classification and feature

Signal Processing83(2003)729–743/locate/sigproSimultaneous classiÿcation andrelevant feature id entiÿcation in high-dimensional spaces:application to molecular proÿling dataC.Bhattacharyya a;e,L.R.Grate b,A.Rizki b,D.Radisky b,F.J.Molina b;c,M.I.Jordan a;d,M.J.Bissell b,I.S.Mian b;∗a Division of Computer Science,University of California Berkeley,Berkeley,CA94720,USAb Lawrence Berkeley National Laboratory,Life Sciences Division,Berkeley,CA94720,USAc Department of Mathematics,University of California Santa Cruz,Santa Cruz,CA95064,USAd Department of Statistics,University of California Berkeley,Berkeley,CA94720,USAe Current address:Department of CSA,Indian Institute of Science,Bangalore560012,IndiaReceived26May2002;receivedin revisedform6September2002AbstractMolecular proÿling technologies monitor many thousands of transcripts,proteins,metabolites or other species concurrently in a biological sample of interest.Given such high-dimensional data for di erent types of samples,classiÿcation methods aim to assign specimens to known categories.Relevant feature identiÿcation methods seek to deÿne a subset of molecules that di erentiate the samples.This work describes L IKNON,a speciÿc implementation of a statistical approach for creating a classiÿer and identifying a small number of relevant features simultaneously.Given two-class data,L IKNON estimates a sparse linear classiÿer by exploiting the simple andwell-known property that minimising an L1norm(via linear programming) yields a sparse hyperplane.It performs well when used for retrospective analysis of three cancer biology proÿling data sets, (i)small,round,blue cell tumour transcript proÿles from tumour biopsies and cell lines,(ii)sporadic breast carcinoma transcript proÿles from patients with distant metastases¡5years andthose with no d istant metastases¿5years and(iii) serum sample protein proÿles from una ectedandovarian cancer putationally,L IKNON is less demanding than the prevailingÿlter-wrapper strategy;this approach generates many feature subsets andequates relevant features with the subset yielding a classiÿer with the lowest generalisation error.Biologically,the results suggest a role for the cellular microenvironment in in uencing disease outcome and its importance in developing clinical decision support systems.?2002Elsevier Science B.V.All rights reserved.Keywords:L1norm minimisation;Molecular proÿling data;Feature selection;Classiÿcation;Cancer biology;L IKNON;Minimax probability machine1.IntroductionMolecular proÿling studies of di erent types of bi-ological specimens are both increasingly widespread∗Corresponding author.E-mail address:smian@(I.S.Mian).andimportant.In cancer biology for example,com-monplace investigations include monitoring the abundances of transcripts and/or proteins in nor-mal andaberrant(tumour)tissues,sera or cell lines [19,6,18,39,25,34,15,36,40,26,29,35].The adoption of proÿling technologies is motivatedlargely by a d esire to create clinical decision support systems for accurate0165-1684/03/$-see front matter?2002Elsevier Science B.V.All rights reserved. doi:10.1016/S0165-1684(02)00474-7730 C.Bhattacharyya et al./Signal Processing83(2003)729–743cancer classiÿcation anda needto id entify robust and reliable molecular targets(“biomarkers”)for interven-tion,diagnosis and imaging.Theÿrst attendant analyt-ical task is classiÿcation andpred iction,estimating a classiÿer from proÿling data which assigns accurately samples to known classes.The secondtask,relevant feature identiÿcation,involves deÿning a small subset of the molecules monitoredwhich best d i erentiate classes.The subject of this work is the tasks of classi-ÿcation andrelevant feature id entiÿcation in the context of two-class molecular proÿling data,i.e. samples are assignedto one of two categories such as normal or tumour specimens.Statistical challenges associatedwith solving these problems includ e the large number of features in an example vector(∼103–104molecular abundances)and the small number of high-dimensional example vectors(∼101–102sam-ples).The classiÿer underlying a clinical decision support system wouldbe expectedto make precise diagnoses for many more and diverse patient samples than hadbeen usedfor its estimation.This requirement for systems with goodpred ictive capability neces-sitates classiÿers which minimise misclassiÿcations on future data,namely those with low generalisation error.For two-class data,the classiÿcation and prediction problem