Cluster Ensemble and Its Applications in Gene Expression Analysis

Transcription

1 Cluster Ensemble and Its Applcatons n Gene Expresson Analyss Xaohua Hu, Illho Yoo College of Informaton Scence and Technology Drexel Unversty, Phladelpha, PA 904 thu@cs.drexel.edu Abstract Huge amount of gene expresson data have been generated as a result of the human genomc proect. Clusterng has been used extensvely n mnng these gene expresson data to fnd mportant genetc and bologcal nformaton. Obtanng hgh qualty clusterng results s very challengng because of the nconsstency of the results of dfferent clusterng algorthms and nose n the gene expresson data. Many clusterng algorthms are avalable and dfferent clusterng algorthms may generate dfferent clusterng results due to ther bas and assumptons. It s a challengng and dauntng task for the genomc researchers to choose the best clusterng algorthm and generate the best clusterng results for ther data sets. In ths paper, we present a cluster ensemble framework for gene expresson analyss to generate hgh qualty and robust clusterng results. In our framework, the clusterng results of ndvdual clusterng algorthm are converted nto a dstance matrx, these dstance matrces are combned and a weghted graph s constructed accordng to the combned matrx. Then a graph parttonng approach s used to cluster the graph to generate the fnal clusters. The experment results ndcate that cluster ensemble approach yelds better clusterng results than the sngle best clusterng algorthm on both synthetc data set and yeast gene expresson data set.. Keywords: cluster ensemble, gene expresson analyss, graph partton Introducton Clusterng s to group analogous elements n a data set n accordance wth ts smlarty. Therefore good clusterng means elements n each cluster are smlar whle elements from dfferent clusters are dssmlar. Unlke classfcaton, clusterng does not requre the class label nformaton about data set because t s nherently a data-drven approach; that s why clusterng plays a very mportant role as the ntal step n data exploratory analyss. In gene expresson analyss, normally not much pror knowledge s accumulated, so genomc researcher tend to apply clusterng algorthms on the gene expresson data sets to gan better understandng and nsghtful Copyrght 004, Australan Computer Socety, Inc. Ths paper appeared at the nd Asa-Pacfc Bonformatcs Conference (APBC004), Dunedn, New Zealand. Conferences n Research and Practce n Informaton Technology, Vol. 9. Y-Png Phoebe Chen. Ed. Reproducton for academc, not-for proft purposes permtted provded ths text s ncluded. genetc and bologcal nformaton. Clusterng s one of most wdely and frequently used data mnng technologes n gene expresson analyss (Azuae 00, Bellaacha, Portnoy, Chen and Elkahloun 00, Ben-Dor and Yakhn 999, Berrar, Dubtzky and Granzow 00, Bloch and Arce 00, Xng and Karp 00, Zeng, Tang, Garca-Fras and Gao 00, Zhao and Karyps 003). Through the use of clusterng algorthms on gene expresson data can answer some challengng bologcal and genetc questons, such as dentfyng the functonalty of genes, fndng out what genes are co-regulated, dstngushng the mportant genes between abnormal tssue and normal tssues etc (Zhao and Karyps 003). There are multple clusterng technques that can be used to analyse gene expresson data. Advantages and lmtatons may depend on factors such as data dstrbuton, pre-processng procedures, number of genes etc. Choosng the best algorthm for a partcular problem may represent a challengng task. Moreover, t s not uncommon to observe nconsstent results when dfferent clusterng methods are tested on a partcular data set. K-Means, Self-Organzng Map (SOM), Herarchcal approaches, Fuzzy C-Means, etc, are very dfferent n some cases (Jan, Murty and Flynn 999). Ths s because clusterng methods have ther own bas and functon crteron. For example, the popular K-means algorthm performs mserably n several stuatons where the data cannot be accurately characterzed by a mxture of K Gaussan wth dentcal covarance matrces. It s well known that no sngle clusterng algorthm that performs best across varous data sets and t s very challengng to choose the best clusterng algorthm for gene expresson analyss. Desgn, performance evaluaton, and applcaton of clusterng algorthm on gene expresson data must take nto account the data characterstcs and randomness arsng from both bologcal and expermental varablty. Instead of focusng on developng sngle clusterng algorthm only work for a narrow-range of data sets, n ths paper, we take a dfferent approach. We present a unfed cluster ensemble framework to combne the clusterng results from varous clusterng algorthms. In our approach, the clusters ensemble problem s converted to a graph parttonng problem. A dstance matrx s frst constructed based on the cluster results from each ndvdual clusterng algorthm; these dstance matrces are combned to form a master dstance matrx. Then a weghted graph s constructed from the master dstance matrx and a graph-based parttonng algorthm s appled to the graph for the fnal clusterng results. The cluster ensemble bulds a robust clusterng portfolo that can perform reasonable well over a wde range of data sets wth lttle hand-tunng.

