A note on Markovian modelling of evolutionaryalgorithms

Pedro Paulo Balhi de Oliveira

Univer sidade do Vale do Paraiha (UNIVAP)Institu to dI' Pesquisa I' Desenvolvimento (IP&D). Av . Shishirna Hifumi Urbanova12244-000 , São Jos é dos Campos, SP, Brazil

Abs tr.act, Various analyses of evolutiouary met.hod s (primarily genetic algorithrns) have beenporformed wit.h Markov chain models , but have systematically left out t he fundamental conceptof I,IH' luln]lfi/1f1/f8s oI genefir operai IJ1·S. Although this not.ion is an es t ab lished fact. in biologicalovolut.iou , on ly a fract.ion of the efforts in evolu tionary computation - which only rarely includep;l'lIdic algoril.hll1s - has IWl'n s- nsit.ive t.o accounti ng for it. in art.ificial syst.ems. Here, weruis» awan-n ess ab out t.he hiological facet of that feature, as well as its syst.ematic neglect in»volut.ionary compu tation , if not en t.ire ly in t.IH' applications , at least in the theoretical effor t soI' iuod elling .

Resumo. Várias aná lises dI' métodos evolu tivos (principalmente algoritmos genéticos) temsido realizadas com mod elos de cad eias de Markov , m as , sistemat icamente, tem deixado de fora oaspecto fund amental d a adapt.at ividadc dos operadores genét icos. Apesar de es te conceito ser umfa.1.o j;í.estabelecido no con texto da evoluç ão biológica , apenas uma pequena parcela dos esforços1'1'11 coruput.açá o evolu ti va - qu e raramente incluem os algoritmos genéticos - tem o levado emco ns ide racâ o . Este a rt.igo visa chamar atençâo pa ra a realidade biol ógica dessa no ção , e parao SPll dl'spr ,·zo s ist.eru át. ico em computa ção evolutiva , se não integralmente na'> aplicações, pelomouos 1111'; esforces t.eóricos dI" modelamento .

1 Iuta-oduction

A uumber of well-known iterat.ive dynamical sys-t.ems a r!"said t.o 1)(' MIl1'lw'I'Úm, un-an ing I.hat t.heirst.at.« in one ,;fpp dpppnds only- upon its st.at.e int.1H' pJ'{' rl'ding st.ep. Those tli at a re discrete definea MIl7'!.:I)'/1 ch.aiu, a suh-r lass t.hat lias been widelys l.lId i.·d iu niauy í.lu-oret.ical a ud pr act .ical asp ecf.s(';f'l' (h ann an HJ7H] and [Oppeuheim cf al. HI77].fOI' inst.anc .-).

Evolut.ion ary co ru pu t.aí.iou t.ochniques , rypifiedhy ).!;1'1 H't.ir algorit.hms , a lso cons ti f.ute an exa rnpleo f Markovian sYSt.I'Hl, wlwr l' it.s st.at.l' is dp.filWd hy1.111' s t.a.1.p oI' I.lw pOJlnlation a t. issu f'. Bnt. whil f'val'io lll' Hllillysps of pvolu t.ionary llwt.h ods (primar-ily p;"IIPI.ic al gorit.h rns) hav l' hf'f'u pprfornwd wi t.hMarkov dl aill IIlOdds , 1.I\('y Imvp sys t.Plllat.irally ll'ft.0111. t.l1 " fUlldall wlIl.al f" al .lII't' of t.11t' Ildll]lf i-l1f7l t'ss oIgnufif O]lITIl f 111'S, t.hat. is , t. lw fact t.hat t.IH' pr oh- .a,hilil.ips ass()cial.pd wit.h t.lw variou s gpnl't.ic 01 )1'1'-

ators Illay rhallgl' a long t.hp. sparrh I)J'OCPss . Andas far as prar.1.icp is concl' l'llpd , f'Vl'n t.hollgh t.hisf"';l.I. lIl'l' is a n·alit.y ill hiologi ral pvolnl.ion , only afrac l.i on of t.lw Plforts in f'vo ln t.ionary wmpnt.;Üion_. whirh on ly rarl'ly i'url lldp gf'lwtir -lias IlPpn spnsitivp 1.0 arco llnt ing for it. in a rt ifirial