is to learn a discriminating surface which separates the classes using a criterion such as gen-eralisation error.Support vector machines(SVMs) [14]are goodclassiÿers which achieve low gener-alisation error by minimising an associatedquantity termedthe margin.SVMs have been employedsuc-cessfully for cancer classiÿcation using transcript proÿles[10,31,36,40].In contrast to SVMs,the newly formulatedminimax probability machine(MPM) minimises directly an upper bound on the generali-sation error[28].As shown here,MPMs provide a viable alternative to SVMs for addressing classiÿ-cation andpred iction problems relatedto proÿling data.MPMs andSVMs cannot d eÿne biomarkers in their own right because each feature in an example vector contributes to delineating the discriminating surface. In transcript proÿling studies,relevant feature iden-tiÿcation has oft been addressed via aÿlter-wrapper strategy[17,31,42].Theÿlter generates candidate gene subsets whilst the wrapper runs an induction algorithm to determine the discriminative ability of a subset.This procedure computes a statistic from the empirical distribution of genes in the two classes and orders genes according to this metric.Forward or backward selection creates subsets by adding or deleting genes successively.Each subset is used to es-timate a classiÿer andto d etermine its generalisation error.A priori,the number of genes andwhich subset will produce a classiÿer with the lowest generalisa-tion error is unknown.Thus,many runs are required to converge upon a subset that constitutes biomark-ers.Although MPMs andSVMs are goodwrap-pers,the choice ofÿltering statistic remains an open question.This study shows the potential of sparse(linear) classiÿers as a framework for addressing simultane-ously the aforementionedproblems of classiÿcation and relevant feature identiÿcation.In so doing,con-siderable prior statistical research is exploited in a new application domain.Here,the focus is sparse hy-perplanes estimatedby minimising an L1norm via linear programming[4,14,16,20,24,38,3].L IKNON,1a speciÿc implementation of this strategy,is usedfor retrospective analysis of data from three exemplars of transcript[26,41]andprotein[35]proÿling stud-ies.L IKNON has non-trivial computational advantages over the prevailingÿlter-wrapper strategy because it creates a classiÿer andid entiÿes relevant features in one pass through two-class data.Reexamination of the transcript proÿles generates biological predictions for subsequent experimental andclinical investigation of two types of cancer andcellular microenvironments. Finally,the results reveal the ability of publishedd ata to answer unanticipatedquestions.2.Materials and methods2.1.Transcript proÿling data:small,round,blue cell tumoursPreviously[26],cDNA microarrays were usedto monitor tumour biopsy andcell line samples from1L IKNON is a wordfor a winnowing basket usedin ancient Greece.C.Bhattacharyya et al./Signal Processing83(2003)729–743731four distinct classes of small,round,blue cell tumours (SRBCTs)of childhood:neuroblastoma(NB),rhab-domyosarcoma(RMS),the Ewing family of tumors (EWS)andnon-Hod gkin lymphoma(NHL)[26].The transcript proÿles consistedof2308nucleic acidse-quences or“genes”monitoredin84samples.These data were used to categorise samples on the basis of their cancer class(EWS,RMS,NHL or NB)andto deÿne96genes which distinguished the four classes. Each class consistedof a mixture of tumour biopsy andcell lines samples,i.e.the origin of a specimen was ignoredd uring categorisation.Here,transcript proÿles for the84SRBCT samples were downloaded from / DIR/Microarray/Supplement/.For each sample,the features in the2308-dimensional example vectors were the log ratios of transcripts in the sample of interest comparedto a common reference[26].To determine whether proÿling data have the potential to answer unanticipatedquestions,the SRBCT sam-ples were partitionedso as to probe the interplay between tissue andcell culture cellular microenviron-ments,andcancer class.The seven,new,two-class data sets formulated by repartitioning the samples were as follows:Partition A,46EWS/RMS tumour biopsies and38EWS/RMS/NHL/NB cell lines;Par-tition B,21EWS/RMS cell lines and30EWS/RMS tumour biopsies;Partition C,28EWS tumour biop-sies/cell lines and23RMS tumour biopsies/cell lines; Partition D,17EWS tumour biopsies and13RMS tumour biopsies;Partition E,11EWS cell lines and 10RMS cell lines;Partition F,17EWS tumour biopsies and11EWS cell lines;andPartition G, 13RMS