A Genetic K-means Clustering Algorithm Applied to Gene Expression Data

Transcription

1 A Geetc K-meas Clusterg Algorthm Appled to Gee Expresso Data Fag-Xag Wu, W. J. Zhag, ad Athoy J. Kusal Dvso of Bomedcal Egeerg, Uversty of Sasatchewa, Sasatoo, S S7N 5A9, CANADA faw34@mal.usas.ca, zhagc@egr.usas.ca Departmet of Computer Scece, Uversty of Sasatchewa, Sasatoo, S S7N 5A9, CANADA usal@cs.usas.ca Abstract. e of the curret ma strateges to uderstad a bologcal process at geome level s to cluster gees by ther expresso data obtaed from DNA mcroarray expermets. The classc K-meas clusterg algorthm s a determstc search ad may termate a locally optmal clusterg. I ths paper, a geetc K-meas clusterg algorthm, called GKMCA, for clusterg gee expresso datasets s descrbed. GKMCA s a hybrdzato of a geetc algorthm (GA) ad the teratve optmal K-meas algorthm (IKMA). I GKMCA, each dvdual s ecoded by a partto table whch uquely determes a clusterg, ad three geetc operators (selecto, crossover, mutato) ad a IKM operator derved from IKMA are employed. The superorty of the GKMCA over the IKMA ad over other GA-clusterg algorthms wthout the IKM operator s demostrated for two real gee expresso datasets.. Itroducto The developmet of DNA mcroarray techques ad geome sequecg has resulted large amout of gee expresso data for may bologcal processes. Gee expresso of tssue sample ca be quattatvely aalyzed by co-hybrdzg cdna fluor-tagged wth Cy5 ad Cy3 (Cy5 for those from a treatmet sample ad Cy3 for those from a referece sample) to gees (called targets) o a DNA mcroarray []. Fluorescece testy ratos are extracted va mage segmetato for all target gees. A seres of ratos collected at dfferet tme pots a bologcal process comprse a gee expresso patter. Gee expresso data from may orgasms are avalable publcly-accessble databases []. e of the ma goals of aalyzg these data s to fd correlated gees by searchg for smlar gee expresso patters. Ths s usually acheved by clusterg them [3-6]. Clusterg methods ca be dvded to two basc types: herarchcal ad parttoal clusterg [7]. They have both bee wdely appled to aalyss of gee expresso data [36]. Geetc algorthms have also bee appled to may clusterg problems [8-]. However, these methods are ot sutable for the aalyss of gee expresso data because of (typcal) sze of a dataset. To our best owledge, there

2 have bee o reports of the applcato of geetc algorthms to clusterg gee expresso data yet. I ths paper, we propose a geetc K-meas clusterg algorthm (GKMCA), whch s a hybrd approach to combg a geetc algorthm wth the teratve optmal K-meas algorthm (IKMA). I our GKMCA, the solutos are ecoded by a partto table. GKMCA cotas three geetc operatos -- atural selecto, crossover ad mutato-- ad oe IKM operato derved from IKMA. The remader of the paper s orgazed as follows. I secto the K-meas clusterg problem ad the IKMA are troduced. I secto 3, the operators corporated to GKMCA are descrbed detal, ad GKMCA s preseted. I secto 4 two DNA mcroarray datasets are troduced. GKMCA s compared to classc K-meas clusterg algorthms (.e. IKMA) ad other GA-clusterg algorthms o these two datasets. Fally, some coclusos are draw secto 5.. IKMA I geeral, a K-parttog algorthm taes as put a set D = { x,x,x} of objects ad a teger K, ad outputs a partto of D to exactly K dsjot subsets D, D,,, DK. I the cotext of clusterg gee expresso data, each object (gee) s expressed by a real umber row vector (called the expresso patter) of dmeso d, where d s the umber of ratos the expresso patter. We wll ot dstgush a object from ts expresso patter. Each of the subsets s a cluster, wth objects the same cluster beg somehow more smlar to each other tha they to objects ay other cluster. f a umber of K-partto algorthms, K-meas s the best-ow oe. We wll ot dstgush a object from ts expresso patter. Let x deote the j th compoet of expresso patter x. For the predefed j umber K of clusters, defe the partto table as the followg matrx W = [ w ] ( =, L, =, L, K)., f th object belogs to th cluster, w = () 0, otherwse. bvously, the matrx W has property that K w = ( =,, ). () = Let the cetrod of the th cluster be m = m,, m ) ( =,, K). The m W * X / Σ = ( d = w. (3) where X = [ x j ] s the expresso matrx determed by the compoet x j s of all expresso patters the dataset. A sum-of-squared-error (the cost fucto of a K - partto) s defed by K = K J ( W ) = J ( W ) = w x m. (4) = =