systern s.The a im of this paper is to raise awareness

about t.he established fact of the adaptiveness of ge-net.ic op erators in bio logical rea lity, and its system-atic neglect., if not entirely in the applications, atleas t in t.he theoretical effor ts of modelling. Firstly,th e pape-r describes a few fac ts fram evolutionary!-\,'ndics , in order t.o point at t.he biological roots oft.he adaptiven ess feature. Then it. go es on to mapt.he t.ypl' of evo lu ti o na ry sys tem that comes out of. l.lw la tter on t.o s t.anda rd knowledge of Markoviansysterns: in doing so , a sketch is made of the direc- .t.ion 1.0 hf' followe.dwh"n analysing th" adapt.ivenessof gl'n l'ti c operat.ors from a Markovian standpoint..It. is beyond present. pllrposes t he modeUing it.self ofSOllW I'vo lu t.io na ry r.om pnt.ation te d 1l1iq ue accord-ing 1.0 t.lw Markovian syst.em t.hat. will be identified .

2 T he generatioll phase in biologicaltion

This sf'ct.ion summarises well es t.ablished farts , avail-ahlp from st.andard t.ext.books in gen etics, such as

and Snustad HJ84] . Init.ially, it is shownt.hat. biologir.al genet.ir operat.ions are adaptive; then,a charrt.erisation is madp of t.he random nat.ure as-

277

sociat.ed to the operations.

2.1 Adapciveness of genetic operarions

Current underst.anding of evolut.ionary process as-sumes that it' is essentialIy an instance of a "generate-and-test" processo That is, evolut.ion is achievedt.hrough a random generation procedure dependentonly upon the organism and a selection mechanismperformed by the external world. Alt.hough there isa general consensus about therole and importanceof selection for evolution, the character of the gen-eration phase still is the source of a fierce debate .Among the: criticisms to it, there is the questionabout ' t he factors affecting its randomness. Hereare some facts from genetics that' are important inthis respect..

Following the kind of informat.ion a gene can.ode for, they can be considered siructural genes,which are effectively coding for t/1P. generation ofsome polypept.ide .associa ted with some phenotypictrait, ar regulatory 'genes , collectivelly referred tohere as conirol qenes, wliich cede for some indi-rert information associated with one o)' more t.raits .There are different kinds of processes a gene cancontrol. For exarnple, in t.he case of the specificmodel of gene expression in prokaryotes (viruses,bacteria and cyanophyceae algae) called operou ,the control genes can be an operuior, if it directlycontrols t.he expression of structural gt-'JlI'S , a reg-ulaior, if it controls the expression of a regulator,or a prometer, if does not effectively cede for any-thing but serves as a physical si te for a part.icular

to bind to. In addition. illthollJ.!;1I litt.leis known about the genet.ic expressioll fuI' eukary-otes (the superior organisms) it is possible t.o iden-tify rcccptors, which are physical binding sites (sim-ilar to promoters) as well as parts of the joint. ill'-tivity that controIs the expression of the prodnrfors(these being allalogous to strurtural genes), intc-grators (whose products are used by the receptors),and sensors (which the expression of the in-tegrators). -So, the gerÍetic expression is organizedas a complex multilevel control structure throughwhich th,e controlled genes can be turned on, turnedoff or even partially expressed, depending on envi-ronmental situations ar the internaI state of theorganismo

A special cIass of control genes is the one as-sociated with the genetic expression itself, parl.im-larly with the mutation processo AIt.hough most ofthe genes are normalJy.fairly stable (that is, onlyvery rarely do t,hey undergo some mutation pro-cess), some genes exhibit an extremely high muta-tiop. rate. Studies (mainly in eukaryotes) related to,t hese genes lead to the concIusion that, rather t.hanbeing autonomous entities, their relatively lowerstability is due to the action of spec.ial cont.rol genes

called muiaiors. Ali interesting aspect of t.heir ac- ,t.ion is that they are able to change alI kiuds ofruut.at.ion that a gene can suffer, 01' even some par-ticular point mutation . Analogously, cerf.ain genescan show a·very low mutation rate, and again, theiranalysis show that their higher stability is actuallymaintained by special control genes called antimu-tators. The concIusion is that, in eukaryotes (andpossibly also in prokaryotes), the rates of sponta-neous mutations are under genetic control of thecells,