2 Our experment results on both synthetc data sets as well as gene expresson data sets ndcate that the clusterng qualty of the ensemble approach sgnfcantly outperforms the best ndvdual clusterng algorthm. The rest of the paper s organzed as follows. In Secton, we gve a bref overvew of the varous clusterng approaches and summarze the related work. We present our cluster ensemble algorthm n detals and expermental tests n Secton 3. We conclude wth our future plan and dscusson n Secton 4. Related Work Classfcaton ensemble approaches such as baggng and boostng have been proved very popular and effectve n supervsed learnng to mprove the learnng accuracy (Detterch 00, Hu 00). Followng the same phlosophy, the goal of cluster ensemble s to combne the clusterng results of multple clusterng algorthms to obtan better qualty and robust clusterng results. Generatng hgh qualty clusterng result s very challengng n gene expresson analyss because of the nose n the expermental data and the nconsstency among the dfferent clusterng algorthms. Even though many clusterng algorthms have been developed (Han and Kamber 00, Hartgan 975, Jan, Murty and Flynn 999), not much work s done n cluster ensemble n data mnng and machne learnng lterature compared wth classfcaton ensemble method. Zeng et al. (Zeng, Tang, Garca-Fras and Gao 00) proposed an adaptve meta-clusterng approach for combnng dfferent clusterng results. In ther research, they converted the ndvdual cluster results nto a dstance matrx and then combne the dstance matrx and apply a herarchcal clusterng to recluster the combned dstance matrx. Strethl et al. (Strehl and Ghosh 00) proposed a hypergraph-parttoned approach to combne dfferent clusterng results. Each cluster n an ndvdual clusterng algorthm s treated as a hyperedge. Ths crsp hypergraph lost much useful nformaton, and t s not sutable for ambguous and nosy envronment. It s very hard to fnd the optmal way of combnng clusters. It s consdered that ths s a natural phenomenon because each obect has varous characterstcs and a group of varous obects can be parttoned n several ways based on the many peculartes. For example, consder people n a unversty. We can group them nto so many groups based on gender, natonalty, poston (faculty, staff, student we can also group faculty and student n fne grade; for faculty group there are full professor, assocate professor, assstant professor, etc and for students there are graduate student and undergraduate student), etc. It s hard to say whch clusterng result s the best. We observe even though varous clusterng algorthms present dfferent types of knowledge concernng the clusterng crteron, most clusterng crtera n varous algorthms are compensatve rather than compettve n gene expresson analyss. We beleve that an effectve combnaton of several clusterng algorthms s an mportant way to mprove the clusterng qualty, but cluster ensemble s dfferent from the classfcaton ensemble. Some of the maor ssues of cluster ensembles addressed n the proposed research are how to combne dfferent clusterng results and how to ensure symmetrcal and unbased consensus wth regard to all the component parttons. The man dffcultes are: () the qualty of a clusterng combnaton algorthm cannot be evaluated as precsely as a combnng classfer, and () varous clusterng algorthms always produce results wth large dfferences due to dfferent clusterng crtera. Drectly combnng the clusterng results wth ntegraton rules such as product, sum and maorty vote cannot generate a good meanngful result. A new mechansm to combne the dfferent cluster results s needed to obtan better clusterng results. We propose a graph-based meta-clusterng approach to extract the nformaton from results of dfferent clusterng technques, so a better nterpretaton of the data dstrbuton can be obtaned. A dstance matrx s constructed to represent the statstcal nformaton of each cluster produced by varous clusterng technques. Our method ncorporates multple cluster-based dstance matrces nto a weghted graph. A graph based clusterng algorthm s used to cluster the graph for the fnal clusterng results. 3 Cluster Ensemble Based on Smlarty-Graph The motvatons for developng cluster ensembles are to mprove the qualty and robustness of results. There are two reasons for ths: () the results of clusterng are easly corrupted by the addton of nose, whch s very common n gene expresson analyss as the expermental measurement may not be very accurate or error may be ntroduced by the data transformaton, () the clusterng results of dfferent clusterng methods can vary sgnfcantly n the same data set, that ndcates that there could be a great potental for mprovement when usng an ensemble for the purpose of mprovng clusterng qualty. The purpose of cluster ensemble s to buld a robust clusterng portfolo that can perform as good as f not better than the sngle best clusterng algorthm across a wde-range of data sets. Dfferent clusterng algorthm may take a dfferent approach. For example, K-means s to group the data set so that the total Mean Square Error to the center of each cluster s mnmum whle graph-based parttonng clusterng s to partton the graph nto K parts based on the mnmum edge weght cuts. Thus a cluster ensemble can be used to generate many cluster results usng varous clusterng algorthms and then ntegrate them usng a consensus functon to yeld stable results. In ths secton we dscuss our novel cluster ensemble approach to combne the clusterng results from varous clusterng algorthms. We present a two-phase clusterng combnaton strategy. At the frst step, varous clusterng algorthms are run aganst the same data sets to generate clusterng results. At the second step, these clusterng results are combned by an auto-assocatve addtve system based on the dstance matrx of graph clusterng. The dagram below summarzes our approach.

