Improved NSGA-II Based on a Novel Ranking Scheme

Transcription

1 Imroved NGA-II Based o a Novel Rakg cheme Ro G. L. D ouza, K. Chadra ekara, ad A. Kadasamy 9 Abstract No-domated ortg Geetc Algorthm (NGA) has establshed tself as a bechmark algorthm for Multobectve Otmzato. The determato of areto-otmal solutos s the key to ts success. However the basc algorthm suffers from a hgh order of comlexty, whch reders t less useful for ractcal alcatos. Amog the varats of NGA, several attemts have bee made to reduce the comlexty. Though successful reducg the rutme comlexty, there s scoe for further mrovemets, esecally cosderg that the oulatos volved are freuetly of large sze. We roose a varat whch reduces the ru-tme comlexty usg the smle rcle of sace-tme trade-off. The mroved algorthm s aled to the roblem of classfyg tyes of leukema based o mcroarray data. Results of comaratve tests are reseted showg that the mroved algorthm erforms well o large oulatos. Idex Terms NGA, Multobectve otmzato, MOEA, Evolutoary algorthms, Pareto-otmal solutos, No-domato. INTRODUCTION M ULTI-OBJECTIVE otmzato has become a useful strategy the soluto of may moder egeerg roblems. Evolutoary algorthms have bee foud to be very successful carryg out the otmzato of multle obectves. Most evolutoary algorthms are robust ad mult-modal whch roves to be a dstct advatage the soluto of such roblems. Amog the exstg evolutoary algorthms for solvg a Multobectve otmzato roblem (MOO [] the most romet oes are the Pareto archved evolutoary strategy (PAE) [2], tregth areto evolutoary algorthm (PEA-2) [3] ad the No-domated sortg geetc algorthm (NGA-II) [4]. All these algorthms are based o the cocet of Pareto-domace. NGA-II has evolved over the last few years wth may ew varats whch have attemted to reduce ts tme-comlexty or mrove ts covergece to the true areto frot [5][6][7]. I ths aer, we troduce a varat of NGA-II whch mroves o the tme-comlexty by maagg the book-keeg a better way tha the basc algorthm. For wat of a better ame, we call ths varat the NGA-IIa. We aly NGA-IIa to the roblem of classfyg tyes of leukema based o the gee exresso dataset by Golub et al [8]. We also comare the erformace of NGA-IIa wth that of the basc algorthm, NGA-II, ad also some of the other algorthms based o areto-domace that have bee troduced recet years. Results show that the ew varat s o ar wth the other varats ad hece Ro G. L. D ouza s wth the Deartmet of Comuter cece ad Egeerg, t. Joseh Egeerg College, Magalore, Ida. K. Chadra ekara s wth the Deartmet of Comuter Egeerg, Natoal Isttute of Techology Karataka, urathkal, Magalore, Ida. A. Kadasamy s wth the Deartmet of Mathematcal ad Comutatoal ceces, Natoal Isttute of Techology Karataka, urathkal, Magalore, Ida. ca be cosdered such stuatos whch warrat a alteratve aroach for valdato of results obtaed by ay of the other method. 2 RELATED WORK Evolutoary algorthms attemt to mmc ature the search for solutos to roblems. I recet tmes they have bee used to solve may real-world otmzato roblems. Early evolutoary algorthms were focused o otmzg sgle obectves. However, most otmzato roblems have multle obectves. Otmzato of multle obectves reures that the relatve mortace of each obectve be secfed advace whch reures a ror kowledge of the ossble solutos. But by usg the cocet of Pareto-domace t s ossble to avod the eed to kow the ossble solutos advace [9]. Ths s oe of the reasos for the oularty of such areto-based aroaches. Multobectve Evolutoary Algorthms (MOEAs) [0] are based for the followg cocets: ) Pareto-Frot: The locus that s formed by a set of solutos that are eually good whe comared to other solutos of that set s called as a Pareto-frot. 2) No-Domato: No-domated or areto-otmal solutos are those solutos the set whch do ot domate each other,.e., ether of them s better tha the other all the obectve fucto evaluatos. The solutos o each areto-frot are areto-otmal wth resect to each other. Foesca ad Flemg [] roosed oe of the frst multobectve evolutoary algorthms called as Multobectve geetc algorthm (MOGA). Kowles ad Core [2] troduced the PAE where the aret ad offsrg as well as the archved best solutos thus far are comared usg areto-domace. Ztzler ad Thele [3] develoed the PEA-2, where the best solutos thus far are stored ad comared usg areto-domace wth the curret oulato.