2.2 The status of the randorn nature of adap-riveness

From what we have seen, the generation phase ofevolution can be considered the result of the jointactiori of two genet.ic IHOCf'SSes: the action of t.hegenet.ir (J)lf'l'itt.ors t.hat cont.rol th - generation of a

genome duriug r-product.ion , and t.he act.ion ofsome p;elles (such as rnutators and ant.imu-t.ator) of th e parental gf'nonlPs over t.he act.ion oft.he gf'net.ic operators. Since t.he action of t.he ge-netic operat.ors is a random !HOCPSS anel the act.ionof the cont.rol gen es constrains the randomness ofth e gpnet.ic operators during reproduct.ion (by in-creasing ar decreasing t.he rate of spontaneous mu-t.ations of a particular parenta! gene), t.his is clearlyt.he case of a random process that progressivelyhas its probability distribution funct.ion reshapedby some of its out.comes.

Simpie examples of this sorf of process can bethe proeess of clrawing balIs from a box cont.ain-ing, say, blaek and whit.e balls, with t.he featlll'f'that each drawn ball is noto put. back in t.hf' box;ar, yet., ali dice, in whic.h some of its nuní-bers wOllld be delel,ed. ac.cording t.o some law /0fit.s out.wmes.. In ('I1.l1er case, the probabilit.y distri-but.ion of t.aking a predetermined balI ou llumberwould change along the time; t.he general sitllat.ionis depicted in Figure 1.

The dOllble arrow ill t lI!:' figure stands for t.hecontro/ that. a portion of the out.come exerts overt.he probability distribution of the randam process,although no explieit. cont.ral has t.o actually exist. .. But. what is t.he nature uf this resulting ran-domness in biological evolllt.ion? The following pointsare aI. issue:

1. The probahilit.y distribllt.ion OVf'r t.he space ofpossible genomes is affect.f'd by t.href' fadors :

• I.he oc.c.urrf'nce of th!" genf't.ir operatorsare not. f'qllally likPly. For example, trans-posit.ion réquil'es more st.eps than a pointmutat.ion; ,

• the ollt.comes of the adion of individualgenetie operators do noto ha,-;e a

278

- RANDOM PROCESS

I outcome 1-

II

Figure 1: A randorn process that progressively liasits prohahility distributiou function reshaped bysom e of its out. coiues.

distribution . For example, to generate a(hypothet.ical) gene BBB from gene AAAthrough point mutat.ions is unlikelier t.hangPIlf'rat.ing AAB ; and

• t,hf' ex istence of gf'nps t.hat are able t.ocout.rol t.he action of t.he gen-f.ic opera-tors over ot.her gt'nt's , as do the mutatorsand ant.imut.ators.

2. The fact t.hat t.IH' act.ion of the gt'net.ic opera-t.ors is not associated wjt li " uniforrn dist.ribu-t.ion function (1)<' it. rplatpd to either one par-t.icular op orator OI' to ali of t.hem together ) isjust a consequence of t.h« constra ints imposedby í.h« world ; in ot.her words , since a. uniforrndistribution would IJ!' a very sp ecial situation ,it, is quite natural t.o expect that it.s chance toorcur Iw vpry low .

Sinct' t.1lf' ,,,ti of t.1lt' joint. act,ion af tllt'gt'llt'tir " I,,'rators ran Iw any new gpnOl11f', t.hpspacp of possihlt' out.conws is 1.i7·tu ft lly i njini tcat, pvpry t.his raract,prist,icst,1H' IH'oepss a pot,t'ntial 7'C"'fT8ibi/it:IJ, in the st'nst't,hat., aI. pach gPlwration, t.ht'rp is a non-zeroprohahilit,y t.hat, allY prpcpding dist.rihut,ionachievpd in t.lw pa.<;t. may Iw rpgPIlt'ratf'd (Marko-via.n syst,PrtlS wit.h this feat.nrt' havf' b t't'1I thf'focm; of sppcific con rprIl , as in[OppPltllt'im et a/. H177] , wht're this propprt.y. is sair! f,o bt' dt'fining of a "rlosed syst.t'm") .

1. Sinc!' not. pvpry Ilf'W gf'nome prf'sents neWgf'nf'Sthat, affpct, t.hf' action of t.he gelwt,ic op eratorsin thf' IH'Xt. rt'pro<iuc.t.ion , then \1ot ali the out-conws modify distribution of the randomI)fOCP.';S; in arldit.ion, sinct' those control genest,haf, rnay Iw brought about represento just seg-IIwnt.s of t.ht' out.conws, t.lw art.ion of an out.-conw is not achievf'd in its t'nt.irf' ext,ension_

3 Are evolutionary systems witb adaptiveoperators special at afl? '

While the genetic operators provide the randomnature of the generation phase.in evolution and,ultimately , are the promot.ers of variability, whatthen is the effective role of the genes that controlt.heir action? In order.to address this question, twoapproaches come to mind: the use of traditional-mathernatics associated with stochastic processes;01' , alternatively, a compnt.ational simulation of thesys tem . However , due to insufficient results in bothsituatious, the answer for the question is not yetclear .

On t.h- on e hand, they seern ·tu have a spe-cial meaning for an evolving system, since they can.be thought t.o be really in charge of the choice ofthe evo lutionary pathways effect ivelypoint her e is that, although in the beginning of theevoluti on th e main agent could be the genetic oper-ators, as evolution went on , the control genes wouldprogressively get. hold of the process (including t.hegeneration of the control genes t.hemselves) . On theother hand, however, since t.hey are always suscep-t.ible to change by the genetic operators, it mightbe thought t.hat. the IWt. behaviour of the whole pro-cess would still be based upon the random action ofthe genetic operators; this would entail ihe controlgenes providing j ust a sort of 2nd order levelactionover the same kind of randorn effect.

The fact is that, if it is true that adapt.ive op-erators provide a special nature for random pro-cesses, it is a st.riking observat.ion that the the-ory of geue í.ic algorithms, as presently conceived,does not seern to correctly account for the biologi-cal reality, since it.s conception of randomness hashpt'n pnt.irf'ly based upon a random search withnon-adapt.ivp fpat.ures. In fact,, adaptive geneticoppral,ors has only het'n an intrinsic, fundamentalconrf'rIl t.o t.he (;erlllan school of evolutionary com-putat.ioll, t.hat. of (, '/lo/1tt io71. stmtrgics, (as test.ified,for inst.anct', in t.1lP. recent monograph [Back 1996]).Quit.p si)!;nificant.ly, t.he only few t'fforts I am awareof, t,hat, t.ry t.o incorporat.t' t,hp notion of adaptiveoperat.ors in genet.ic algorithms, have come froml11ellllwrs of the Gerlllan t.radition (see [Back 1991]and [Biirk and Schiitz 1996]) . But. while in cvo/u-tio7l.a7·y p7'ogramming, the third largest school ofthought in f'volutionary cOlllputat.ion, explicit at-tf'mpts have been made by practitioners from wit.hint,he school (a.<; in [Fogel rt ai. 1991]), these effortshavf' been rat.her marginal so far.

4 Towards tbe Markovian analysis of evo-llltionary eomputatioIl tecbniques

algorithms have bf'en analysed by Markovchain models in various sf'ttings. Among these ef-

279

forts , we sho uld cite : steady-st.ate behaviour wit.heit.he r in finite ([VOSf' HHn], [Suzuki U)!):l] and[Suzuki HHH5)) or fin ite population size([Eil)l'n r]. ai. 1991] and [Nix and VOSf'transient-hehaviour wit.h finit.e population 1JJ t.hes pecific co u te x t of funct.i on op t.im isation([Of' Jong cf nl. H1H4] and[OI' Jong and Sp ears UJ96]) ; and var ious other as -pects with fini te populat.ion such as gen etir drift.([Goldbt'rg and Sf'grf's t. l!Hn]) , selection([Uoldherg and Segrest 1987] and [Mafoud 199:l] ),and niching ([Horn UJ!:l:m .

But whil e genetir a lgorith ms hav e had beenapproached as Markov chains from ali those per-spectives , neither evolution strategies nor evolu-t.ionary programming techniques hav- apparentlybeen analy sed as such so faro And th e lat.ter t.f'ch-niques are precisely t.he ones in wh ich th e adapt.ive-ness of gf'lwt.ic op erat.ors has heen addressed with inits comrnunity of pr act.itioners!

From what. we have di scu ssed con cemiug t \11'nat.ur e of the adapti veuess of gf'Il!'t.ic operators andt.1lt'ir as soci at.l'd r all dolllllt' SS, Wf' cn.n t.hf'n providf'a skl't.chy idf'nt.ificat.ion , in t.Pnns of Markov-chainnHHldling, of t'vo ln t,io nary sy st,f'ms t.hat. rf'ly 011 t.!mt.ft'at.II1'f' . Wf' ns f' s t.andard concf'pt.s p rl'sl-'ntNI in t.1l!'lit.t'r at.IIl'I' , hllt. providf' an Ap J-wndix for localf'ncl' .

Sut. I)('forl' , Il't. ns ti rst. o bsl'rvt' t.ha t. t.1l!' fol-lowing s t.;II.f'I'lll'nt.,:; hold fo r t.lw gl'l ll' I'éÜion p ha.'w inl'voln t.ion:

I . Since any nf'W gl' lwrat.io n depends on ly on t.lwIHirt'nt.al gt'nonl l:'S, t.ht' condi t.ional prohabilit.yof I'f'aching any point. in t.hl' f'volnt.iona ry spac l'frOlIl iI givl-'n point. do l's not. dt'pl' llCI on t.he l'VO-lu t.ion ary pat.hway t.aken np tn t his point.;

([Oppl'nl1l'im ri ai. Hl77)) . While these two fea-tures are sufficient to characterise t.he usual Marko-vi an models that ap pear in th e litera t.u re, t.hey arenot. enough to cliaracteris e th e adaptiveness of t.hegenet.ic op erators . In order to fulfil t.hat., ali wehave t.o do is t.o transform th e general scherne inFigure 1 int.o th e equivalent scheme of Figure 2.

In t.he figure , t he cont ro l exerted by some con-trol gl'IWS over t.he acti on of the gen et.ic op érat.orswas mad e explicit so th at t.IH' Sf' gf'nes can be re-gard ed as anot.her out.corn e o f th e sa rne randornIlfOCI'SS. Thus, t.he whole st. ru cture acquires th e.n a-t.ur e of a multivariaic IlfOCl'SS (actually, bivariate)in which t.he t.wo variables involved are corre lutcd(since t.ln- co nt.rol gl'nl'S a re jnst part of the ent. ir egf'noJ1lI') .

111 conc lusion , evolut. iouary systems feat.uringadapti ve gf' lw t.ic. operat.ors can jw rl'g a rdl'd as aninst.ance of a multinariui c, trniporullu h.onioqcnrousMa7'ko7' cluiui untli corrrluird dynawir.'i([OPIH'nlH'illl ri al. UI77)). Al ternat.iv el y, by as-surn ing 1.111' evolu t.ionary search space as an Eu-clidean space, as required hy random walks, onf'm u ld a lso rl'g ard t.hosf' sysÚm s a.<; a 1/I'1tifiva7·Útlf .nlJ1/.-h07l/.0.Qf1/ fO'US mndo7// wnlk mith rll1Trlnied dy-lIn7l/. irs ([K annan 1979)) . Whidwvf'r t.1l!' ca."l' , anumlwr of rl'sult.s wo u ld 1)(' prompt.ly availah lt' framt.1l!' lit.l'r at .u rl' of st.och a..,I.ic syst.l'ms , a nd a lot. of l'f-fort. m ight. 1)(' saVl'd .

I l'x p ress In y grat.it.udl't.o ( :NPq , Br a zil's Cons l'lho Naci on al de Ol'st'II- 'volvinwnt.o C il'n t.ífico I' T I'CIlol ógi co fo r support.-illg t.his pi ecf' of work under t.he res t'arch g rant.No. ;lUU465/95-5 .

R eferellees

[Back IBUri] T . Back . E71o luii o7/. lt7':tj Itl,q ll1·ith7l/.s inih f07':tj und 111·ud in ·. Oxford trni vf'rsit.y Prl'ss ,Nl'w Yo rk , IHB6 .

2. Rq lJ'Odn ct.ioll oc cu rs in disCl'l'I f' ti n lf': [Nix and VOSI' A .E. Nix and M .O . VOSI'.:l . Alt.h ough it. is possi bl l' t.o rt'ach any pa r t.icul ar "Morll' ling gPlwt.ic algorit.hms wit.h Markov

point. in t.ht' t'volu t.ionary s pacl' at. f'ach rI'IHO- ch ains" , A nnllls of M Ilthl'7I/.ll tirs ll7/.d A 7'iifi ci lllduct.ion , t.lw f'asi nl'ss t.o accom plis h t.ha t. will !n t rl1ig fna , 5 :7H-R8, l H!-J2 .dl·p t'lHi on t.hf'ir p an 'Ilt.a.1 gl:'nomf's .Sinc" t \11'parl'l)t.al gf'nomps a rl' cont.inllo lls ly chang..d al ong [BÚk 1!:lU1] T . Back . "Se lf-adapt.at.ion in gt'n et.ict.1l!' l'v olllt.lon , tl w probabilit.y t.o reach a SIW- algorit.hms" . In : F ..J. VareIa a nd P . Bonrginf',riti'c point. .in tl1l' f'volllt.ionary spacl' will ht' a l'd it.ors. p1'IJacdings of Ih. e Fá 'st E1l1'lJ]lt'ltnfU;lct.ion of t.illl l'· . (.'0 71jáenCI' on A7'l ic fil'iul Lifl ': TO'llJ I(.7·d Il P7'!tf-

, lirf of Ani lJ1w 7I/.ous ,"·ys/r 7l/.s, MIT P rl'ss , ( :a lll-4 . By a s irn ila r a l'gu1I1t'nt., t.1l!' probahi li t.y t.o j oint.1y hr idgl' , MA , 2():l-:n1, 1!)!J1.ach il'v l' a ny two poi nt.s iu 1.l1l' f'vo lnt.ionary spacl'in d ifferl'nt ti nll's , will abo Iw d'·PPlld l'lIt. ont.iJiw, ill' t.hi s ca.';!' , o n t.lw t.wo t.inw vari ahl l'sI.hat. ch arac t.t'r isl' f'ar h of t.1](-' t.wo poill t.s .

. TI1l' Markov ch ain implil'd fram t.lw firs t. t.wost.at.f'mf'\1t.s is refill!'d hy st.at.t'mel11.s a lld 1 so ast.o as st'rt. t.hat. it. has t.1l!' n at11J'1' of a 1f111]lIl1'ull:1J h.o-wo,qfn rou s st.ocha.stic p ro cl'SS

[Biick and Sdliitz 19Hr5] T . a nd M.. "In t.f'll igf' ntIlll.lt.at.ioll rat,l' cont.rol in canollical gl-'Ill'ti c al-gorit.h rns" . In : Z.W. Ras and M. Michalewicz ,t'd it.ors . FOll1ldalion of !n t rll igenl S'llst f7l/.S 9th

280

Intcrnat ionol Symposi1t1l/., Springer , Berl in,1!)8-l()7, l!-l!J(L

[De Jong and Sp ears HHJ(j] K.A . De Jong andW.M. Spears. "Analyziug (;As using markovmodels wit.h semanti ca lly ordered .states andlurnped st.ates" . In : R .h: . Belew aud M. Vose,editors, Procrrdinçs of tlic 4th Wo7"!.:shop onih.c Foun.dations o] Gcnct ic A lqoritlinis , Mor-

Kauffman , 1!)!)(j.

J oug fi ai. 1!)!)4] K .A . De J onga ud W .M . Sp ears and D.F . Gordon. "Usi ngMnrkov cliaius 1.0 a nal yze UAFOs". P1'OC('ed-ill.IJs of l lu : :17·d Worksh.0]1 01/. Foundutions ofGcuciir A lçoritlnu», 11!)-I:n , 1994.

[Eiben ri ai. I!)Hl] A.E . Eiben: E.H . Aarts audK.M . Vau Hee. "G lob al conv ergence of genet.iralgorithms: an infinite Markov chain anal-ysis" . In : Han s-Panl Schweffel , editor . P1'O-.ccrdin.q» o] th e r ' lni , W01'kshO]1 on ParallclProblcui f1"l111/. Naturc , Lect.ure Notes inCompu t.er Scien ce 4H(j , Springer- Verlag , Hei-

1!IH 1.

ci 111. 1$1 $1 1] D.B . Fogel, . L..J. Fogel anel.l .W . At.mar . "Me t.a-evo lu t. ioua ry prograrn-ming" . In: R..R. . Chen, edito r. Proceeduujs o]::!5th A si lonuir ( :01/fr1'('1/.((, on Signals, S:lJst rmsan.d ( :om]l1ltr7"s, Pacifir Greve, CA, !)40-!)4!),IDlll.

[( ; ;lI'duer aud Suust.ad HlH4]E..I. (;,m!Iwr and -D .P. Snust.ad . Prin.cipl rs ofgrllt"tics. John Wiley <v. SClIlS , New York , 7t.h('(\., I!)Ht\ .

a nel 1987] D.E . a ndP. "Fin it.e Mar kov ehain analysi s of

algo rit.hllls" . P1'OCI"rdillgs of thr ::!nd/1/1r1'1/.1lt io1/.1I1 (:o1/f('1'c'nn' O1/. (;n/.dÚ' AIgo-7"illlms , 1--8, 1987 .

[Uak('n I !178] 11 . Hakpll. ,""':1/11('1'.'1C'ii I'S : li1/. i1l 11'lJdur-/Úm . Spri I!J7H.

[Botfuwist,er and Bii,rk F . HotfnlPis-tn and T . Bii.ek . "( ; (,IH't.ie In :F .J . Varela a nel P. Bourgine, edi t.or s. P1'Ocrrd-illgs of th c FiTsl EU7"o]lCC/.7/. (.'o1/fr1't'n cr on A1·ti-fÚ'Ú/ I LiJc- : TOW(/'1'd a l'mctir r of Autollo1/WUS......ystrms, MIT PreSf:: , ( :amhridgp, MA, 227-

l!HII.

[Bom .1. Hortl . "F in it.e Markov ehain analy-sif:: of gl'net.ie a lgorit.hms wit.h nieh ing" . Pro-cCfdi7/.gs of 111. 1' 5th- Intc1'1wtio7/.al (.'cmfC1'C1 /·rc'O7/. GC1/.rl ic AIgorithms, Morgan Kaufmann ,San Mat.eo , CA , 110-117, 1!)!):J .

[Kannan 1!)7!)] D . Kaunan . A7/. introduct ion tosioch.asiic processes. North-Holland , NewYork,

[Mafoud I!)!):!] S. Mafoud . "F inite Markov chainmodels of an al tern ative selection strategyfor gpnet.ie algorithms", (lontple» Syst ems,7(2) :1!)!)-170 , 19!):! .

[Oppenheim r.t ai. 1!)77] I. Oppenheim;K.E. Shuler and G.H . Weiss, editors . Stoch.as-i ic processes in ehemical ]lhysics : ihc Mast erEqu uium: MIT Press , Cambridge, MA , 1977.

[Suzuki .J . Suzuki. "A Markov chain a na ly-sis of a genet.ic algorithrn" . Pro ceedin.qs of th e5th Int crnational Conjcrrnce on. Gcn etic A I-qorittnns,Morgan Kaufmann, San Mateo , CA,14(j-l!):J , 199:3 .

[Suzuki HJ!)(5] J. Suzuki . "Further results on theMarkov chain modei of and their app li-cat.ion 1.0 SA-like strategy". In: R.K. Belewand M.Vose, editors. Procccdin.qs of ih e 4thW01'k8hop on th. e Foundaiions of Genciic A I-qoriiluns, Morgan K auffman, 1!)!)6 .

[Vose H)!)2] M.D . Vose. "Mo de ling rsim ple gen et.iealgorithms" , Procccdin.qs.o] tlie Foundaiions ofGcnciic A lqoritlims W07'ksho]J , (j::l-74 , l!)n .

Appendix

Defiu it ion 1 - Markov Process([OppPIl!wim rt ai. 1!)77, I!)]):

A sto ch.asiic ]l1'OC('SS in uihich. ili c ualuc of ih cran.doni uariablc X (t, .) depcn.ds O1/.ly ou ih.c ualucX(I"_I), i .L , ]J1'e'I /Úms '/l alu (' s X(I) for f; < t"_1 donol (/.,!f('rt X (t ,,) . In ma.the1//.atieal te1'1l/.s:'IV,. (;/:, . , /.,'; ;1:,' -1, t" _I ; ;l:,._ :!, /."_:! . . . ;1: I , ti) ='/Il ( :I:,., 1,';:1:,'_1 ,/"_ 1), Th is mralls tlwt th- e condi-lio1/.u.1 Jl1'Obab ility dcnsily 'IV,. of X (I) which has avalu(' bctween ;/:,. a1/.d :I:,. + at ti1//.C t,. de]lcndsonly 011 lhe value of X(t) at time t' ·_I .

DefinÍtÍon 2 - Markov ChaÍn([Oppenheim ri a.1. 1977, Hi]):

A Ma1'kO'l' ]l1'ocess wh- e1'(: th (' unils of time a7'edisf1 ·rte multi]lles of u. single 1tnit T (i .e., t., = sT)and that th r7'e is at 1//.ost a dcnumcrablr infinity of1!(t/urs wh-ich X (t) can aSS1t1//.C.

Defin ÍtÍOll 3 - Tem p ol'ally H omoge lleons P l'O-cess ([Oppenheim et ai. 1!)77, page (56]) :

A stochastic ]ITOeCSS in whi ch th e conditionalp1'O/mbility W(1I , t/nl , tIl is afunction oft-t'l only,the jo int ]17'obability P2(111 , tI ;11., t) is a fun ction ofboth tI and t and th e singlct ]l1'Obab ildy P( 11. , t) is afun rt ion of t .

281

Definition 4 - Mulüivariate Process with COl'-relatod D yu amics (hased upon re lat .cd definit.ionp;ivpll in [Oppeuheim c! ai. 1977, paw-' llfij) :

lt is a stocliast i c ]I7'orcs s uihos« 1'It1/.d01/l ';'a7'i-ulilc is arllwl/:I} (L ran.doui r- di uicnsion al vccior soI/UlI ih.c cou. di tunutl prolmlnlit ir» IV (1/.; , t j-m; , s ] 1/.1'('

11 ut indrpriid cn I an.d lh.us , ih.c ! ollo 'IIJ in,q 1'3p'1TssÚmdoes 11 oi h.old:

I '

1!v'!I ' l(N, I/!VI, s) =rr 1,,\/(1/; ,1/'1// ; , s);=1

utlu r« I,V = IV I aud N an.d Ivl 1/.1'(' nrciors ,

Defirrit.iou õ - Master Equa t .iou ([OPPl'll lil'illl 1'1 ai , I!In,chapt.or :1] aud [11 akou 1 pap;(' HHj) :

'1'11.1 ' cquiüio n. lh.al. drs rrilu: tli r tl'1117l1J1'II1 dl'-·I' l'i071111.I'1/.I o] [tnnl . an.d ('O 1/.di IÚI1Iai probatnlit ic» o]sl.och asl.u: Jl7 'O f/ 'SSI'S; dl'J/I'1/.di1/.g '11.7/1111 tlir 71,a1'11. 7'1' o]111.1' st.atc s7/a fl ', it fll1I IJ I' «ctuulh; ci th rr a se! oIdi;{jál'1/.l ial- di;ffr.1'I'1/.rr 07' i n! rq1'lJ-di.lfl'I 'l'nl ial I'lt ua-l.urus [in : 111.1' Jl1'011II.hilil i l'S,

D efirri t ion 6 - Nou-Homogeueous Ranclom Walk([K all nrtll I pap;<' :W]) :

'1'11.1' sl och.así.u: 7/7'OU'SS X = {X" ' 1/. :::: n} d/,-,pll nl hy: X" = XII + .l , + + ' " + ./" ,11 :::: I111111'1'1':

• XII is a .!i,udllCI'IIJ7' i n. Ih c d- diuunsion li I /,'11 -clidtan SJl(L('C R« :

• {./" , 1/. :::: I} 7S a scquruc: oI i dr ti li calht dis-triln ücd 7'1/7/,do 111 1'1/.7-i ablcS tluü can. t.akc '/III.!UI'Stu Hd ,' tuul

• 111.1' jU1II7/s./" are 1/.01 ÚI/ '1/.l i('(/.Il,lj dis/l'ih /l/l'd ,

- RANDOM PROCESS

!ou t.come-I ou tcome-z

(ent.ire I!;euome ) [regu lal.ory gen es t.hata.ffeet. I!;(·net.ic ope-raf.orx]

I. F ig ur« 2 : 1\ rdilwd clia,rar l,l'r isa.t. io ll o f a ran domprorr -ss t.hat. prop;r l'ssivl' ly lias its probahi lit.y disl.ri-but.ion Iun ct.iou n 'sli apl'd hy sOllW of iLs OIlt.COIII('S,

282