3 Fgure Cluster Ensemble Archtecture In our approach, a dstance matrx s frst constructed based on the cluster results from each ndvdual clusterng algorthm; these dstance matrces are combned to form a master dstance matrx. Then a weghted graph s constructed from the master dstance matrx and a graph-based parttonng algorthm s appled to the graph for the fnal clusterng results. Graph-based clusterng uses varous knds of geometrc structure or graphs for analyzng data. Dfferent graphs reflect varous local structure or nherent vsual characterstc n the data set. Clusterng dvdes the graph nto connected components by dentfyng and deletng nconsstent edges, and each subgraph consstng of connected components refers to a cluster. In the subsectons below, we explan our cluster ensemble step by step. We frst dscuss the cluster valdaton ndces, whch help answer the tough queston of how many clusters n the data sets, bref descrbe the clusterng methods ntegrated n our framework, and then we explan the cluster ensemble mechansm and clusterng result evaluaton. 3. Clusterng Methods Many clusterng algorthms have been developed from computer scence and other dscplnes such as data mnng, machne learnng, pattern recognton and statstcs, to name a few. Clusterng algorthms can be roughly classfed nto herarchcal methods and non-herarchcal methods. Non-herarchcal method can also be dvded nto four categores; parttonng methods, densty-based methods, grd-based methods, and model-based methods (Han and Kamber 00, Jan, Murty and Flynn 999). Herarchcal methods proceed successvely by ether mergng the smaller clusters nto large clusters or splttng the larger clusters. The methods yeld a dendrogram or a tree of clusters representng how the clusters are related. Parttonng methods generate ntal k clusters and mprove the clusters by teratvely reassgnng elements among k clusters. The number of k and teraton s user nput. K-means and K-medods (Parttonng Around Medods (PAM) and Clusterng LARge Applcatons (CLARA)) (Jan, Murty and Flynn 999) belong to ths category. Self-Organzng Map (SOM) as a model-based method was developed for better speech recognton by Teuvo Kohonen n the early 980s (Kohonen 000). Fuzzy C-means as one of fuzzy clusterng methods has been developed by Bezdek (Bezdek 98, Bezdek and Pal 99) by generalzng Dunn s dea (Dunn 974). In our experment we ntegrated three clusterng algorthms; K-means, Self-Organzng Map (SOM), and Fuzzy C-means as our ntal mplementaton and more complementary clusterng algorthms can be added wthout any changes to the archtecture of the ensemble framework. 3. Clusterng Ensemble Algorthm Based on mult-obectve programmng: a smple strategy of desgnng clusterng ensemble algorthm s based on mult-obectve programmng that seeks a soluton to satsfy multple clusterng crtera. Mult-obectve programmng can be transformed to sngle obectve programmng by a weghtng method, whch s employed n our algorthm. Our algorthm to ensemble clusters s. Algorthm : Cluster Ensemble Based on Smlarty-Graph (CESG) Input: () the data set X={x,x,x 3,,x n ), () the upper bound of the cluster number k, (3) edge threshold value δ (4) a set of clusterng algorthms C (q) Output: the fnal clusterng result C (opt) Method: Step : Run the ndvdual clusterng algorthm C (q) multple tmes on the same data set under dfferent cluster numbers (clusters vares from to k). Step : Pck up the optmal number of clusters for each data set usng three cluster valdaton ndces (Slhouette ndex, Dunn ndex, and C ndex). If the number s not consstent, use votng strategy to choose the number wth the maorty as the number of the clusters. Step 3: Construct a dstance matrx (DM) for the clusterng results for each clusterng algorthm. (DM represents the smlarty of two data x and x ponts) Step 4: Combne the dstance matrxes by addng them nto one master dstance matrx (MDM) Step 5: Construct a weghted graph based on the dstance matrx. (There s an edge between data pont x and x f the value MDM of x and x s greater than