3 where = J ( W ) w x m, ad s Eucldea dstace measure of a vector. = The objectve of K-meas algorthms s to fd a optmal partto expressed by * W * = [ ] whch mmzes J (W ),.e. w J ( W*) = m{ J ( W )}. (5) W The optmzato problem (5) s NP-hard ad may be solved by a heurstc algorthm called the teratvely optmal K-meas algorthm (IKMA) []. 3. GKMCA GKMCA show Fgure s a hybrd algorthm of GA to IKMA, cludg the three geetc operators GA ad a IKM operator derved from IKMA. I ths secto we specfy codg, selecto operator, crossover operator ad mutato operator ad IKM operator before we preset GKMCA. Codg: A partto table s used to express a soluto to a clusterg. Thus, the search space cossts of all W matrces that satsfy () (). Such a W s coded by a teger strg s W cosstg of tegers from the set {,, K }. Each posto the strg correspods to a object ad the value the posto represets the cluster umber where the correspodg object belogs. I the followg, we wll ot dstgush W from ts code s. A populato s expressed by a set of partto W tables represetg ts dvduals, deote by W p or Wp. Selecto operator-- W p = Selecto( Wp, X, N ) : For coveece of the mapulato, GKMCA always assgs the best dvdual foud over tme a populato to dvdual ad copes t to the ext populato. perator W p = Selecto( Wp, X, N ) selects ( N ) / dvduals from the prevous populato accordg to the probablty dstrbuto gve by N Ps( W ) = F( W ) / F( W ) (6) = where N (a odd postve teger) s the umber of dvduals a populato, W s the partto table of dvdual, ad F W ) represets the ftess value of dvdual the curret populato defed as F( W ) = TJ J ( W ), where J (W ) s calculated by (4), ad TJ s the total squared error curred represetg the objects (,x,x by ther ceter m x / = x =,.e. TJ = = x m. Note that there are ( N ) / + dvduals W p. Crossover operator-- Wp = Crossover( Wp, RW, : The teto of the crossover operato s to create ew (ad hopefully better) dvduals from two selected paret dvduals. I GKMCA, of two paret dvduals, oe s dvdual (.e. the optmal dvdual foud over tme), ad aother s oe of the selected

4 ( N ) / dvduals from the paret populato other tha dvdual by selecto operator. Here crossover operator adopts sgle-pot crossover methods for smplcty. Note that after crossover operato populato Wp has N dvduals. Geetc K-meas Clusterg Algorthm (GKMCA) Iput: Expresso matrx, X Number of objects, RW Number of attrbutes, CL Number of clusters, K Mutato probablty, Pm Populato sze, N Number of geerato, GEN utput: Mmum sum-of-squared-errors of clusterg foud over evoluto, JE.. Italze the populato, Wp /* Wp s a set of partto table of a populato */. Re-order dvduals such that the frst oe s the optmal populato Wp, ad set W * = Wp(), JE ( 0) = J ( W*), ad g =. 3. Whle ( g GEN ) 4. W p = Selecto( Wp, X, 5. Wp = Crossover( Wp, RW, 6. Wp = Mutato( Wp, Pm, RW, CL, 7. [ Wp, J ( Wp)] = IKM ( Wp, X, RW, CL, 8. Fd the optmal dvdual populato Wp, deote by W 9. If ( J ( W*) > J ( W ), the W * = W, ad set JE ( g) = J ( W ) else JE ( g) = JE( g ) 0. Re-arrage dvduals such that Wp ( ) = W. g = g +. Ed whle 3. Retur JE correspodg to the partto table W * by (4). Fgure. Geetc K-meas Algorthm (GKMCA) Mutato operator-- Wp = Mutato( Wp, Pm, RW, CL, : Each posto a strg s radomly selected wth a mutato probablty P m, ad the value of the selected posto s uformly radomly replaced by aother teger from the set {,, K }. I [8], the value of the posto s chaged depedg o the dstace of the cluster cetods from the correspodg object. Actually such a complex techque may ot be ecessary because IKM operator s used. To avod ay sgular partto (cotag a empty cluster), after prevous operato, mutato operator also radomly assgs K dfferet objects to K dfferet clusters order to assure that every cluster has at least oe object. IKM operator-- [ Wp, J ( Wp)] = IKM ( Wp, X, RW, CL, : IKM operator s obtaed by IKMA [] where each dvdual W populato Wp s