tumour biopsies and10RMS cell lines (for NHL andNB,only cell lines were available). The seven two-class data sets were analysed using L IKNON anda Fisher scoreÿlter-MPM/SVM wrapper strategy.2.2.Transcript proÿling data:sporadic breast carcinomasPreviously[41],cDNA microarrays were usedto monitor5192genes in97sporadic breast carcinoma samples.These data were used to deÿne70genes which discriminated between patients with distant metastases¡5yr andthose with no d istant metas-tases¿5yr.Here,transcript proÿles for the97sporadic breast carcinoma samples were downloaded from /publications/vantveer.htm.For each sample,the features in the5192-dimensional example vectors were the log ratios of the transcripts in the sample of interest comparedto a common reference[41].The two-class data set,46patients with distant metastases¡5yr and51patients with no distant metastases¿5yr,was analysedusing L IKNON.2.3.Protein proÿling data:ovarian cancerPreviously[35],SELDI-TOF mass spectroscopy was usedto generate spectra for serum samples from una ectedandovarian cancer patients.The protein proÿles consistedof15,154Mass/Charge(M=Z)val-ues measuredin200samples.These d ata were usedto deÿne5“proteins”which di erentiated non-maligant from ovarian cancer samples.Here,protein proÿles for the200serum samples were downloaded from http://clinicalproteomics.steem. com/.For each sample,the features in the15,154-dimensional example vectors were SELDI-TOF mass spectrum amplitudes representing15;154M=Z values in the sample of interest[35].Each M=Z value repre-sents a low molecular weight molecule.The two-class d ata set,100una ectedand100ovarian cancer serum samples,was analysedusing L IKNON.2.4.L IKNON:simultaneous classiÿcation and relevant feature identiÿcationConsider two-class data,{(x1;y1);:::;(x N;y N)}, consisting of N example vectors,x i∈R P.The label,y i∈{+1;−1},indicates whether the ex-ample vector x i is equatedwith class1or with class2.For the two-class proÿling data described above,the number of example vectors,N,andtheir dimensionality,P,are(i)small roundblue cell tu-mours,N=84;51;51;30;21;28and23,and P=2308, (ii)sporadic breast carcinomas,N=97and P=5192 and(iii)ovarian cancer,N=200and P=15;154. Each feature x p in a P-dimensional example vector corresponds to an observed transcript level or M=Z value.If two-class data can be separated by a linear deci-sion boundary,the discriminating surface has the form732 C.Bhattacharyya et al./Signal Processing 83(2003)729–743of a hyperplane,w T x =b ,parameterisedin terms of a weight vector,w ∈R P ,ando set term,b ∈R .A clas-siÿer is the hyperplane which satisÿes the N inequal-ities y i (w T x i −b )¿0∀i ={1;:::;N }.The learning problem is to estimate the optimal weight vector w ∗ando set b ∗.Given this hyperplane,a vector x is assignedto a class basedon the sign of the corre-sponding decision function.If sign(w T∗x −b ∗)=+1,x is id entiÿedwith class 1,otherwise it is assignedto class 2.The problems of classiÿcation andrelevant fea-ture identiÿcation can be solved concurrently by considering a sparse hyperplane,one for which the weight vector w has few non-zero elements.Recall that the class of a vector x is assignedaccord ing to sign(z ).z =w T x −b =P p =1w p x p −b =w p =0w p x p −b:If a weight vector element is zero,w p =0,then featurep in the example vector does not decide the class of x andis thus “irrelevant”.Only a feature for which the element is non-zero,w p =0,contributes to sign(z )andis thus useful for d iscrimination.Thus,the prob-lem of deÿning a small number of relevant features (biomarkers)can be thought of as synonymous with identifying a sparse hyperplane.Learning a sparse hyperplane can be formu-latedas an optimisation problem.Minimising the L 0norm of the weight vector, w 0,minimises the number of non-zero elements.The L 0norm is w 0=number of {p :w p =0}.Unfortunately,min-imising an L 0norm is NP-hard.However,a tractable,convex approximation is to replace the L 0norm with the L 1norm [16].Minimising the L 1norm of the weight vector, w 1,minimises the sum of the abso-lute magnitudes of the elements and sets most of the elements to zero.The L 1norm is w 1= P p =1|w p |.The optimisation problem becomes minw ;bw 1s :t :y i (w T x i −b )¿1∀i ∈{1;:::;N };(1)w 1=P p =1|w p |;|w p |=sign(w p )w p :Problem 1can viewedas a special case of minimis-ing a weighted L 1norm,min w P p =1a p |w p |,in whichthe vector of weighting coe cients a is a unit vec-tor,a p =1;∀p ∈{1;:::;P }.In other words,all genes