4 some threshold value δ, MDM s also the weght of the edge lnk x and x ) Step 6: Cluster the graph nto the optmal number of clusters based on the cluster number chosen at Step End In Step 3, there are so many ways to construct the dstance matrx based on cluster results from ndvdual clusterng algorthm. We propose a soluton based on statstcal theory. Here we assume that our data set s n Gaussan dstrbuton as n (Zhao and Karyps 003). Cluster-based dstance matrx DM (q) for the clusterng result C (q). DM (q) s a par-wse dstance matrx defned between two data ponts accordng to the clusterng result. Ths dstance s able to effcently extract the statstcal nformaton from the obtaned cluster structure. The matrx sze s nxn. Snce ts sze s ndependent of the clusterng approach, t provdes a way to algn the dfferent clusterngs onto the same space even for some stuatons where the numbers of clusters are dfferent for dfferent clusterng algorthms. Assume that nput data set are X={x, x, x 3, x n }, and the cluster algorthm generates m clusters for the data set X. Clusterng result s S={s, s, s 3,, s m }, where s s the th clusters consstng of some data ponts n X. For example, X={x, x, x 3, x 9 } and S={s, s, s 3 }, s = {x, x 5 }, s ={x, x 3, x 7 }, s 3 ={x 4, x 6, x 8, x 9 }. We assume that probablty densty functon of s s gven by p(x s ), the posteror probablty of cluster s gven x can be expressed as: p( x s ) P( s ) P( s x ) = m, where p( x s ) P( s ) P( x s k = exp[ ) = ( k m k T x µ ) (π ) ( x µ )], m s the number of clusters. s a matrx of co-varances among attrbutes n cluster, µ s the mean vector of the data ponts n the cluster s. For example, to calculate P(s x ), gven the followng elements n a cluster s = ={x, x 3, x 7 }, (assume each x has three condtons), Att_ Att_ Att_3 x x x x 3 x 3 x 3 x 3 x 33 x 7 x 7 x 7 x 73 s shown as = where s covarance between Att_ and Att_ and s varance of Att_ n s For each data pont x, we calculate the correspondng probablty vector PX = {P(s x ), P(s x ), P(s m x )}, where =,..m P(s x ) =, the probablty vectors form a probablty space of dmenson of m, wth each dmenson correspondng to one cluster. The probablty space contans nformaton from both the nput data and the cluster results. So we beleve the smlarty of any two ponts PX l and PX m n the probablty space s a good measurement to reflect the dstance of the correspondng ponts x l and x m n the orgnal space. Then for any two ponts, x l and x m, n the data set, ther dstance s defned as the dstance between PX l and PX m, namely, DM (q) (x l,x m ). Many dfferent dstance measures such as Eucldean dstance, Mahalanobs dstance or correlaton dstance can be used to calculate DL(PX l,px m ). We defne the smlarty of two ponts (x and x ) n the data set as DM ( q) = PX PX PX PX N ( ) ( PX ) PX PX PX N N In step 6, a graph-based clusterng algorthm s appled to the weghted graph for the fnal clusterng result. Many graph-based parttonng algorthms can be used for ths purpose, such METIS, HMETIS (Kayyps and Kumar 998a). Clusterng dvdes the graph nto connected components by deletng edges based on some constrant such as mnmum cuts, and each subgraph consstng of connected components refers to a cluster. In our experment, we chose the graph parttonng-based algorthm METIS (Kayyps and Kumar 998a, Kayyps and Kumar 998b) because of ts good performance and scalablty. 3.3 Clusterng Result Evaluaton To evaluate the qualty of cluster s a non-trval and often ll-posed task. Generally speakng, there are nternal crtera and external crtera. Internal crtera formulate qualty as a functon of the gven data and/or smlartes. For example, the mean squared evaluaton crteron (for k-means) and other measures of compactness are popular evaluaton crtera. Measure can also be based on solaton such as the mn-cut crteron, whch uses the sum of edges weghts across clusters (for graph portonng). When usng nternal crtera, clusterng becomes an optmzaton problem, and a clusterer can evaluate ts own performance and tune ts results accordngly. External crtera on the other hand mpose qualty by addtonal, external nformaton not gven to the clusteres, such as category labels. Ths s sometmes more approprate snce groupngs are ultmately evaluated externally by humans. For example, when obects have already been categorzed by an external source,.e., when class labels are avalable, we can use nformaton theoretcal measure to quantfy the match between the categorzaton and the clusterng. In our cluster ensemble, external crtera ft very well wth our archtecture. We use the Mnkowsk Score (Ben-Hur and Guyon 003) as our

