A Probabilistic Analysis of a Simplified Biogeography-Based Optimization Algorithm

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Electrical Engineering & Computer Science

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Summer 2011

A Probabilistic Analysis of a Simplified

Biogeography-Based Optimization Algorithm

Daniel J. Simon

Cleveland State University, [email protected]

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Simon, Daniel J., "A Probabilistic Analysis of a Simplified Biogeography-Based Optimization Algorithm" (2011).Electrical Engineering & Computer Science Faculty Publications. 4.

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Original Citation

Simon, D. (2011). A probabilistic analysis of a simplified biogeography-based optimization algorithm. Evolutionary Computation, 19(2), 167-188.

A

Probabilistic

Analysis

of

a

Simplified

Biogeography-Based

Optimization

Algorithm

Dan

Simon

DepartmentofElectricalandComputerEngineering,ClevelandStateUniversity, Cleveland,Ohio,44115

Abstract

Biogeography-basedoptimization(BBO)isapopulation-basedevolutionaryalgorithm (EA)thatisbasedonthemathematicsofbiogeography.Biogeographyisthestudyof thegeographicaldistributionofbiologicalorganisms.Wepresentasimplifiedversion ofBBOandperformanapproximateanalysisoftheBBOpopulationusingprobability theory.Ouranalysisprovides approximatevaluesfor theexpected numberofgen­ erationsbeforethepopulation’sbestsolutionimproves,andtheexpectedamountof improvement.Theseexpectedvaluesarefunctionsofthepopulationsize.Wequantify three behaviorsas thepopulationsizeincreases:first,weseethatthebestsolution intheinitialrandomlygeneratedpopulationimproves; second,weseethattheex­ pectednumberofgenerationsbeforeimprovementincreases;andthird,weseethatthe expectedamountofimprovementdecreases.

Keywords

Biogeography-basedoptimization,evolutionaryalgorithms,Markovanalysis.

1

Introduction

1.1 Biogeography-BasedOptimization

Mathematicalmodelsofbiogeographydescribethemigration,speciation,andextinc­ tion ofspecies(MacArthurand Wilson,1967;Lomolino etal.,2009).Species migrate betweenislands.Islandsthatarewellsuitedasresidencesforbiologicalspeciesaresaid tobehighlyhabitable.Featuresthatcorrelatewithhabitabilityincludefactorssuchas rainfall,diversityofvegetation,diversityoftopographicfeatures,landarea,andtem­ perature.Islandsthatarehighlyhabitabletendtohavemanyspecies,whilethosethat arenotveryhabitablehavefewspecies.

Highlyhabitableislandshaveahighemigrationrate.Emigrationoccursasanimals rideflotsamtoneighboringislands,orswimtoneighboringislands.Inthecaseofinsects andbirds,emigrationcanoccurbyflyingorbeingcarriedbythewind.Thereasonthat emigrationoccursfromhabitableislandsisduetotheaccumulationofrandomeffects ontheirlargepopulations.

Highly habitable islandshave a low immigration rate because they are already nearlysaturatedwith species,andthereforecannoteasilysupportnewspecies.Con­ versely, islandsthatare nothabitablehavea high immigrationrate becauseof their sparsepopulations,whichallowsroomformanyadditionalspecies.Theimmigration ofnewspeciestoislandsmightraisethehabitabilityofthoseislandsbecausehabitability isproportionaltobiologicaldiversity.

Figure 1: Illustrationof twocandidateproblemsolutionsusingsymmetricimmigra­ tion and emigration curves. S₁ is a relatively poor solution while S₂ is a relatively goodsolution.S₁ hasahighimmigrationandalowemigrationrate,andS₂hasalow immigrationandahighemigrationrate.

The application of biogeography to optimization was first presented in Simon (2008) andis anexample ofhow a naturalprocess canbe modeledto solve general optimization problems.Biogeography isnature’s way ofdistributing species,and is analogoustogeneralproblemsolving.Supposethatwehavesomeproblem,andthat wealsohaveacertainnumberofcandidatesolutions.Agoodsolutionisanalogousto a highlyhabitableisland,anda poorsolutionisanalogoustoalesshabitableisland. Goodsolutionsaremore likelytosharetheirfeatureswithothersolutions,and poor solutionsaremorelikelytoacceptsharedfeaturesfromothersolutions.Thisapproach to problemsolvingiscalledbiogeography-based optimization(BBO).Aswith every other evolutionaryalgorithm (EA), wemightalso incorporate mutationand elitism, althoughthesearenotessentialfeaturesofBBO.

Figure1illustratesmigrationmodels.Theimmigrationrateλ andtheemigration rateμ ofasolutionarefunctionsofitsfitness.Theimmigrationcurveshowsthatthe least fit solution has the largest immigrationrate and smallest emigration rate. The mostfitsolutionhasthesmallestimmigrationrateandthelargestemigrationrate.S₁

inFigure1representsapoorsolutionwhileS₂representsagoodsolution.

WehaveshownthemigrationcurvesinFigure1asstraightlines,butingeneralthey mightbemorecomplicatedcurves.Wealsoassumethateachsolutionhasidenticalmi­ grationcurves,butingeneralthemigrationcurvescouldbeadjustedonaper-solution basis.Nevertheless,thesimplemodelshowninFigure1givesageneraldescriptionof migration.

ThereareseveraldifferentwaystoimplementthedetailsofBBO.Figure2outlines theoriginalBBOalgorithm (Simon,2008),whichiscalledpartial immigration-based BBO. Inthis approach,for eachfeature ineach solution,weprobabilistically decide whetherornottoimmigrate(i.e.,replacethatsolutionfeature).Ifimmigrationisselected for a given feature, then the emigrating solution is probabilistically selected based onfitness (e.g.,usingroulettewheelselection). Migrationandmutationoftheentire

Figure 2: One generation of a BBO algorithm, where y is the entire population of solutions,yk isthekthsolution,andyk(s) is thesthfeatureofyk.

Table 1:ComparisonbetweenGAandBBO.

GA BBO

Setofsolutions Population Archipelago Solution Chromosome Island Solutionfeature Allele Species Recombination Crossover Migration

populationtakeplacebeforeanyofthesolutionsarereplaced,whichrequirestheuse ofthetemporarypopulationz inFigure2.

TheBBOmigrationstrategyissimilartotheglobalrecombinationapproachofevo­ lutionarystrategies(ES;B¨ack,1996;B¨acketal.,1997),inwhichmanyparentscancon­ tributetoasingleoffspring.GlobalrecombinationhasalsobeenadaptedtoGAs(Eiben, 2000,2003),butBBOdiffersfromGAsinoneimportantaspect.InGAs,recombination isusedtocreatenewsolutions,whileinBBO,migrationisusedtochangeexistingsolu­ tions.GlobalrecombinationinESisareproductiveprocesswhichcreatesnewsolutions, whileBBOmigrationisanadaptiveprocessthatmodifiesexistingsolutions.Aquanti­ tativecomparisonbetweenBBOandotherEAsisincludedinSimon(2008),where14 benchmarkfunctions,eachwith20dimensions,werestudied.ItwasshownthatBBO andthestudGA(sonamedforitsselectionofthebestindividualinthepopulationas oneoftheparentsforeverycrossoveroperation)performedthebestoutofeightEAs.

TheEAwhichismostlikeBBOisaGAwithglobaluniformrecombination.How­ ever,therearestilldifferencesbetweenthetwo.Aconceptualcomparisonandcontrast betweenGAsandBBOisshowninTable1and isdiscussedinmoredetail inSimon etal.(2009,2010).NoteinTable1thatweproposethetermarchipelagotorefertoaset ofcandidatesolutionsinBBO.However,inthispaper,inordertoretainfamiliarEA terminology,weusethetermpopulationtorefertoasetofcandidatesolutionsinBBO.

1.2 PaperGoalsandOverview

The goalsofthis paper aretwofold. Ourfirstgoal istopresent asimplified version of BBO,which we do in Section 2. Our second goal is to use probability theory to

2

analyzesomeofthemathematicalpropertiesofsimplifiedBBO(SBBO),whichwedo inthefollowingsections. Section3derives theprobabilitythatthebestindividualin theSBBOpopulationimprovesfromonegenerationtothenext.Section4derives an approximateMarkovtransitionmatrixforSBBO.Section5derivestheexpectedamount ofimprovementinthebestindividualoveroneSBBOgeneration.Wegivesupporting simulation results in Section6, and provide concluding remarks and directions for futureworkinSection7.

A more traditional Markov analysis, along the lines of that discussed for GAs inNix andVose(1992),Davis andPrincipe(1993),and ReevesandRowe(2003),has been presented for thestandard BBOalgorithm in Simon et al.(2009,2010).In that approach,eachMarkovstateisapopulationdistribution.Thelimitationsoftraditional MarkovanalysesofEAsisthatthetransitionmatrixgrowsfactoriallywiththeproblem size.EachdimensionofthetraditionalMarkovmatrixis(n + N − 1)-choose-N,where

n isthecardinalityof thesearchspaceandN isthepopulationsize.Asanexample, a 10-bit optimizationproblem(n = 1,024)with apopulation sizeN = 10resultsina transitionmatrix thathas ontheorderof1024 elements!TraditionalMarkov analysis providesinsightsintoverysimpleproblems,butbecauseofcomputationallimitations cannotbeusedforrealisticproblems.

InthispaperweavoidthecurseofdimensionalitybydefiningtheMarkovstates differentlythanthosethataretraditionallyused.WepartitiontheMarkovstatespace depending on which individual is mostfit and which individual has been selected for immigration.Each dimension of our Markov matrix is only N2, is independent of thesearchspacecardinality,andiseven independentofwhetherthesearchspace iscontinuousordiscrete.ThelimitationofourapproachisthatourMarkov analysis is only approximate, but the advantage is that we can deal with realisticallysized problems.

A

Simplified

BBO

Algorithm

An SBBOalgorithm canbe formulatedbyalwaysusingthebestsolutionastheemi­ gratingisland,andusinganyothersolutionwithequallikelihoodastheimmigrating island.Theimmigratingislandischosenfromauniformprobabilitydistributionand isthusindependentoffitness.ThisisconceptuallysimilartothestudGAinwhichthe bestchromosomeisalwayschosenasoneoftheparents,andtheotherparentischosen usingstandardfitness-basedselection(KhatibandFleming,1998;Silvaetal.,2005).In SBBO,themigrationcurvesofFigure1aremodifiedtothoseshowninFigure3.One generationofSBBOcanbedescribedasfollows:

1. Findthefittestsolution.Callthissolutionxi. 2. Pickarandomsolutionfeatures.

3. Selecttheimmigratingislandxj fromauniformprobabilitydistribution(j = i). 4. xj (s)← xi(s).

WithSBBOwedonotneedtoevaluateallfitnessvalueseachgeneration.Weonly needtokeeptrackofthemostfitisland.Aftereachgeneration,wecomparethenew fitnessoftheimmigratingislandwiththepreviousbestsolution.Thisrequiresonlyone fitnessevaluationpergeneration.

3

Figure 3: IllustrationofSBBOmigrationcurvesinann-islandpopulation.Allsolutions havean equalprobabilityofimmigrationexceptforthemostfitsolution,whichhas a zero probability of immigration. The most fit solution has a 100% probability of emigration.

Probability

Analysis

Fromthispointon,weassumethatwearetryingtosolveaminimizationproblem.High fitnesscorrespondstolowcost.Wetrytominimizecost,whichisalwaysnon-negative, andmaximizefitness.

Supposethatthecostofasolutionxi isdenotedasf (xi).Wewrite

xi =[xi1 · · ·xis ] (1) wheres isthedimensionoftheproblemandalsothenumberoffeaturesineachsolution. Iff (xi)isseparable,thenitcanbewrittenas

f (xi)=f1(xi1)+ · · · +fs(xis) (2) Wemaketheassumptionofseparabilityinordertoallowtractabilityinthefollowing analysis.Iff (xi)isnotseparable,thenEquation(2)willnothold,butitmaystillhold approximately.Theapproximationerrorinourresultsfromthispointonaredirectly correlated withtheapproximationerrorofEquation(2) andare discussedfurtherin Section6.1.

Ifxi isarandomlychosenisland,thenf (xi)isarandomvariablewithmeanf ¯ and varianceσ2_{.Theparametersf¯ andσ canbeapproximatedfromapopulationofislands}

{xi}.Intheabsenceofanyotherinformation,weassumethatthefj (·)functionvalues ontheright-handsideofEquation(2)areindependentidenticallydistributedrandom variables.Thentheexpectedvalueandvarianceofeachfj (·)canbewrittenas

¯ f /s

fj = ¯

2 2

σ_j =σ /s (3)

Nowsupposethatwemigrateasinglesolutionfeaturefromthelowest-costislandtoa randomly-chosenisland.Thecontributionofasinglesolutionfeaturetothecostofthe immigratingislandcanbeapproximatedastherandomvariable

whereweareusingthenotationy ∼ (μ, σ2)toindicatethaty isarandomvariablewith meanμ andvarianceσ2. Ify isauniformrandomvariable,thenitcanalsobewritten asfollows:

f f

y ∼ U[f_min/s − σ 3/s, f_min/s + σ 3/s] (5)

where weare usingthenotation y ∼ U[a, b]to indicatethat y is a uniformrandom variablewhoseprobabilitydensityfunction(pdf)isnonzeroonlyfroma tob (Papoulis andPillai,2002).

Theimmigrating islandcanhaveanyfitness f ∈ [fmin, fmax].Therefore, beforea solutionfeature isreplacedintheimmigratingisland, thatfeaturecontributesa cost thatcanbeapproximatedastherandomvariable

z ∼ (f/s, σ2/s)

f f

∼ U[f/s − σ 3/s, f/s + σ 3/s] (6)

where the secondexpression for z applies if itis uniform.SBBO replacesa random solutionfeatureintheimmigratingislandwitha featurefromtheemigratingisland. Theprobabilitythatthisfeaturereplacementresultsintheimmigratingisland’sfitness becominglowerthantheemigratingisland’sfitnesscanbewrittenas

ps = Prob(z − y > f − fmin)

= Prob(y < z − If) (7) whereIf isthedifferencebetweenthetotalcostoftheimmigratingandemigrating islands before migration.Figure4 pictures thisprobability for twouniform random variables. The probability that thepreviously best island is replaced by a new best islandduetomigrationisequaltotheprobabilitythatarandomvariabletakenfrom thez distribution, minusarandomvariable takenfrom they distribution,isgreater thanf − f_min.

Equation(7)canbewrittenas

zmax z−If

ps = pdf(y)pdf(z)dy dz (8) zmin ymin

However,theupperlimitoftheinnerintegralmustbegreaterthanthelowerlimitof theouterintegral.Thisimpliesthat

z − If > ymin

z > y_min+ If (9)

whichchangesthelowerlimitoftheouterintegralofEquation(8)accordingly.Also, Equation(8)appliestoaspecificimmigratingislandwithfitnessf.Iftheimmigrating island ischosenrandomly,thenthefitnessoftheimmigratingislandcanrangefrom

Figure 4: Sampleprobabilitydistributionfunctionsofanemigratingandimmigrating solutionfeature.Theprobabilitythattheimmigratingislandisbetterthantheemigrat­ ingislandaftermigrationisequaltotheprobabilitythatz − y > f − f_min.

f_mintof_max.TheseideasleadtoamoregeneralexpressionforEquation(8)as

fmax zmax z−f+fmin

ps = pdf(y) pdf(z) pdf(f)dy dz df (10) f_min y_min+f₋f_min y_min

Thepdf’sintheaboveequationareentirelyproblem-dependent.Givenapopulation,

f_minandf_maxareknown.Thepdfoff canbeapproximatedbygeneratingalargeenough populationofsolutions.Similarly,thepdfofy andzcanbeapproximatedbyquantifying thecostcontributionofalargesampleofrandomlyselectedsolutionfeatures.

Equation(10)givestheprobabilitythatthebestindividualinapopulationisre­ placed aftera single generationofSBBO.Thisprobabilityisapproximatelyconstant fromonegenerationtothenext,assumingthatthepopulationdoesnotchangemuch betweengenerations.Sotheprobabilitythatthereisnoimprovementintheglobalbest ofthepopulationaftereachgenerationis(1− ps).Therefore,onaverageweexpectthe bestsolutionofthepopulationtoimproveafterk generations,where

(1− ps)k = 0.5 log(0.5)

k = (11)

log(1− ps)

AstheSBBOpopulationincreases,wehavea betterchanceofbeginningtheop­ timization process with a smaller initial cost. But a larger population also typically causesthedifference(f_max− f_min)toincrease,whichinturncausesps todecrease.This causesk,theexpectednumberofgenerationsbetweenimprovementsinEquation(11), toincrease.Weseethatalargerpopulationsizeresultsinabetterinitialsolutionbuta longertimebetweenimprovementsinSBBO.ThisisillustratedqualitativelyinFigure5. Small populations collapse to a singlesolution morequickly thanlarge popula­ tions.Thisisintuitivelyobviousandisdiscussedquantitativelyinthenextsection.As illustratedinFigure5,improvementsarenotonlymoreoftenforsmallerpopulations, butalsolargerinmagnitude.ThisisexploredquantitativelyinSection5.

4

Figure 5: IllustrationoftheeffectofpopulationsizeonSBBO.Alargepopulationhas a better initial solution, but more generations between improvements, and smaller improvements.Asmallpopulationhasapoorerinitialsolution,butfewergenerations betweenimprovements,andlargerimprovements.

Markov

Analysis

of

the

Simplified

BBO

WedefinetheMarkovstatesofthen-islandSBBOasfollows.

• Statei ∈ [1, n]:Theithislandisthemostfitpopulationmemberandtheimmigrat­ ingislandhasnotyetbeenselected.

• Statek∈ [n+ (i− 1)(n− 1)+ j],wherei ∈ [1, n] andj ∈ [1, n− 1]:Theithisland isthemostfitpopulationmemberandislandmij hasbeenselectedforimmigration, where

j j < i

mij = (12)

j + 1 j ≥ i

Weseethatann-memberpopulation hasn 2 _{states.Asa simple illustrativeexample,} consider a three-member population. The nine Markov states would consist of the following,whereweuseI todenotetheimmigratingisland.

1. ThefirstislandisthemostfitandI hasnotyetbeenselected. 2. ThesecondislandisthemostfitandI hasnotyetbeenselected. 3. ThethirdislandisthemostfitandI hasnotyetbeenselected.

4. Thefirstislandisthemostfitandthesecondislandhasbeenselectedforimmi­ gration.

5. Thefirstislandisthemostfitandthethirdislandhasbeenselectedforimmigra­ tion.

6. Thesecondislandisthemostfitandthefirstislandhasbeenselectedforimmi­ gration.

7. Thesecondislandisthemostfitandthethirdislandhasbeenselectedforimmi­ gration.

8. Thethirdislandisthemostfitandthefirstislandhasbeenselectedforimmigra­ tion.

9. Thethirdislandisthemostfitandthesecondislandhasbeenselectedforimmi­ gration.

TheMarkovtransitionprobabilitiesforthegeneralSBBOalgorithmcanbesummarized asfollows.

• If theSBBO isin statei ∈[1, n], then wehave an equallylikely probability of transitioning to staten+i(n−1)+j for each j ∈[1, n−1]. The probability of transitioningtoeachofthesestatesis1/(n−1).

• IftheSBBOisinstaten+i(n−1)+j forsomei ∈[1, n]andsomej ∈[1, n−1], then we have a probability ps of transitioning to state mij, and a probability (1−ps)of transitioningto statei.Theprobability ps isgiven inEquation (10) andistheprobabilitythatmigrationtoislandmij resultsinthatislandbecoming morefitthanislandi,whichistheemigratingisland.Theprobability(1−ps) is theprobabilitythataftermigrationtheemigratingislandisstillbetterthanthe immigratingisland.

ThestatetransitionmatrixisdenotedasQ,whereQij istheprobabilityoftransitioning fromstatej tostatei.InordertowriteQ,wefirstintroducesomeauxiliaryvariables.

1 ₍n-1)×1 Ln-1= [ 1 · · · 1 ]T ∈R n−1 ⎡ ⎤ ps ⎢ ⎥ ⎢ . . ⎥ ⎢ . _⎥ ⎢ ⎥ ⎢ ps ⎥ ⎢ ⎥ ⎢ ⎥ Pi = _⎢⎢(1−ps) · · · (1−ps) (1−ps) · · · (1−ps)⎥_⎥←ithrow (13) ⎢ ps ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ._. ⎥ ⎣ . ⎦ ps

wheretheunspecifiedelementsofPi arezero.Pi isobtainedbytakingthe(n−1)×(n−

1)identitymatrix,multiplyingitbyps,andtheninsertingastheithrowavectorwhich haseachelementequalto(1−ps).Pi isthereforeamatrixofdimensionn×(n−1).

Giventhesedefinitionswecanwritethen 2×n 2MarkovtransitionmatrixforSBBO inblock-matrixformas ⎡ _{0 · · · 0 P} ⎤ 1 · · · Pn ⎢_L ⎥ ⎢ n-1 ⎥ ⎢ ⎥ Q= _{⎢ . ⎥} (14) . ⎣ . ⎦ Ln-1

wheretheunspecifiedelementsareequaltozero,andthenumberofLn-1matricesis equalton.

Q2 describestheprobabilityoftransitioningfromonestatetoanotherstateafter twostatetransitions.ThiscanbecalculatedfromEquation(14)as

⎡Ln j=1Pj L( j) n-1,n 0 · · · 0 ⎤ Q2= ⎢ ⎢ ⎢ ⎢ 0_. L (1) n-1,n P1 . · · · L(1)_n-1,n Pn . ⎥ ⎥ ⎥ ⎥ (15) ⎢ ._. ._. ._. ⎥ ⎣ ⎦ 0 L(_nn_-1)_,n P₁ · · · L(_nn_-1)_,n Pn (j)

where the(n −1)×n matrixL_n-1,n containsallzerosexceptforthejthcolumn,each ofwhoseelementsisequalto1/(n −1).Theupperleftn ×n blockofQ2describesthe transitionprobabilityfromeachofthefirstn statestoeachofthefirstn statesaftertwo statetransitions.WeusethesymbolT todenotethistransitionmatrix.

n (j)

T = PjLn-1,n (16)

j=1

THEOREM1: T isaregulartransitionmatrixforps ∈(0, 1).

PROOF: SeeAppendixA.

THEOREM2: 1 Tk = (N In +ps1n)k (n −1)k ⎛ ⎞ 1 k _{k-j -1 k-j} k = ⎝NkIn +ps n Nj p s 1n ⎠ (17) (n −1)k j=1 j −1

where1n isthen ×n matrixthatcontainsallones,andk-choose-j isdenotedas

k k!

= (18)

j j! (k −j)!

PROOF: SeeAppendixA.

NotethatcomputingTk _{requiresaboutlog}

2k matrixmultiplicationswiththefirst expression ofEquation(17),butdoesnotrequireanymatrixmultiplicationswiththe secondexpressionofEquation(17).

THEOREM3: SBBOreachesapopulationwithuniformfitnessasthenumberofgenerations

approachesinfinity.

EXAMPLE: AppendixBgivesasimpleexampleofthetheoryuptothispoint.

5

Expected

Improvement

in

the

Simplified

BBO

Section3 gave theprobability ofimprovementinoneSBBOgeneration. Thissection discusses theexpected improvement inthe bestpopulation member, assuming that immigrationimprovedthesolutiontotheoptimizationproblem.

Supposethatthebestindividualhascostf_min,anditemigratesasolutionfeatureto anindividualwithcostf .Theimprovementinthecostoftheimmigratingindividualis denotedasIf ,whichweconsidertobepositiveifthecostdecreases.Beforemigration, thereplacedsolutionfeatureintheimmigratingislandcontributedacostz totheisland asgiveninEquation(6).Theemigratingsolutionfeaturecontributesacosty asgivenin Equation(4).ThechangeincostIf duetoimmigrationisthereforearandomvariable approximatelygivenas

If = z − y

2

∼ (f/s − fmin/s, 2σ /s)

∼ (μ₁, σ₁2) (19)

where μ₁ and σ₁ are auxiliary variablesdefined bytheabove equation.We want to compute theexpectedvalue ofcost improvement,giventhatimprovement occurred duetomigration.Thiscanbewrittenas

E(If |f − If < f_min)= E(If |If > f − f_min) fmax

= If pdf/(If )dIf (20) f ₋f_min

Theupperlimitintheintegralisf_maxbecauseweareassumingthatf (x)≥ 0forallx, whichimpliesthatIf ≤ f_max.Thefunctionpdf/(If )isthesameasthepdfgivenin Equation(19),exceptthatitisscaledsothatithasanareaofonebetween(f − f_min) andf_max.

IfEquation(19)isaGaussianpdf,thenEquation(20)canbewrittenas

( ) fmax _x −_(x − _μ 1)2 E(If |If > f − f_min)= √ exp dx f −fmin cxσ1 2π 2σ12 [ ( ) ( )] 1 f_max− μ₁ f − f_min− μ₁

wherecx = erf √ − erf √

2 σ₁ 2 σ₁ 2

x

2 ( ₂)

anderf(x)= √ exp −t dt (21)

π ₀

Equation (21) isfor a specific value of f . Sincethe immigratingisland is randomly selected,wetaketheexpectedvalueofEquation(21)withrespecttof toobtain

( ) fmax fmax _x −_(x − _μ 1)2 E(If |If > f − f_min)= √ exp dx pdf/(f )df (22) 2σ 2 fmin f −fmin cxσ1 2π 1

6

Thefunctionpdf/(f )isthepdfoff ,butscaledsothatithasanareaofonebetweenf_min

andf_max. Iff hasameanoff ¯ andavarianceofσ 2,asdiscussedinthetextfollowing Equation(2),andiff isGaussian,thenEquation(22)canbewrittenas

fmax fmax _x (−_(x − _μ ) 1)2 E(If |If > f − f_min)= exp f_min f ₋f_min cxcyσ1σ 2π 2σ₁2 (₋ ) (f − f ¯)2 × exp dx df 2σ 2 [ ( ) ( )] 1 f_max− f ¯ f_min− f ¯ cy = erf √ − erf √ (23) 2 σ 2 σ 2

Simulation

Results

SBBO was simulatedonsome benchmarkfunctions inorder toconfirmthetheoret­ ical results of the preceding sections. For each function, we ran 100 Monte Carlo SBBO simulations with populationsizesof 4, 20,and 100. Werecorded theaverage number of generations that was required to improve the best population member. Wealsorecordedtheaverageimprovementofthebestpopulationmember.Wecom­ pared these numberswith theexpectednumber of generationsbefore improvement asgiven inEquation(11),and theexpectedimprovementamountasgiven inEqua­ tion (23). Table 2 shows the results for four-dimensional benchmarks, and Table 3 showstheresultsfor10-dimensionalbenchmarks.Moreinformationaboutthebench­ markfunctions, includingtheirdomains,canbefoundin B¨ack(1996),Aluffi-Pentini et al.(1985), Yao et al.(1999),Cai and Wang (2006), Feng et al.(1998),and Li et al. (2008).

NotethatEquations (11)and(23)inTables2and 3arecalculatedonthebasisof pdf’s,f_min,f_max, andf ¯.Thesequantitiesarenotavailableanalytically,andsotheymust beapproximatedonthebasisofrandompopulations.Wethereforeused100random populationstoapproximatethesequantitiessothatwecouldcalculateEquations(11) and(23).

WemakethefollowingobservationsaboutTables2and3.

1. Foragivenbenchmark,thenumberofgenerationsbeforeimprovementincreases asthepopulationsizeincreases.Thisconfirmstheintuitiveresultthattheproba­ bilitypergenerationoffindingabettersolutiondecreasesasthepopulationsize increases.

2. Theexpectednumberofgenerationsbeforefindingabettersolutionascalculated from Equation (11) matches reasonably well with the simulation results. The simulationresultscanvarywidelyfromruntorunasshownbythelargestandard deviations,butthetheoreticalresultsarewellwithinonestandarddeviationfrom themeanofthesimulationresults.

3. TheexpectedimprovementamountfromEquation(23)matchesreasonablywell withthesimulationresults.Also,asshownqualitativelyinFigure5,weseefrom Tables2and3thatforeachbenchmarktheexpectedimprovementdecreasesas thepopulationsizeincreases.

Table 2: Theoreticalandsimulationresultsforfour-dimensionalbenchmarks.Thefour right-mostcolumnsshowtheexpectednumberofgenerationsuntilthefirstimprove­ ment in thebest problem solution, and the expected amount of that improvement. Simulationresultsarebasedon100MonteCarlorunsandareshownasmean± 1SD.

Benchmark Population Expectednumberofgenerations Expectedimprovement function size Eq.(11) Simulation Eq.(23) Simulation Ackley 4 3.7 3.9± 4.5 0.48 1.3± 2.0 (nonseparable) 20 30 20± 25 0.85 1.0± 1.2 100 11 54± 59 1.2 0.90± 1.1 Griewank 4 6.2 6.1± 3.4 11 11± 8.5 (nonseparable) 20 20 19± 21 9.8 4.9± 6.2 100 82 85± 144 0.0 1.6± 2.6 Penalty1 4 7.4 5.1± 3.8 3.3E7 (3.2± 4.6)E7 (nonseparable) 20 17 21± 21 3.4E6 (1.3± 3.0)E6 100 49 64± 83 2.6E5 (1.3± 5.4)E5 Penalty2 4 6.6 6.4± 4.8 4.8E7 (3.8± 5.5)E7 (nonseparable) 20 16 17± 15 1.7E7 (6.1± 9.2)E6 100 53 74± 66 2.4E6 (1.7± 4.9)E6 Rastrigin 4 3.6 3.9± 3.0 5.1 5.2± 5.7 (separable) 20 20.4 22± 29 4.3 4.1± 3.8 100 66 97± 146 2.6 2.9± 3.0 Schwefel2.21 4 3.6 2.6± 0.9 6.8 14.3± 9.7 (nonseparable) 20 25 25± 29 10 7.7± 6.1 100 186 138± 165 0.0 3.6± 3.9 Schwefel2.26 4 4.9 4.8± 3.9 156 148± 112 (separable) 20 9.8 12± 13 98 94± 68 100 43 43± 62 90 70± 48 Shubert 4 4.9 3.8± 2.7 1.2 1.3± 1.1 (separable) 20 27 22± 29 0.69 0.52± 0.50 100 46 47± 51 0.93 0.24± 0.24 SineProduct 4 5.8 4.8± 3.4 0.15 0.13± 0.12 (separable) 20 21 19± 17 0.097 0.055± 0.065 100 34 45± 46 0.064 0.012± 0.018 Sphere 4 4.0 3.9± 2.7 3.6 3.9± 3.7 (separable) 20 25 24± 21 2.1 1.9± 2.0 100 36 45± 46 0.0 0.5± 1.0 Step 4 7.4 8.2± 6.0 1,496 1,522± 1,241 (separable) 20 17 17± 19 981 911± 890 100 78 58± 57 68 235± 291 Weierstrass 4 5.3 5.0± 5.4 0.59 0.68± 0.37 (separable) 20 15 18± 16 0.48 0.45± 0.36 100 53 60± 75 0.36 0.31± 0.33 6.1 ApproximationErrors

Althoughweseegoodcorrelationbetweenthetheoreticalandsimulationresults,there are still noticeable differences between the two sets of results. This is due to the approximations thatwere usedin obtainingEquations (11)and (23).Theseapproxi­ mationsincludethefollowingassumptions.

1. Weassumedseparability,asseeninEquation(2).Manyproblemsareseparable, butmanyothersarenot.Forreal-worldproblemswecannotalwaystellwhether

Table 3: Theoreticalandsimulationresultsfor10-dimensionalbenchmarks.Thefour right-mostcolumnsshowtheexpectednumberofgenerationsuntilthefirstimprove­ ment in thebest problem solution, and the expected amount of that improvement. Simulationresultsarebasedon100MonteCarlorunsandareshownasmean± 1SD.

Expectednumberofgenerations Expectedimprovement Benchmark Population

function size Eq.(11) Simulation Eq.(23) Simulation Ackley 4 13 13± 14 0.074 0.18± 0.21 (nonseparable) 20 65 40± 44 0.0 0.36± 0.38 100 20 113± 113 3.6 0.38± 0.36 Griewank 4 11 11± 9.4 16 18± 14 (nonseparable) 20 46 34± 35 11 9.6± 9.4 100 116 124± 122 10 6.1± 9.2 Penalty1 4 13 15± 21 3.6E7 (3.4± 3.1)E7 (nonseparable) 20 44 44± 42 1.6E7 (1.4± 2.0)E7 100 192 161± 144 4.1E6 (1.5± 3.5)E6 Penalty2 4 13 15± 15 7.0E7 (7.5± 7.3)E7 (nonseparable) 20 43 43± 36 2.2E7 (3.0± 3.6)E7 100 197 134± 132 4.2E6 (4.5± 9.9)E6 Rastrigin 4 8.6 9.9± 9.9 7.8 6.9± 6.3 (separable) 20 38 38± 46 4.8 5.5± 4.5 100 128 126± 132 4.0 4.7± 4.8 Schwefel2.21 4 13 16± 14 1.6 7.3± 5.7 (nonseparable) 20 98 71± 65 3.3 5.1± 5.0 100 17 272± 233 4.9 3.6± 3.0 Schwefel2.26 4 15 15± 16 147 145±116 (separable) 20 45 49± 42 114 136± 112 100 109 102± 136 139 104± 88 Shubert 4 11 11± 12 2.8 3.3± 3.1 (separable) 20 41 36± 39 1.4 1.4± 1.7 100 133 129± 108 1.1 0.92± 1.0 Sineproduct 4 9 9± 8 0.19 0.21± 0.22 (separable) 20 48 47± 44 0.15 0.15± 0.15 100 125 130± 114 0.14 0.056± 0.081 Sphere 4 11 11± 12 5.1 5.1± 4.7 (separable) 20 38 40± 35 3.7 3.9± 4.2 100 140 150± 150 2.9 1.3± 1.7 Step 4 9.3 9.3± 10.6 2,081 2,076± 1,802 (separable) 20 76 62± 79 1,280 1,014± 1,262 100 115 133± 122 392 948± 1,095 Weierstrass 4 11 11± 11 0.48 0.58± 0.47 (separable) 20 37 39± 44 0.33 0.48± 0.35 100 109 107± 127 0.58 0.37± 0.30

ornottheproblemisseparable.Wemadetheassumptionofseparabilityinorder toallowtractabilityinouranalysis.Table4givestheaverageapproximationerrors fromtheprevioussection,andshowsthat(asexpected)theapproximationerrors aresmallerfortheseparablecostfunctionsthanforthenonseparablefunctions. 2. Wemadetheassumptionof identicalpdf’s foreachsolutionfeature asseen in

Equation(3).Inotherwords,weassumedthateachsolutionfeaturecontributes anequalamounttothefitnessofthecostfunction.

Table 4: AveragepercenterrorsbetweenthetheoreticalandsimulationresultsinTa­ bles2and3.Asexpected,theerrorsaresmallerfortheseparablefunctionsthanforthe nonseparablefunctions.

Nonseparablefunctions Separablefunctions

Dimensions Numberofgenerations Improvement Numberofgenerations Improvement

4 26% 67% 14% 30%

10 29% 34% 5% 25%

3. WemadetheassumptionofnormalityinSection5.Thatis, weassumedthata uniformlydistributedpopulationofsolutionsgivesanormallydistributedfitness function.Althoughthisisonlyanapproximation,itisareasonableoneinviewof thecentrallimittheorem(PapoulisandPillai,2002),whichsaysthatacombination ofindependentrandomvariablestendstowardaGaussianrandomvariable. Inspiteoftheseassumptionsandapproximations,thesimulationandtheoretical resultsintheprevioussectionarestillcloseenoughthatthetheorycanbeusedtoat leastqualitativelypredictSBBOperformance.However,futureworkshouldbepursued torelaxsomeoftheseassumptions.

Forexample,givenareal-worldproblemwithanunknownfunctionalform,separa­ bilitycanbetestedbyevaluatingthecostfunctionatvariousvaluesoftheindependent variables. Furthermore, individual contributionsof independent variablesto fitness, andcorrelationsbetweenindependentvariables, canbetested.Dependingonthere­ sultsofthesestatisticaltests,Equation(2)canbemodifiedforaspecificproblem.For example,itcouldberewrittenas

f (xi)= α1f1(xi1)+ α2f2(xi2)+ ρ12f1(xi1)f2(xi2) (24)

where α₁,α₂, andρ₁₂ aredeterminedfromstatisticaltests.TheanalysisinSections3 andfollowingcouldthenberepeatedforthisformofthecostfunction.

TheassumptionofGaussiancostfunctionscouldalsoberelaxed,eitherforother general casesor forspecific optimizationproblems.Inthegeneral case,theanalysis in Section 5 could be repeatedfor fitness distributionsother thanGaussian. Incase wehaveaspecificoptimizationproblem,thefitnessdistributioncouldbenumerically obtainedbygeneratingarandompopulationofsolutions.Wecouldthenfitaweighted sumofGaussianstotheempiricaldistribution(Alspach,1974).Thedoubleintegralon theright-handsideofEquation(23)wouldthenbereplacedwitha weightedsumof integrals.KernelsotherthanGaussiancouldalsobeusedtoapproximatethecostfunc­ tionpdf(Simonoff,1998;DevroyeandLugosi,2001),whichwouldchangeEquation(21) andthefollowingequations.

Insummary,theanalysisinthispaperhasbeenconductedwithspecificassump­ tions(separability,identicalpdf’s,andnormality),butthemoreimportantpointisthat our analysis providesa general frameworkthatcanbe usedeven when thespecific assumptionsinthispaperdonothold.Theworkpresentedinthispaperopens upa widerangeoffutureresearchpossibilities.

Table 5: Average MATLAB computational effort (in seconds) on a 2.0 GHz PC for thecalculationofthetripleintegralofEquation(10)andthedoubleintegralofEqua­ tion(23)foreachbenchmark.Thecomputationtimeinthetableisdominatedbythat ofEquation(10). Population Size 4 Dimension 10 4 149 98 20 99 81 100 77 117 6.2 ComputationalEffort

Thecomputationaleffortrequiredtoobtainthetheoreticalresultsinthispaperdepends onthelevelofaccuracyrequired.Thebulkofthetheorycomputationisdominatedby thenumericalcalculationofthetripleintegralinEquation(10)andthedoubleintegral in Equation(23).In thispaper weusedthe MATLABintegrationfunctionsdblquad andtriplequad. Thecomputertimerequiredforthesefunctionsstronglydependson their errortolerance parameters.Asmaller errortolerance givesmoreaccurate inte­ grationresultsbutrequiresmoreintegrandevaluations.For theresultsinthispaper, weusedanerrortoleranceof10-8 _{(thedefaultMATLABtolerance is10}-6_{).Ironically,} asthepopulationsizeand thenumberofdimensionsincrease,thecomputationalef­ fortoftendecreases,asshowninTable5.Thisisbecausealargerpopulationsizeand a higherdimensionality resultsin awiderdistribution ofthecostfunction,thatis,a largerstandarddeviationforpdf(f )inEquation(10),andlargervaluesforσ andσ₁in Equation(23).Thisinturnresultsinsmootherintegrandsandfasterconvergenceofthe integrationroutines.

However, the reason for our theoretical analysis isnotto reduce computational effortrelativetosimulation.Infact,computationaleffortcannotevenbefairlycompared betweentheoryandsimulation.Theoryonlygivestheexpectednumberofgenerations tothefirstimprovement,andtheexpectedamountofimprovement.Simulation,though, typicallyrunsforaspecifiednumberofgenerations,oruntilsometypeofconvergence isachieved.

The reason for our theoretical analysis is to propose new tools for thestudy of BBOpopulations,andmoregenerally,forpopulationsofothertypesofEAs.Sincethe characteristicsofrandompopulationdistributionsdependonthespecificproblem,the problemdimension,andthepopulationsize,theresultsofouranalysiswillalsochange withtheseparameters.Ourtoolsincludeanalyticalexpressionsthatcouldthereforebe studiedtolearnhowBBObehavior(and perhapsotherEAbehaviorinfuturework) dependsontheseparameters.Inaddition,changestotheBBOsetup(elitism,mutation, migration curve shape, etc.) could possibly be incorporated into ouranalysis in the futuretoseehowthesechangesaffectBBObehavior.

Theseissuescanallbestudiedbysimulationalso,butsimulationresultsaredifficult togeneralize.Forexample,asimulationmighttellusthatapopulationsizeofn pro­ videsagoodperformance/computationtradeoffforaparticularproblem.Butanalysis might tellus thata populationsize ofn providesa goodperformance/computation tradeoff for all problems with a prescribedlevel of separability and with a specific fitnessdistribution.

Figure 6: Normalized(f_max− f_min)averagedover14benchmarksand100simulations: four-dimensionalresults.

Figure 7: Normalized(f_max− f_min)averagedover14benchmarksand100simulations: ten-dimensionalresults.

6.3 Uniformity

Figures6and7showhowthenormalizedfitnessrangeofSBBOchangeswiththenum­ berofgenerations.Theorem3predictedthatSBBOwouldreachauniformpopulation, andFigures6and7confirmthis.Inagreementwithintuition,thenumberofgenerations beforereachinguniformityisproportionaltoboththepopulationsizeandtheproblem dimension.Interestingly,thefitnessrangeincreasesatthebeginningofthesimulations asf_min rapidlydecreases.Thisisespeciallytrueforsmallpopulations(comparewith Figure5). Laterinthesimulations,alloftheindividualsimproveuntiluniformityis eventuallyachieved.

Notethatauniformpopulationisnotnecessarilydesirable.Thereisnoguarantee that a uniform population is optimal. It is onlywith mutation that we canbe sure of obtaining an optimal solution for a givenoptimization problem. In fact,this isa conclusionthatcanbegenerallyinferredfromthispaper.SeeReevesandRowe(2003) foradiscussionofthisideaasappliedtoGAs.

7

Intheory,thenumberofgenerationsuntilSBBOreachesuniformitycouldbepre­ dictedbyTheorem3.Thistheoremgivesustheratioofthelargestandsmallestelements oftheMarkovtransitionmatrix,sowhenthatratiofallsbelow somesmallthreshold afteracertainnumberofgenerations,wecanassume thatthepopulationisuniform andnomoreimprovementispossible.Inpractice,however,thisisdifficulttocalcu­ latebecauseTheorem3assumesthatps isconstant,whereasps actuallychangeswith eachgeneration. TheuseoftheMarkovtheoryinSection4topredictthenumberof generationsuntiluniformityisreachedhasthereforebeenleftforfuturework.

Conclusion

AsimplifiedBBO(SBBO)algorithmhasbeenpresentedinthispaper.SBBOhasbeenan­ alyzedusingprobabilitytheoryinordertofindthreerelatedquantities:theprobability pergenerationthatitspopulationoptimumimproves,thestatetransitionmatrixofthe algorithm,andtheexpectedamountofimprovementinthepopulationoptimum.These quantitiesareinexactinlightofassumptionsandapproximationsthatweremade,but theymatchbenchmarkfunctionsimulationresultsreasonablywell.

Further workcould focus onreducing theapproximation errors thatwere used to obtain the theoretical results in this paper. In this paper we assumed separable fitnessfunctions,equalfitnesscontributionfromeachindependentvariable,andnormal distributionsforthefitnessandforthecontributionofeachparametertototalfitness. Thedevelopmentinthispapercouldbegeneralizedsothatitmatchesotherproblem characteristics,orsothatthoseassumptionscouldberelaxed.Futureworkcouldalso includetheextensionofouranalysistoothertypesofEAs.

Acknowledgment

ThisworkwassupportedbyNSFGrant0826124intheCMMIDivisionoftheEngineer­ ingDirectorate.

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Appendix

A

PROOF OF THEOREM1: From Equation (16)weseethat T isa sumof n matrices.If

ps ∈ (0, 1),theneachelementinthejthcolumnofthejthmatrixinthesumisgreater thanzero.Sowhenthesematricesareaddedtogether,theresultisamatrixthatcontains allnonzeroelements.

Theorem1canalsobeprovenwithamoreintuitiveargument.Tij istheprobability thatislandj isreplacedbyislandi asthemostfitislandafteronemigration.Butthe probabilitythatislandj migratestoislandi isnonzero,1/(n − 1)tobeexact,andthe probabilitythatthismigrationresultsinislandi replacingislandj asthemostfitisland isgivenbythenonzeroquantityofEquation(10).

ByexpandingthesuminEquation(16)itcanbeseenthat 1

T = (N In + ps1n) (25)

n − 1

whereN = n − 1− nps,In isthen × n identitymatrix,and1n isthen × n matrixthat

containsallones. QED

PROOFOFTHEOREM2: ItcanbeseenfromEquation(25)thatEquation(17)holdsfor

k = 1.SupposethatEquation(17)holdsforsomevalueofk.Then

⎛ ⎞ k ( ) 1 k Tk+1= ⎝NkIn + ps n k-jNj-1p sk-j 1n ⎠(N In + ps1n) (n − 1)k₊1 j₌1 j − 1 ⎛ k ( ) 1 _⎝ k Nk+1 k-jNj k-j = In + ps n p s 1n + Nkps1n (n − 1)k+1 j=1 j − 1 ⎞ k ( ) 2 k-j+1_Nj-1 k-j k ⎠ + ps n p s _{j − 1} 1n j=1 ⎛ k ( ) 1 _⎝ k Nk+1 k-jNj k-j = In + ps n p s 1n (n − 1)k₊1 j=1 j − 1 ⎞ k₊1 ( ) k-j+1_Nj-1 k-j+1 k ⎠ + ps n p s _{j − 1} 1n j₌1 ⎛ _{( )} 1 _⎝ k k Nk+1 k-jNj k-j = In + ps n p s 1n (n − 1)k+1 j=1 j − 1 ⎞ k+1 ( ) k-j k-j k k+1 + ps n Njp k 1n + n p 1n ⎠ s _j s j=1 ⎛ k [( ) ( )] 1 _⎝ k k Nk+1 k-jNj k-j = In + ps n p s + 1n (n − 1)k₊1 j=1 j − 1 j k k+1 + n p s 1n (26)

RecallfromChuan-ChongandKhee-Meng(1992,p.69)that

( ) ( ) ( )

k k k + 1

+ = (27)

j − 1 j j

UsethisinEquation(26)toobtain

⎛ ⎞ k ( ) 1 k-j k-j k + 1 k k+1 Tk+1 = ⎝Nk+1In + ps n Njp s 1n + n p s 1n ⎠ (n − 1)k₊1 j₌1 j ⎛ ⎞ k ( ) 1 _⎝ k + 1 Nk+1 k-jNj k-j ⎠ = In + ps n p s 1n (n − 1)k+1 j=0 j ⎛ ⎞ k+1 ( ) 1 _⎝ k + 1 Nk+1 k+1-j ⎠ = In + ps n Nj-1pks+1-j 1n (28) (n − 1)k+1 j=1 j − 1

whichcompletestheproofofEquation(17)byinduction. QED

PROOF OFTHEOREM3: Equation(17)showsthatthediagonalelementsofTk _{are all}

equal,andtheoff-diagonalelementsareallequal.Thedifferencebetweenadiagonalele­ mentandanoff-diagonalelementis[(n − 1− nps)/(n − 1)]k.Thisdifferenceapproaches 0 ask → ∞.Therefore,

limTk = 1n/n (29)

k→∞

Inother words,asthenumber ofgenerations approachesinfinity, eachisland inthe population hasanequal probabilityofbeingthemostfitmemberofthepopulation.

Thismeansthateachislandisequallyfit. QED

Appendix

B

HerewegiveasimpleexampletoillustratethetheoryofSections1–4.Supposewehavea two-dimensional(s = 2)fitnessfunctionthatisuniformlydistributedbetweenf_min = 1 and_√f_max= 2. Theny in Equation_√ (5) is uniformlydistributed betweeny_min= 1/2− 1/ 8 andy_max= _√1/2+ 1/ 8,andz in_√ Equation(6)isuniformlydistributedbetween

z_min= f/2− 1_√/ 8 andz_max= f/2+ 1/ 8.Themagnitudesofthepdf’sofy andz are bothequalto 2betweentheirminimumandmaximumvalues.ps canbecalculated fromEquation(10)as

√

Now suppose thatwe haven = 3 members in our SBBOpopulation.The transition matrixofEquation(14)isthenequalto

⎡ _{0 0 0 (1− p} ⎤ s) (1− ps) ps 0 ps 0 ⎢ _{0 0 0} ps 0 (1− ps) (1− ps) 0 ps ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 ps 0 ps (1− ps) (1− ps)⎥ ⎥ ⎢ ⎢0.5 0 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ Q = _⎢0.5 0 0 0 0 0 0 0 0 _⎥ ⎢ _{0 0.5 0 0 0 0 0 0 0} ⎥_⎥ ⎢ ⎢ ⎥ ⎢ 0 0.5 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎣ 0 0 0.5 0 0 0 0 0 0 ⎦ 0 0 0.5 0 0 0 0 0 0 (31) Notethateachcolumnof Q sumsup toone.Thetransitionmatrix of thefirstthree statescanbecalculatedfromtheupperleftblockofEquations(15)or(16)as

⎡_{(1− ps) p} ⎤

s/2 ps/2

T = ⎣ ps/2 (1− ps) ps/2 ⎦ (32)

ps/2 ps/2 (1− ps)

where weagain noticethateachcolumnof T sumstoone.Wecaneasilyverifythat limk→∞ Tk = 1n/3,whichconfirmsTheorem3andEquation(29).

Post-print standardized by MSL Academic Endeavors, the imprint of the Michael Schwartz Library at Cleveland State University, 2014