are presumedto be equally goodrelevant feature can-didates.Prior knowledge about the (un)importance of feature p can be encoded by specifying the value of a p .If the data are not linearly separable,misclassiÿca-tion can be accounted for by adding a non-negative slack variable i to each constraint andintrod ucing a weightedpenalty term to the objective function minw ;b;w 1+CN i =1is :t :y i (w T x i −b )¿1− i ;i ¿0∀i ∈{1;:::;N }:(2)The term Ni =1 i is an upper boundon the number of misclassiÿcations.The parameter C represents a tradeo between misclassiÿcation and sparseness.The higher the value of C ,the less sparse the solution.Here,setting C =1classiÿedcorrectly all the points in the data sets encountered.However,the value of C can be chosen more systematically via cross validation.Problem (2)can be recast as a linear program-ming problem by introducing extra variables u p and v p where w p =u p −v p and |w p |=u p +v p .These variables are the p th elements of u ;v ∈R P .The L 1norm becomes w 1= Pp =1(u p +v p )=u +v and the problem can rewritten in a standard form as follows:min u ;v ;b;(u +v )+CN i =1is :t :y i ((u −v )T x i −b )¿1− i ; i ¿0∀i ∈{1;:::;N }u p ¿0;v p ¿0∀p ∈{1;:::;P }:(3)Problem (3)minimises a linear function sub-ject to linear constraints.This type of linear pro-gramming problem has been well studied inoptimisation theory.There are e cient algorithms for solving problems involving N ∼104constraints and (2∗P +1)∼104variables.The code for L IKNONC.Bhattacharyya et al./Signal Processing 83(2003)729–743733is available at /∼jordan/liknon/.2.5.Fisher score ÿlter-MPM/SVM wrapper:independent classiÿcation and relevant feature identiÿcationGiven linearly separable two-class data,the task is to determine a hyperplane w T z =b which separates example vectors belonging to class 1(x )andclass 2(y ).Both MPMs andSVMs attempt to minimise the generalisation error,i.e.misclassiÿcation on future data.The MPM framework seeks the hyperplane for which the misclassiÿcation probabilities for class 1,P (w T x 6b ),andclass 2,P (w T y ¿b ),are low.The SVM framework seeks the unique discriminating hy-perplane which maximises the margin separating the classes.MPMs andSVMs are comparable in com-plexity (detailed descriptions of these techniques are available in Appendix A ).Preliminary results (data not shown)indicated that the two-class proÿling data examinedhere were ind eedlinearly separable.Hence,the use of L IKNON andSVMs with linear kernels was justiÿed.MPMs and SVMs only address the problem of clas-siÿcation andpred iction.In the ÿlter-wrapper strat-egy,relevant feature identiÿcation is an independent,data preprocessing step.For simplicity and illustrative purposes,SVM/MPM wrappers were employedin conjunction with a Fisher score ÿlter.Given example vectors assignedto class x or class y ,the Fisherscore for feature p is given by F p =( xp − y p )2=( x p + y p ); xp and y p are the means of feature p in the respective classes,whereas x p and y p are standard deviations.Higher values signify more discriminative features.Given P features rankedin descending order according to their score F 1;:::;F P ,the Fisher score top-r rankedfeatures are those ranked1;:::;r .A particular value of r signiÿes a spe-ciÿc feature subset for use in estimating a classiÿer.Forwardselection creates feature subsets by progres-sively increasing the value of r in a user-deÿned manner.Although the recursive feature selection approach utilises a separating hyperplane w [23],it is closer to a ÿlter-wrapper strategy than to L IKNON .Features are ord eredbasedon |w p |,the absolute magnitude of the elements of the weight vector (the range ofvalues for each feature are assumedto be the same).Backwardelimination creates feature subsets by re-cursively removing the bottom 10%of features.The feature subsets are usedas input to a wrapper of choice.putational experiments:L IKNON and Fisher score ÿlter-SVM/MPM wrapperL IKNON creates a classiÿer andid entiÿes relevant features in a single pass through two-class data.The Fisher score ÿlter-MPM/SVM wrapper strategy has distinct feature subset generation and classiÿcation steps.A relevant subset is equatedwith the feature subset of smallest cardinality that yields a classiÿer with the lowest generalisation error.These simultane-ous and independent classiÿcation and relevant feature identiÿcation strategies were compared by means of the leave-one-out error,a surrogate for generalisation error.Given the choice of leave-one-out