5 cluster qualty ndcator. Below s our formula for the clusterng qualty evaluaton. A clusterng soluton for a set of n elements can be represented by an nxn matrx C where C = ff x and x are n the same cluster accordng to the soluton and C =0 otherwse. A measure of Mnkowsk Score (MS) between the clusterng results C (h) from a partcular clusterng algorthm CA h wth a reference clusterng T (or alternatvely, the true clusters f the cluster nformaton n the data set s known n advance) s defned as MS (T, C (h) ) = T-C (h) / T, where T = sqrt( T ) The Mnkowsk score s the normalzed dstance between the two matrces. Hence a perfect soluton wll obtan a score zero, and the smaller the score, the better soluton. We abbrevate the set of cluster groupngs from r dfferent clusterng algorthms as Ψ = { C (q) q {,,r}}. The average MS score of combned clusterng result C wth the Ψ s defned as MS ( ANMI ) ( C, Ψ) = r 3.4 Expermental Results r q= MS( C, C ( q) We conduct some experment study on both data sets from the UCI machne learnng repostory ( and yeast gene data set ( UCI Data Sets Irs data set, Pen dgt data set, and Vowel data set were used for our experment. The followng table shows clusterng result of the three clusterng algorthms and cluster ensemble. IRIS data set(3 clusters) K-means SOM C-mean Ensemble The followng table represents the element (or pont) compostons of rs data set n the true clusters and the best clusters usng cluster ensemble. Element composton IRIS data set Cluster Cluster Cluster3 True clusters ) Best clusters usng cluster ensemble Group Group 0 0 Group3 0 0 Group 0 0 Group Group3 0 Group 0 0 Group 0 Group The followng tables show clusterng results of the three clusterng algorthms and cluster ensemble technology for pen dgts data set and vowel data set. Pen Dgts data set(0 clusters) K-means SOM C-mean Ensemble Vowel data set ( clusters) K-means SOM C-mean Ensemble Yeast gene data set There are 6 genes n the data sets but not every gene s classfed nto a certan functon famly. In our experment we consdered the genes n a functon famly as one cluster and created 6 data sets (cluster, 3, 4, 5, 6, 7). Table shows 6 functon famles of yeast gene and how we construct the sx data sets (C, C3, C4, C5, C6, and C7) for our cluster ensemble comparson. For example, C3 means the cluster set has 3 clusters (ATP synthess, mtoss, and vacuolar proten targetng here) Functon Famles # of genes ATP synthess 9 mtoss 9 vacuolar proten targetng 9 slencng 0 fatty acd metabolsm 0 meoss phospholpd metabolsm TCA cycle proten processng 7 DNA repar 9 proten foldng 30 nuclear proten targetng 3 sgnalng 3 maor facltator superfamly 3 mrna splcng 34 chromatn structure 4 DNA replcaton 4 Cluster Sets C3 C5 C4 C C6 Table Some of Yeast gene functon famly Table shows the clusterng results ncludng cluster ensemble n Mnkowsk scores (MS) for each cluster set. As clearly ndcated by the MS values of the clusters, the cluster ensemble method made sgnfcant mprovement of qualty of the clusterng results over the ndvdual clusterng algorthm on all the sx gene data sets. For example, the best ndvdual clusterng algorthm for C3 s K-means (MS=0.890), whle the cluster ensemble has MS= For C5, the best ndvdual clusterng algorthm s SOM (MS =.4) and the cluster ensemble reduced them to MS =.059. Cluster K-mean SOM set # s C-means Ensemble C C C C C7

