using principled local causal methods instead , within the frameworks of lgl . in the remainder we describe further an evaluation of hhc - bach ( fig3 , implemented using gll - pc without symmetry correction ), a configuration of lgl , in gene expression microarray data for inference of gene regulatory / transcriptional networks . we additionally compare it with the best state - of - the - art comparator methods for that domain summarized in fig1 . real gold - standards and microarray datasets were obtained for both yeast and e . coli ( fig1 ). the yeast gold - standard was built by identifying the promoter sequences that are both bound by transcription factors ( tfs ) according to chip - on - chip data and conserved within the saccharomyces genus [ 16 , 17 ]. binding information is essential because tfs must first bind to a gene to induce or suppress expression , while conservation information is important because true - positive tf - dna interactions are often conserved within a genus . this study used a set of yeast gold - standard networks that ranged from conservative to liberal . to obtain this range , networks were chosen with different chip - on - chip binding significance levels α = 0 . 001 or 0 . 005 , and were required to have a binding sequence that is conserved in c = 0 , 1 or 2 of the related saccharomyces species ( fig6 ). hence , the most conservative gold - standard , yeast ( c ), was built from tf - dna interactions with α = 0 . 001 , such that bound dna sequence was conserved in at least 2 yeast relatives . a compendium of 530 yeast microarray samples was taken from the many microbe microarray database [ 18 ]. the e . coli gold - standard network was obtained from regulondb ( version 6 . 4 ), a manually curated database of regulatory interactions obtained mainly through a literature search [ 19 ]. chip - qpcr data has shown regulondb to be approximately 85 % complete [ 20 ]. evidence for each regulatory interaction in regulondb is classified as “ strong ” or “ weak ”, depending on the type of experiment used to predict the interaction . for example , binding of a tf to a promoter is considered strong evidence , whereas gene - expression based computational predictions are considered weak evidence . for the purposes of our study , we created a conservative gold - standard of only strong interactions , and a liberal gold - standard that includes both strong and weak interactions . to ensure that our results are directly comparable with the dream2 challenge , we also included an earlier version of the regulondb gold - standard . a compendium of 907 e . coli microarray samples was taken from the many microbe microarray database [ 18 ]. we also used gene expression data from the dream2 challenge that was a subset of the previous dataset . in addition to using real gene expression data with approximate gold - standards , we also used simulated data where gold - standards are known exactly ( fig1 ). we focused here exclusively on cutting - edge simulation methods that produce artificial data that resembles real biological data . the resimulated gene expression dataset ( reged ) is based on a high - fidelity resimulation technique for generating synthetic data that is statistically indistinguishable from real expression data [ 21 ]. the reged network was induced from 1 , 000 randomly selected genes in a lung cancer gene expression dataset [ 22 ]. this network displays a power - law connectivity [ 23 ] and generates data that is statistically indistinguishable from real data according to an svm classifier [ 24 ]. moreover , statistical dependencies and independencies are consistent between the real and synthetic data according to the fisher &# 39 ; s z test . note that the reged dataset was used in the causality and prediction challenge [ 21 ]. the genenetweaver ( gnw ) simulation method attempts to mimic real biological data by using topology of known regulatory networks [ 25 , 26 ]. stochastic dynamics that are meant to model transcriptional regulation were applied to the extracted networks to generate simulated data . we assessed all global network reverse - engineering methods by first computing positive predictive value ( ppv ) and negative predictive value ( npv ). ppv measures the probability that a regulatory interaction discovered by the method exists in the gold - standard ( i . e ., the precision of the output graph ), while npv measures the probability that an interaction not predicted by the method does not exist in the gold - standard . then , ppv and npv were combined in a single metric by computing the euclidean distance from the optimal method with ppv = 1 and npv = 1 :√{ square root over (( 1 − ppv ) 2 +( 1 − npv ) 2 )}{ square root over (( 1 − ppv ) 2 +( 1 − npv ) 2 )}. implementations of hhc - bach , regardless of parameters , constituted all of the top - five performing techniques ( fig1 ; hhc - bach is denoted as lgl - bach ). this method output few regulatory interactions relative to the size of the gold - standard . however , a larger percentage of these were true - positive interactions than for any other method . moreover , for most datasets & gt ; 98 %- 99 % of the regulatory interactions not predicted by hhc - bach did not exist in the gold - standard . such a relatively accurate list of putative interactions can be fruitful for biologists because it limits the case of experimentally validating the false - positive interactions of a reverse - engineering method . of note , graphical lasso performed the best on the gnw ( a ), gnw ( b ), and ecoli ( d ) datasets . however , it performed poorly on all other datasets , and therefore ranks only seventh among all methods . chip — chromatin immuneprecipitation ( biological methodology to investigate interaction between proteins and dna ) hhc - bach — hiton hill - climbing with bach scoring ( method for discovery of global causal networks ) hhc - oo — hiton hill - climbing with optimal order ( method for discovery of global causal networks ) lars - en — least angle regression with elastic net ( method for regression and feature selection ) lgl - bach — local - to - global with bach scoring ( method for discovery of global causal networks ) lgl - oo — local - to - global with optimal order ( method for discovery of global causal networks ) pc ( x )— parents and children set of a variable x ( set of variables / nodes in the graph ) sa - clr — synergy augmented context likelihood of relatedness ( method for discovery of global causal networks ) tpc ( x )— tentative parents and children set of a variable x ( set of variables / nodes in the graph ) tpda — three phase dependency analysis ( method for discovery of global causal networks ) 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 ; 2013 . 2 . douglas l : the importance of ‘ big data ’: a definition . gartner ( june 2012 ) 2012 . 4 . provost f , fawcett t : data science for business : what you need to know about data mining and data - 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