5 a tal partto. I [8-0], several dfferet K-meas operators were employed, ad ther fuctos are smlar to that of IKMA. However, those K-meas algorthms are ot teratvely optmal. 4. Expermets ad Dscusso 4. Datasets Expermets o two datasets are performed to demostrate the performace of GKMCA, compared to IKMA ad other GA-clusterg algorthms. The frst dataset (α factor) cotas 6 gees whch were detfed as cell-cycle regulated the α factor-sychrozed expermet [4], wth o mssg data the 8 arrays. It may be created from the related data at The secod dataset (Fbroblast) cotas 57 gee expressos selected by authors from a expermet studyg the respose of huma fbroblasts to serum [5]. The orgal data may be obtaed at 4. Expermet Results Accordg to cell-cycle dvso process [4, 5], we too 4 = for umber of clusters both datasets expermets. I GKMCA, we too populato sze N =, mutato probablty Pm = 0. 0, ad umber of geeratos GEN = 0. Expermet results (ot exhbted here because of space lmtatos) shows that the cluso of the IKM operator greatly mproves the covergece rate of the algorthms. I fact, GAs wthout ths type of operator [] coverge slowly, or ot at all. Thus such GAs are ot applcable to DNA mcroarray datasets of practcal sze. Table. Performace comparsos of GKMCA ad IKMA. GKMCA: the sum-of-squarederrors of the fal clusterg from GKMCA the K -Meas Average ad STD: the average of the sum-of-squared-errors ad the stadard dervato of the resultat clustergs for 40 depedet rus, respectvely Datasets Average K-Meas STD GKMCA α factor Fbroblast I order to compare GKMCA ad IKMA, GKMCA was ru aga o two datasets 5 tmes (results from them are the same for each dataset) whle IKMA was ru o two datasets 40 (= N *GE tmes. I these expermets, for each dvdual IKM operator GKMCA oly performs two repeat-loops whle each IKMA performed much more tha two repeat-loops to reach ts covergece. Expermet results are lsted Table. From the Table, t ca be observed that GKMCA clearly outperforms the IKMA that GKMCA s less sestve to the tal codtos,

6 ad that the sum-of-squared-errors of the resultat clustergs from GKMA s less tha the average of the sum-of-squared-errors of the clustergs from 40 rus of IKMA. 5. Cocluso I ths study, a geetc K-meas clusterg algorthm (GKMCA) s proposed for clusterg tass o large-scale datasets such as gee expresso datasets from DNA mcroarray expermets. GKMCA s a hybrd algorthm of the teratve optmal K- meas algorthm ad a geetc algorthm. Some specal techques were employed GKMCA to avod ay sgular clusterg ( the mutato operator) ad to speed up the rate of covergece ( the IKM operator). GKMCA was ru o two real gee expresso datasets. Expermets results show that ot oly ca GKMCA fulfl the clusterg tass o gee expresso datasets, but also ts performace s better tha that of IKMA ad some exstg GA-clusterg algorthms. Refereces. Ese, M. B. ad Brow, P.. DNA Arrays for Aalyss of Gee Expresso. Methods Ezymol 303: 79-05, Sherloc, G., et al. Davd Botste ad J. Mchael Cherry, The Staford Mcroarray Database, Nuclec Acds Research, 9: 5-55, Ese, M. B., et al. Cluster Aalyss ad Dsplay of Geome-Wde Expresso Patters. Proc Natl. Acad. Sc. USA, 95: , Spellma, P. T., et al. Comprehesve Idetfcato of Cell Cycle-regulated Gees of the Yeast Saccharomyces cerevsae by Mcroarray Hybrdzato. Mol. Bol. Cell., 9: , Iyer, V R., et al. The trascrptoal program the respose of huma fbroblasts to serum. Scece. 83: 83-87, Che, G., et al. Cluster aalyss of mcroarray gee expresso data: applcato to ad evaluato wth NIA mouse 5K array o ES cell dfferetato. Statstca Sca, : 4-6, Hartga, J. (975) Clusterg Algorthms. Wley, New Yor, NY. 8. Krsha, K. ad Murty, M. M. Geetc K-meas algorthm, IEEE Trasactos o Systems, Ma, ad Cyberetcs---Part B: Cyberetcs, 9: , Maul, U. ad Badyopadhyay, S. Geetc algorthm-based clusterg techque, Patter Recogto, 33: , Frat, P. et al. Geetc algorthms for large-scale clusterg problems, The compuer Joural, 40: , Hall, L.., zyurt, I. B., ad Bezde, J. C. Clusterg wth a geetcally optmzed approach, IEEE Trasactos o Evolutoary Computato, 3: 03-, Rchard,. D., Peter, E. H., ad Davd, G. S., Patter Classfcato, New Yor: Wley, 00.