2 92 Paret oulato geerato t, Pt Mutato ad Crossover But the NGA whch was frst roosed by rvas ad Deb [2] 995 roved to be a ladmark the hstory of MOEAs. The tme-comlexty of the smle NGA s O(MN 3 ) where M s the umber of obectves ad N s the sze of the dataset. oo after, Deb ad hs studets [4] develoed a faster varat of NGA, called NGA-II, whose tme-comlexty s O(MN 2 ). Durg the last few years several researchers have come u wth varats of NGA-II whch have a tmecomlexty of O(MN log M- N). The works by Fag [5] ad Tra [6] are worth metog. Kumar et al. [3] have develoed a memetc verso of the NGA-II. Jese [7] has aalyzed the techues that could be used to mrove the rutme of MOEAs ad has also roosed a mroved varat of NGA-II. We comare our varat wth ths varat by Jese (whch we call as NGA-IIb) the Results secto of ths aer. Combato Offsrg oulato Qt Combed oulato Rt No-domated sortg orted & ed oulato electo based o ad crowdg dstace Paret oulato for the ext geerato Pt+ Fg.. Processg of oulatos NGA-II over oe geerato. 3 NON-DOMINATED ORTING GENETIC ALGORITHM Frst we descrbe the workg of the NGA-II algorthm as gve [4]. Our varat s dfferet oly the way the o-domated sortg s erformed. Ths varato we dscuss the ext secto. I the followg dscusso, we use the terms soluto ad dvdual to mea the same thg, sce dvduals the oulato rereset solutos to the roblem that s beg otmzed. I NGA-II, we frst create the offsrg oulato Qt (of sze N) usg the aret oulato Pt (of sze N), as show Fg.. The usual geetc oerators such as sgle-ot crossover ad bt-wse mutato oerators are used ths rocess. Next, we combe the two oulatos to form a termedate oulato Rt of sze 2N. Thereafter, we evaluate the ftess of each offsrg the 2N oulato usg the multle obectve fuctos. At ths stage, we carry out o-domated sortg rocedure over the 2N oulato to ad dvde the dvduals to dfferet o-domated frots. The detals of the o-domated sortg ad assgmet are gve Fg. 2. Thereafter, we create the ew aret oulato Pt+ by choosg dvduals of the odomated frots, oe at a tme. We choose the dvduals of best ed frots frst followed by the ext-best ad so o, tll we obta N dvduals. ce the termedate oulato Rt has a sze of 2N, we dscard those frots whch could ot be accommodated. I case there s sace oly for a art of a frot the ew oulato, we use a crowded-dstace oerator to determe the dvduals amog those the frot that are from the least crowded regos. We choose such dvduals so as to fll u the reured umber the ew oulato Pt+. Detals of crowed-dstace oerator ca be obtaed from [7]. The comlete NGA-II rocedure s gve below: BEGIN Whle geerato cout s ot reached Beg Loo Combe aret Pt ad offsrg oulato Qt to obta oulato Rt of sze 2N. Perform No-domated ort o Rt ad assg s to each areto frot wth ftess F. tartg from Pareto frot wth ftess F, add each areto-frot F to the ew aret oulato Pt+ utl a comlete frot F caot be cluded. From the curret areto-frot F, add dvdual members to ew aret oulato Pt+ utl t reaches the sze N. Aly selecto, crossover ad mutato to ew ar-

3 93 et oulato Pt+ ad obta the ew offsrg oulato Qt+. Icremet geerato cout. Ed Loo END. As stated [4], ths algorthm has a rutme comlexty of O(MN 2 ) whch s obvous from Fg. 2. I ths fgure, reresets a set of solutos that the soluto domates ad reresets the domato cout (the umber of solutos whch domate the soluto ). The symbol < dcates domato ( < dcates that domates ) ad F reresets the areto-frot of solutos. I Jese [7], a mroved verso of ths algorthm s fast _ o _ dom ated _ sort( for each P 0 for each Q F Q f ( ) the else f ( ) the f whle F 0the Q F F for each F for each f 0the Q Q Fg. 2. No-domated sortg rocedure adoted NGA-II [4]. reseted. I ths verso, NGA-IIb, all the areto-frots are costructed smultaeously by sweeg the solutos oe-by-oe a way that guaratees that f s soluto s swet after soluto s, the s caot domate s. Ths s acheved by resortg all of the solutos o the obectve values such a way that the above codto wll always hold f <. Due to the resortg, whch ca be erformed O(N logn) tme, ths algorthm has a rutme comlexty of O(MN log M- N). For the rest of the detals of ths method, lease refer to [7]. 4 THE IMPROVED NGA-II I our work, we have used a varato of the odomated sortg rocedure show Fg. 3. As see the fgure, we erform sortg of dvduals based o each of the obectves, oe after the other, tll all obectves are cosdered. Durg ths sort, we kee track of the dex of each dvdual, so that we kow the osto value of ay gve dvdual each sorted array. Ths formato s crtcal sce t hels us to the frots the ext ste. We assg the of each dvdual by summg u the osto value of that dvdual all the obectves. ce smlar osto values where assged to dvduals havg smlar obectve values, the sum of the osto values becomes euvalet to the whch the dvdual would have obtaed through o-domated comarso. I Fg. 3, reresets the array of dvduals sorted o each obectve, osto(, ) returs the osto of, sort( returs a stadard sort of P based o obectve value, (ob) reresets the obectve value of dvdual ad set_osto(, Q, os) wll set the osto of as os Q. We estmate the rutme comlexty of the above rocedure as follows: the sort_o_obectve rocedure ca be comleted O(N log 2 N). Ths has to be reeated for M obectves, hece the overall comlexty s O(MN log 2 N). The arrays sorted o obectves have a sze of O(N). But we wll eed M such arrays ad hece the sace comlexty creases to O(MN), whch ca be ute a desrable trade-off eve for moderate szes of M ad N. 5 THE CLAIFICATION OF TYPE OF LEUKEMIA Here we reset the multobectve roblem to whch we aly the algorthms descrbed above. The task for the otmzer s to detfy the otmal gee subsets whch classfy mcroarray gee exresso data. Both the algorthms were ru smlar codtos o the 50-gee Leukema dataset of Golub et al. [8]. Ths dataset cotas the exresso data of 729 huma gees take from 38 samles of atets, 27 of whom were sufferg from Acute lymhoblastc leukema (ALL) ad of whom were sufferg from Acute myelod leukema (AML). We adot three dfferet coflctg obectves: to mmze the umber of gees used classfcato whle matag accetable classfcato accuracy exressed

4 94 faster _ o _ for each sort _ o _ obectve( P, ) for each obectve M dom ated P osto(, ) sort _ o _ obectve( P, ) os 0 Q sort( for each P f ( ( ob) ( ob)) os os set _ osto(, Q, os) retur Q _ sort( Fg. 3. A faster o-domated sortg rocedure as comared to the oe of NGA-II. as trag error ad testg error [4]. The mult-obectve otmzato roblem s formulated as follows: ) The frst obectve fucto - The gee subset detfcato task s to mmze the umber gees a subset or to mmze the gee subset sze for a classfer. 2) The secod obectve fucto - Mmze the umber of class redcto msmatches the trag samles. These are calculated usg the Leave-Out-Oe-Cross- Valdato (LOOCV). 3) The thrd obectve fucto - To mmze the umber of class redcto msmatches the test samles, these are calculated usg the classfer costructed based o all samles the trag set. We have used a weghted-votg aroach to redct the class of a samle based o such formatve gee subsets ad a set of samles wth kow class labels, ad the LOOCV rocedure s used to determe the umber of msmatches the trag samles. Thereafter, the classfer s costructed usg such formatve gee subsets ad all the trag samles. The, the erformace of the classfer s estmated usg the remag samles a test-set. The NGA-II descrbed above s used for hadlg the above three coflctg obectves. Durg each geerato, the algorthm fds gee subsets whch classfy the data to the two sub-tyes of leukema. As the evolutoary algorthm roceeds, rogressvely better gee subsets are detfed. 6 REULT AND DICUION We carred out a comaratve study betwee the basc algorthm (NGA-II) [4], the varat due to Jese (NGA-IIb) [7] ad our ow varat NGA-IIa. The otmzato was carred for gee subset selecto as descrbed secto 5 above. Tests were carred out o a Petum-IV, 2.0 GHz mache wth GB RAM, rug Fedora Core 6.0 oeratg system Processg Tme (secs) NGA-II NGA-IIb NGA-IIa Poulato ze, N Fg. 4. Grah showg the varato of rocessg tme secods wth the oulato sze, N. Fg. 4 shows the varato of average rocessg tme for covergece, CPU secods, take over te rus each case whe the oulato sze s gradually creased, stes. It s evdet from the grah that the NGA-IIa we roose s faster tha the NGA-IIb roosed by Jese. The basc NGA-II erforms eve worse sce ts comlexty s O(MN 2 ). Though the dfferece s eglgble for small oulatos, there s a marked dfferece for oulato szes of 000 ad more. Ths s sgfcat sce such roblems freuetly demad the use of large oulato szes order to yeld reasoable results. 7 CONCLUION I ths work we have roosed a varat of the NGA- II algorthm whch has a lower rutme comlexty. The lower comlexty s acheved by tradg off sace agast tme the o-domated sortg stage. A comaratve study betwee the basc NGA II, ad a varat of ths due to Jese ad our mroved algorthm shows that our algorthm erforms better whe oulato szes are larger tha 000. We are curretly workg o alyg ths mroved algorthm to other stadard test roblems, as well as other datasets.

5 95 ACKNOWLEDGMENT Ths aer s a exteded verso of our aer A Tme- Effcet Varat of the No-Domated ortg Geetc Algorthm, whch has bee acceted for oral resetato at the Frst IFIP Iteratoal Coferece o Boformatcs, to be held at urat, Ida, from 25 th to 28 th March, 200. REFERENCE [] J. Adersso, "A urvey of Multobectve Otmzato Egeerg Desg," Techcal reort LTH-IKP-R-097, Det of Mechacal Egg, Lkg Uversty, wede, 2000,. 34. [2] J. Kowles ad D. Core, The Pareto archved evoluto strategy: A ew basele algorthm for multobectve otmzato, Proceedgs of the 999 Cogress o Evolutoary Comutato. Pscataway, NJ: IEEE Press, 999, [3] E. Ztzler ad L. Thele, Multobectve otmzato usg evolutoary algorthms A comaratve case study, Parallel Problem olvg From Nature, V, A. E. Ebe, T. Bäck, M. choeauer, ad H.-P. chwefel, Eds. Berl, Germay: rger-verlag, 998, [4] K. Deb, A. Prata,. Agarwal, ad T. Meyarva, A Fast ad Eltst Mult-obectve Geetc Algorthm: NGA-II, IEEE Tras. Evol. Com., vol. 6, o. 2, Ar. 2002, [5] H. Fag, Q. Wag, Y. Tu, & M.F. Horstemeye, "A effcet o-domated sortg method for evolutoary algorthms," IEEE Tras. o Evolutoary Comutato, Vol. 6, Issue 3, Fall 2008, , [6] K. D. Tra, "A Imroved No-domated ortg Geetc Algorthm-II (ANGA-II) wth adatable arameters," Itl. Jour. of Itellget ystems Techologes ad Alcatos, Vol. 7, No. 4, et 2009, (23), Iderscece, [7] M. T. Jese, "Reducg the ru-tme comlexty of multobectve EAs: The NGA-II ad other algorthms," IEEE Trasactos o Evolutoary Comutato, 7:502-55, [8] T.R. Golub, D.K. lom, P. Tamayo, C. Huard, M. Gaasebeek, J.P. Mesrov, H. Coller, M.L. Loh, J.R. Dowg, M.A. Calgur, C.D. Bloomfeld, ad E.. Lader, Molecular Classfcato of Cacer: Class Dscovery ad Class Predcto by Gee Exresso Motorg, cece, vol. 286, 999, [9] D. Parrott, L. Xaodog, V. Ceselsk, "Mult-obectve techues geetc rogrammg for evolvg classfers," The 2005 IEEE Cogress o Evolutoary Comutato, Vol. 2,. 4-48, [0] K. Deb, Mult-obectve Otmzato usg Evolutoary Algorthms, Wley, Chchester, UK, 200. [] C. M. Foseca ad P. J. Flemg, Geetc algorthms for multobectve otmzato: Formulato, dscusso ad geeralzato, Proceedgs of the Ffth Iteratoal Coferece o Geetc Algorthms,. Forrest, Ed. a Mateo, CA: Morga Kauffma, 993, [2] N. rvas ad K. Deb, Multobectve fucto otmzato usg odomated sortg geetc algorthms, Evol. Comut., vol. 2, o. 3, , Fall 995. [3] P. K. Kumar, harath., R. G. D'ouza, K. Chadra ekara, "Memetc NGA A Mult-Obectve Geetc Algorthm for Classfcato of Mcroarray Data," adcom, , 5th Iteratoal Coferece o Advaced Comutg ad Commucatos (ADCOM 2007), IEEE Comuter ocety, [4] K. Deb ad A.R. Reddy, Classfcato of Two-class Cacer Data Relably Usg Evolutoary Algorthms, Publ. of Kaur Geetc Algorthms Lab., Ida, KaGAL Reort No , Ro G. L. D ouza s a Research cholar at the Deartmet of Comuter Egeerg at Natoal Isttute of Techology Karataka, Ida. He s curretly o sabbatcal leave from t Joseh Egeerg College, Magalore. Hs research terests clude oft Comutg, Comuter Networks, ad Boformatcs. He s a member of IEEE ad IEEE Comutatoal Itellgece ocety. Dr. K. Chadra ekara s a Professor of Comuter Egeerg at Natoal Isttute of Techology Karataka, Ida. Hs research cludes Comuter Networks, Deedable Network / Dstrbuted comutg, Autoomc comutg ad Commuty Iformatcs. He has 20 years of teachg ad research ad oe year Idustry exerece. He has ublshed more tha 86 ublcatos Iteratoal ad Natoal roceedgs ad authored two books. He was the Orgazg Char of 4th Iteratoal Coferece ADCOM 2006, Iteratoal ymosum o Ad Hoc ad Ubutous Comutg IAHUC'06. He has served as a member of Program Commttee varous Iteratoal cofereces ad also revewer may Jourals. He has suervsed sosored roects ad IT cosultat to some cororates ths rego of Ida. Dr. A. Kadasamy s a Professor at the Deartmet of Mathematcal ad Comutatoal ceces, Natoal Isttute of Techology Karataka, Ida. I the recet ast, he has bee a Vstg Faculty at IE, chool of Egeerg & Techology, Asa Isttute of Techology, Thalad. Hs research exerece sas over 2 years ad he has a teachg exerece of more tha 6 years. Hs research terests clude Comutatoal Techues ad Algorthms, Comutatoal Flud Dyamcs, Otmzato Techues, tochastc Processes, Geetc Algorthms.