error as the per-formance metric,L IKNON needs to be run twice for a given two-class data set:ÿrst to identify relevant features (a small subset l of the P input features)andsecondas a classiÿer which uses the resultant l -dimensional vectors as e of the conven-tional error,number of misclassiÿcations on a test set,would require one pass through data.Results (data not shown),indicated that for SRBCT Problem A,all leave-one-out partitionings gave the same set of L IKNON relevant genes as when all N example vectors were used.For each of the seven partitionings of the SRBCT samples,Fisher scores for the P features in the ex-ample vectors were computed.The Fisher score top-r rankedfeatures were usedto generate 13gene subsets where r =1;2;4;8;16;32;64;128;256;512;1024;2048and2308.Thus,L IKNON andthe Fisher score ÿlter deÿned 14feature subsets that di ered only in their dimensionality and precise nature of the genes.The leave-one-out error of SVMs/MPMs trainedusing example vectors derived from every subset was ascertained.For a two-class data set,the leave-one-out error was determined as follows.The N example vectors were divided into an estimation set consisting of N −1ex-ample vectors anda test set composedof the remain-ing example.The MPM/SVM or L IKNON classiÿer was734 C.Bhattacharyya et al./Signal Processing83(2003)729–743usedto pred ict the(known)class of the example in the test set.This estimation andevaluation proced ure was repeated N times so that the class of each example was assignedby a classiÿer estimatedusing all other examples.The leave-one-out error is the number of misclassiÿcations out of N.3.Results3.1.Transcript proÿles:small round blue cell tumoursTwo approaches for performing classiÿcation andrelevant feature id entiÿcation given two-class high-dimensional molecular proÿling data were eval-uated.L IKNON creates a classiÿer andascertains a small number of relevant features simultaneously. The widely usedÿlter-wrapper strategy estimates a classiÿer for every feature subset generatedby an in-dependentÿltering step.The performances of L IKNON anda Fisher scoreÿlter-MPM/SVM wrapper were assessedby means of the leave-one-out error,a com-mon proxy for generalisation error when there are few example vectors.MPMs are a viable alternative to SVMs for solving the classiÿcation andpred iction problem in aÿlter-wrapper strategy.Table1presents the leave-one-out error when the seven problems were analysedusing these classiÿers as the wrapper.Irre-spective of the feature subset,MPMs andSVMs had similar performance andgeneralisedequally well. Whereas SVMs andMPMs couldoperate d irectly in high-dimensional spaces,the original study used the 10dominant Principal Component Analysis compo-nents of the2308-dimensional example vectors as input to artiÿcial neural networks(ANNs)[26].Since MPMs andSVMS solve convex optimisation prob-lems,they avoidthe local minima problems which plague ANNs.L IKNON is computationally less demanding than the ÿlter-MPM/SVM wrapper strategy in identifying rel-evant features.Table2tabulates the relevant features giving zero leave-one-out error for the seven two-class SRBCT data sets.For a given data set,similar num-bers of Fisher score andL IKNON relevant genes are requiredandthese are generally one to two ord ers of magnitude smaller than the2308input features.However,whilst L IKNON requiredone pass through data,theÿlter-wrapper approach required many runs to pinpoint its subset.For Partition A,the23L IKNON relevant features gave zero out of84leave-one-out error,whereas the top-16or top-32Fisher score ranked genes gave low,but not zero out of84leave-one-out error(Table1).L IKNON relevant features shouldbe regard edas a small,though not necessarily unique set of biomark-ers.Fig.1shows a histogram of Fisher scores for all 2308genes overlaidwith the Fisher scores of relevant genes.L IKNON relevant genes are not necessarily asso-ciatedwith high Fisher scores yet they yieldclassiÿers with zero leave-one-out error.Higher Fisher scores correspond to larger di erences in the empirical dis-tributions of transcript levels(more discriminative features)so classiÿers trainedwith top-rankedgenes might be expectedto generalise well.The results reinforce the notion that many distinct relevant feature subsets canÿt the data equally well(see for example[12]).From a numerical perspective,the84SRBCT tran-script proÿles are su ciently informative that biolog-ical questions not considered in the original study can be posedandanswered(see also[32]).3.2.Cellular microenvironment and SRBCT classiÿcationA biological assessment of the L IKNON relevant features reveals that the tissue or cell culture origin of a sample a ects the nature andnumber of rele-vant genes.Table3lists these genes for the seven two-class data sets.Four of these compared tumour biopsies with tumour-derived cell lines in the context of di erent numbers of SRBCT classes,four(Par-tition A:EWS,RMS,NHL,NB),two(Partition B: EWS,RMS)andone(Partition F:EWS;G:RMS). There were23relevant genes for Partition A,21for B,12for F and13for G.Tissue andcell culture microenvironments are manifestations of variations in cell shape andcell-extracellular matrix interac-tions.This di erence is re ected in relevant genes such as actin 2,SMA3(smooth muscle actin3), andcollagen type III .The relevant genes represent goodtargets for stud ying how tumour cells escape quiescence andevad e cell cycle arrest in vivo and in vitro.C.Bhattacharyya et al./Signal Processing83(2003)729–743735 Table1Prediction of SRBCT transcript proÿles using MPMs and SVMs.Rank A(N=84)B(N=51)C(N=51)D(N=30)E(N=21)F(N=28)G(N=23) SVM MPM SVM MPM SVM MPM SVM MPM SVM MPM SVM MPM SVM MPM 110107835221117222 299133321002111 465234221000001 894244120021000 1684300022062205 325221114212011401 64315320600001003 12812230331001201 25601000131011000 51200000132021100 102400000031012200 204821000121102200 230831************ The seven two-class data sets and numbers of example vectors in each class(total N)are Partition A,46EWS/RMS tumour biopsies and38EWS/RMS/NHL/NB cell lines;Partition B,21EWS/RMS cell lines and30EWS/RMS tumour biopsies;Partition C,28EWS tumour biopsies/cell lines and23RMS tumour biopsies/cell lines;Partition D,17EWS tumour biopsies and13RMS tumour biopsies; Partition E,11EWS cell lines and10RMS cell lines;Partition F,17EWS tumour biopsies and11EWS cell lines;andPartition G,13 RMS tumour biopsies and10RMS cell lines.For each partition,the table gives the leave-one-out error out of N for an SVM or MPM estimated using the feature subset indicated.Theÿrst12feature subsets are the Fisher score top-r rankedgenes where r takes on the value given.Theÿnal“subset”corresponds to all features in the original2308-dimensional example vectors[26].Table2Identiÿcation of relevant genes in SRBCT transcript proÿling data using L IKNON anda Fisher scoreÿlter-MPM/SVM wrapper strategy Name Class1samples Class2samples N ClassiÿerSVM MPM L IKNONA EWS/RMS tumour EWS/RMS/NHL/NB cell line8425651223B EWS/RMS cell line EWS/RMS tumour512561621C EWS tumour/cell line RMS tumour/cell line511688D EWS tumour RMS tumour3064648E EWS cell line RMS cell line21222F EWS tumour EWS cell line284412G RMS tumour RMS cell line234813For each of the seven two-class data sets,the total number of example vectors N is listed.“SVM”and“MPM”give the Fisher score feature subset of smallest cardinality that yielded a classiÿer with zero out of N leave-one-out error(taken from Table1).“L IKNON”gives the cardinality of the relevant features identiÿed;each feature subset yielded a L IKNON classiÿer with zero out of N leave-one-out error.Transcriptional di erences between tumour biop-sies andcell lines confoundattempts to d eÿne biomarkers for classifying SRBCTs.Three Partitions comparedEWS andRMS in the context of tumour biopsies andcell lines(Partition C),tumour biopsies (Partition D)andcell lines(Partition E).The relevant genes for EWS tumour biopsies andRMS tumour biopsies(Partition D)may constitute clinically useful biomarkers forÿne-grainedcancer class d iagnosis and/or imaging.Of the96EWS,RMS,NHL and NB cancer class markers identiÿed originally[26],11 are markers for the cellular microenvironment of the sample.The results reiterate the view that information pro-vided by interactions with neighbouring cells,the composition andorganisation of the surround ing736C.Bhattacharyya et al./Signal Processing 83(2003)729–743200400600800100012001400160018000.511.522.5200400600800100012001400160005001000150020002500020040060080020040060080000.51 1.52 2.53 3.5200400600800Fig.1.Histograms of Fisher scores for all 2308genes in the seven SRBCT binary problems.The abscissa represents the Fisher score and the ordinate the number of genes with that score.As might be expected,most genes have low scores and far fewer genes have high scores.Open triangles mark the Fisher scores of the L IKNON relevant genes.For each problem,the number of triangles is the same as the entry in the “L IKNON ”column of Table 2.。