6 C C Table Clusterng Results of Yeast gene data sets 4 Concluson In ths paper we present a novel cluster ensemble approach for combnng clusterng results from multple cluster algorthms. The experment results on UCI machne learnng data and gene expresson data ndcate that the cluster ensemble approach can generate better qualty and robustness clusters compared wth sngle best clusterng algorthm. Clusterng ensemble s a new and very promsng research area. There are a lot of open problems for future research. We plan to expand our ensemble approach to ntegrate feature selecton for clusterng very hgh dmensonal data set and add some nference mechansm to automatcally nfer vald nformaton from the clusterng results and hope to report our fndngs n the future. 5 References Azuae, F. (00): In Slco Approaches To Mcroarray-Based Dsease Classfcaton and Gene Functon Dscovery. Annals of Medcne 34(4): Bellaacha, A., Portnoy, D., Chen, Y. and Elkahloun, A. G. (00): E-CAST: A Data Mnng Algorthm For Gene Expresson Data. BIOKDD Bloch, K. M. and Arce, G. R. (00): Nonlnear Correlaton For The Analyss Of Gene Expresson Data. Proceedngs of the 00 Workshop on Genomc Sgnal Processng and Statstcs, Ralegh, North Carolna. Boley, D., Gn, M., Gross, R., Han, E., Hastngs, K., Karyps, G., Kumar, V., Mobasher, B. and Moore, J. (999): Parttonng-Based Clusterng For Web Document Categorzaton. Decson Support Systems 7: Daves, D. and Bouldn, D. (979): A Cluster Separaton Measure. IEEE Transactons on Patter Recognton and Machne Intellgence (): 4-7. Hu, X. (00): Usng Rough Sets Theory And Database Operatons To Construct A Good Ensemble Of Classfers For Data Mnng Applcatons, IEEE ICDM, Jan, A. K., Murty, M. N. and Flynn, P. J. (999): Data Clusterng: A Revew. ACM Computng Surveys 3(3): Rchard, M.D. and Lppmann, R.P. (99): Neural Network Classfers Estmate Bayesan A Posteror Probabltes. Neural Computaton 3(4): Xng, E. and Karp, R. (00): CLIFF: Clusterng of Hgh-Dmensonal Mcroarray Data Va Iteratve Feature Flterng Usng Normalzed Cuts. Bonformatcs 7: Zeng, Y., Tang, J., Garca-Fras, J. and Gao, G.R. (00): An Adaptve Meta-Clusterng Approach: Combnng The Informaton From Dfferent Clusterng Results, CSB00 IEEE Computer Socety Bonformatcs Conference Proceedngs Detterch, T. G. (000): Ensemble Methods n Machne Learnng. Frst Internatonal Workshop on Multple Classfer Systems, Lecture Notes n Computer Scence -5. Ben-Hur, A. and Guyon, I. (003): Detectng stable clusters usng prncpal component analyss. In Methods n Molecular Bology, M.J. Brownsten and A. Kohodursky (eds.) Humana press, Zhao, Y. and Karyps, G. (003): Clusterng n Lfe Scences, In Functonal Genomcs: Methods and Protocols Khodursky, A. and Brownsten M. (eds). Humana Press. Berrar, D., Dubtzky, W. and Granzow M. (00): A Practcal Approach To Mcroarray Data Analyss. Kluwer Academc Publshers, 00. Bezdek, J. C. and Pal, S.K. (99): Fuzzy Models For Pattern Recognton: Methods That Search for Structures n Data. New York, IEEE Press. Bezdek, J. C. (98): Pattern Recognton Wth Fuzzy Obectve Functon Algorthms. New York, Plenum Press. Han, J. and Kamber, M. (00): Data Mnng: Concepts And Technques. San Francsco, Morgan Kaufmann Publshers. Hartgan, J. A. (975): Clusterng Algorthms. New York, John Wley & Sons, Inc. Kohonen, T. (000): Self-Organzng Maps. New York, Sprnger. Ben-Dor, A., Shamr, R., and Yakhn, Z. (999): Clusterng gene expresson patterns. Journal of Computatonal Bology 6: Dunn, J. (974): Well Separated Clusters And Optmal Fuzzy Parttons. Journal of Cybernetcs 4: Goodman L. and Kruskal, W. (954): Measures of Assocatons For Cross-Classfcatons. J. Am. Stat. Assoc. 49: Hubert, L. and Schultz, J. (976): Quadratc Assgnment As A General Data-Analyss Strategy. Brtsh Journal of Mathematcal and Statstcal Psychology 9: Kayyps, G. and Kumar, V. (998): A Fast and Hgh Qualty Multlevel Scheme for Parttonng Irregular Graphs. SIAM Journal on Scentfc Computng 0(): Kayyps, G. and Kumar, V. (998): Multlevel k-way Parttonng Scheme for Irregular Graphs. Journal of Parallel and Dstrbuted Computng 48(): Strehl, A. and Ghosh, J. (00): Cluster Ensembles - A Knowledge Reuse Framework For Combnng Multple Parttons. Journal on Machne Learnng Research (JMLR) 3: