萤火虫算法

168J.Senthilnathetal./SwarmandEvolutionaryComputation1(2011)164–171

Table2

Averageclassificationerrorpercentagesusingnatureinspiredtechniquesontest

Numberoffireflies(N)=20Attractiveness(β0)=1

Lightabsorptioncoefficient(γ)=1Numberofgenerations(T)=100.

Formostoftheapplications,thesameparametervaluesaresuggestedbyYang[18].Afterthefirefliesaredeployedrandomlywithinthesearchspace,theparameterβ0=1whichisequivalenttotheschemeofcooperativelocalsearchwiththebrightestfireflystronglydeterminedtheotherfirefliespositions,especiallyinitsneighborhood.Theparametervalueofγ=1determinesthevariationoflightintensitywithincreasingdistancefromthecommunicatedfirefly,resultsinthecompleterandomsearch.

Thenumberoffunctionevaluationsinthefireflyalgorithmcanbeobtainedasfollows:letNbethesizeofinitialpopulation,andTbethemaximumnumberofgeneration.Thenthenumberof

functionevaluationsforeachiterationisN (N 1)

N (N 12.Thetotalnumberoffunctionevaluationsgeneratedis)

wehaveused100asthemaximumnumber2

×T.Inourstudies,ofgenerations.Thenumberoffunctionevaluationforeach13classificationdataset,(withN=20andT=100)inonesimulationrunis19000.5.2.2.AnalysisofClassificationErrorPercentageusingFA

In[9,14],theClassificationErrorPercentage(CEP)measureisusedwithallthe13benchmarkdatasets.Falcoetal.[14]comparedtheperformanceofthePSOalgorithmwiththeother9methodsnamelyBayesNet[26],MultiLayerPerceptronArtificialNeuralNetwork(MLP)[27],RadialBasisFunctionArtificialNeuralNetwork(RBF)[28],KStar[29],Bagging[30],MultiBoostAB[31],NaiveBayesTree(NBTree)[32],RippleDownRule(Ridor)[33]andVotingFeatureInterval(VFI)[34].KarabogaandOzturk[9]implementedtheABCalgorithmandanalyzedCEPwithalltheabovementionedmethods.Inthisstudy,inadditiontothesemethods[9,14]wehaveanalyzedtheCEPmeasureoftheFAtomakereliablecomparison.

FromthetrainingdatasettheknowledgeintheformofclustercentersisobtainedusingtheFireflyAlgorithm(FA).FortheseclustercentersthetestingdatasetsareappliedandtheCEPvaluesareobtained.Theresultsofthenatureinspiredtechniques—FA,ABCandPSOfortheproblemsaregiveninTable2whereCEPvaluesarepresented.TheFAoutperformstheABCandPSOalgorithmsinall13problems,whereasABCalgorithm’sresultisbetterthanthatofPSOalgorithminall12problemsexceptforoneproblem(theglassproblem)intermsofclassificationerror.Moreover,theaverageclassificationerrorpercentagesisalsobetterforallproblemsinthecaseofFA(11.36%)comparingtothatofABC(13.13%)andPSO(15.99%).

FromTable3,wecanobservethattheCEPmeasureoftheFAand11methodsthataregivenin[9,14]arepresented,andtherankingisbasedontheascendingorderofaverageclassification

erroroftheclassifiersoneachproblemarealsogivenintheparenthesis.Ataglance,onecaneasilyseethattheFAgetsthebestsolutionin8oftheproblemsamong13problemsused.Tobeabletomakeagoodcomparisonofallthealgorithms,Tables4and5arereported.Table4showstheaverageclassificationerrorsofallproblemsandtherankingisbasedontheascendingorderofaverageclassificationerror.Table5showsthesumofthealgorithmsrankingofeachprobleminascendingorder.

FromTable4,wecanobservethatbasedonaverageCEPvalues,FAisthebestincomparisonwiththatofMLPartificialneuralnetworktechniqueandABC,whileMLPperformedbetterincomparisonwiththeABC.However,eveniftheresultsinthetablearecomparable,webelievethatitmaycausesomesignificantpointstobedisregardedsincethedistributionoftheerrorratesarenotproportional.Therefore,thegeneralrankingofthetechniquesinTable5isrealizedbycalculatingthesumoftheranksofeachproblemfromTable4.Fromthisranking,onceagaintheFAisthebest,whiletheABCalgorithmatthesecondposition,andtheBayesNettechniqueatthethirdposition.TheclassificationerrorrateandrankingsfromthetablesshowthatclusteringwiththeFAofferssuperiorgeneralizationcapability.Notethat,hereweareusingonlytheresultsofalltheothermethodsgiveninearlierstudies[9,14]excepttheFAalgorithm.

5.2.3.AnalysisofclassificationefficiencyusingFA

Intheprevioussection,wepresentedtheresultobtainedusingCEP.This(CEP)ingthesameclustercenterstheaverageandoverallefficiencyforentiredatasetareobtained.

i.Significanceofindividualefficiency:Foratestingdataset,themainsignificanceofindividualefficiencyistoanalyzetheclass-levelperformanceofaclassifieralgorithm.FromTable6,wecanobservethattheindividualclassificationefficiencyofthetestingsamples,hereCancer-Int,Iris,ThyroidandWineisgettingclassifiedwithoutanymisclassificationsandhencehasanindividualefficiencyof100%.InthecaseofBalancedatasettheindividualefficiencyofClass2is66.7%.InCreditandDiabetesdatasetclass1ismisclassifiedasclass2withindividualefficiencyof70%whereasinHeartandDermatologydatasetClass2haslessindividualefficiencyof73.1%and50%respectively.

FormTable3wecanobservethat,fortheclassificationproblem—Heart(2classproblem),theFAperformedbetterthanalltheotherclassifierwiththeCEPvalue13.16.Thisdoesnotmeanthattheindividualefficiencyofeachclasstobegood.Toillustratethisinmoredetail,letusconsiderHeartdataset,fromTable6wecanobservethattheClass1hasimpressiveindividualefficiencyof94%whereasinClass2mostofthesamplesbelongingtoClass2ismisclassifiedasClass1withindividualefficiency73.1%.Henceitisimportanttoconsidertheindividualefficiencytoanalyzetheclass-levelperformanceofaclusteringalgorithm.

ii.Performanceofaverageandoverallefficiency:Forentiredataset,itisalwaysnecessarytoknowtheglobalperformanceofaalgorithm.Thiscanbeachievedbyusingaverageandoverallefficiency.AswecannoticefromTable6,fortheentiredatasettheaverageandoverallefficiencyusingthefireflyalgorithmfor13benchmarkdatasets.TheBalancedatasethasanaverageandoverallefficiencyof74.9%and80.8%respectivelywhereasCancerdatasetwithaverageandoverallefficiencyof91%and92.5%respectively.AnaverageandoverallefficiencyofCancer-Int,Dermatology,IrisandWinedatasetare97.9%,81.9%,94.7%and90.6%respectively.TheaverageefficiencyofCredit,Diabetes,E.Coli,HeartandThyroidare75.5%,73.4%,88.5%,77.1%and92.6%.TheoverallefficiencyofCredit,Diabetes,E.Coli,HeartandThyroid