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Package'emma'February19,2015TypePackageTitleEvolutionarymodel-basedmultiresponseapproachVersion0.
1-0Date2011-10-22AuthorLauraVillanova,KateSmith-MilesandRobJHyndman.
MaintainerLauraVillanovaDependsR(>=2.
9.
2),earth,clusterSimImportsmethodsDescriptionTheevolutionarymodel-basedmultiresponseapproach(EMMA)isanovelmethodologytoprocessoptimisationandproductimprovement.
Theapproachissuitabletocontextsinwhichtheexperimentalcostand/ortimelimitthenumberofimplementabletrials.
LicenseGPL(>=2)LazyLoadyesRepositoryCRANDate/Publication2011-10-2617:59:36NeedsCompilationnoRtopicsdocumented:emma-package2ackley3distance4emma5emmacheck9emmat012emmatn14peaks19plot.
emma2012emma-packageIndex21emma-packageDesigningexperimentsforprocessoptimizationDescriptionTheevolutionarymodel-basedmultiresponseapproach(EMMA)isaprocedureforprocessopti-mizationandproductimprovement.
Itisparticularlysuitedtoprocessesfeaturingirregularexper-imentalregionduetoconstraintsontheinputvariables(factors),multipleresponsesnotaccomo-datedbypolynomialmodels,andexpensiveortime-consumingexperiments.
EMMAiterativellyselectsnewexperimentalpointsthatincreasinglyconcentrateonthemostpromisingregionsoftheexperimentalspace.
Theselectionofthenewexperimentalpointsisperformedonthebasisoftheresultsachievedduringprevioustrials.
Amultivariateadaptiveregressionsplines(MARS)modelandaparticleswarmoptimization(PSO)algorithmareusedtodrivethesearchoftheoptimum.
DetailsPackage:emmaType:PackageVersion:1.
0Date:2011-02-22License:GPL(>=2)LazyLoad:yesAuthor(s)LauraVillanova,KateSmith-MilesandRobJHyndmanMaintainer:LauraVillanovaReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
Examplesin.
namex1","x2")nlevx",nd=10,na=5,weight,C,w1=0.
7,w2=0.
4,c1i=2.
5,c1f=0.
5,c2i=0.
5,c2f=2.
5,b=5,pr.
mut,graph="no",fn1=peaks)plot(emma.
peaks,fn=peaks,n=50,C=20)ackleyAckleytestproblemDescriptionGeneratestheAckleybenchmarkfunction.
TheAckleyfunctionisacommonlyusedtestproblemforglobaloptimizationprocedures.
Usageackley(x)ArgumentsxAmatrixcontainingthevaluesoftheinputvariables.
ValueVectorofthesamelengthasxgivingthevaluesoftheAckleyfunction.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferenceshttp://www-optima.
amp.
i.
kyoto-u.
ac.
jp/member/student/hedar/Hedar_files/TestGO_files/Page295.
htm4distanceExamplesx1x2x1xxpand.
grid(x1,x2)zx(ackley(x),nrow=length(x1))nrzx1,x2,z,col=color[facetcol],theta=0,phi=10,expand=1,xlab="x1",ylab="x2",zlab="f(x1,x2)",ticktype="detailed")distanceDistancemeasurefromthetargetDescriptionComputesascalardistancebetweenthetarget(asetofdesirablevaluesfortheresponses)andtheresponsesvaluesthathavebeeneitherobservedorestimatedforeachpointintheexperimentalspace.
Suchadistanceisusedtoidentifyadditionalexperimentalpointstobeinvestigated.
Usagedistance(xpop,xspace,yspace,weight,opt)ArgumentsxpopAdataframecontainingthefactorvaluesfortheexperimentalpointsinvesti-gated;therownamesuniquelyidentifyeachexperimentalpoint(ID).
xspaceAdataframecontainingthefactorvaluesfortheexperimentalpointsdeningtheentireexperimentalregion;therownamesuniquelyidentifyeachexperi-mentalpoint(ID).
yspaceAdataframecontainingtheresponsevalues(eitherobservedorestimated)forthepointsintheexperimentalregion.
weightAnumericalvector,ofthesamelengthasthenumberofresponses,containingtheweightsassignedtotheeachresponse;thesumoftheweightsmustbeequalto1.
optAcharactervector,ofthesamelengthasthenumberofresponses,deningifeachresponseneedstobeminimizedormaximized.
Theallowedvaluesare'mn'(minimize)and'mx'(maximize).
emma5DetailsThefunctionnormalizestheresponsevalueswithrespecttotheestimatedlimitsoftheresponsespace,sothattheresponsevaluesliebetween0and1.
Subsequently,thefunctionidentiesthetargetandcomputesascalardistancebetweenthetargetandtheresponsevalues.
ValuefitThescalardistancesbetweenthetargetandtheresponse(s)valuesfortheexper-imentalpointsinxpop.
obj.
nnScalardistancefromthetargetforthebestexperimentalpointidentiedbyEMMA.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesFriedmanJ.
H.
(1991)'Multivariateadaptiveregressionsplines'(withdiscussion),TheAnnalsofStatistics19,1:141.
VillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165emmaEvolutionaryModel-basedMultiresponseApproachDescriptionEMMAdesignstheexperimentsusingaprocedurebasedontheParticleSwarmOptimization(PSO)algorithm.
Firstly,EMMAselectsasetofinitialexperimentalpoints(seeemmat0)thatdenetheinitialpositionoftheparticles;subsequently,foragivennumberofiterations,theparticlesaremovedandnewexperimentalpointsareselected(seeemmatn).
Usageemma(in.
name,nlev,lower,upper,out.
name,opt="mn",nd=10,na=5,weight,C=20,w1=0.
7,w2=0.
4,c1i=2.
5,c1f=0.
5,c2i=0.
5,c2f=2.
5,b=5,pr.
mut,graph,fn1=NULL,fn2=NULL,fn3=NULL,fn4=NULL,nresp)6emmaArgumentsin.
nameAvectorcontainingthenamesoftheinputvariables(factors).
nlevAnumericvectorofthesamelengthasin.
name,containingthenumberoffactorlevels.
lowerAnumericvectorofthesamelengthasin.
name,containingthelowervaluesofthefactors.
upperAnumericvectorofthesamelengthasin.
name,containingtheuppervaluesofthefactors.
out.
nameAvectorcontainingthename(s)oftheoutput/responsevariable(s).
optAcharactervectorofthesamelengthasthenumberofresponses,indicatingforeachresponsefunction,iftheresponsemustbeminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointstobeselectedwhent=0.
naAnumericvalueindicatingthenumberofexperimentalpointstobeselectedwhent>0.
weightAnumericalvectorofthesamelengthasthenumberofresponses,reectingtherelevanceofeachresponse.
Useweight=1ifonlyoneresponseisinvestigated;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
CAnumericvalueindicatingthemaximumnumberofiterations.
w1TherstnumericvalueusedtocalculatetheinertiaweightparameterofthetimevariantPSOalgorithm;thedefaultisw1=0.
7.
w2ThesecondnumericvalueusedtocalculatetheinertiaweightparameterofthetimevariantPSOalgorithm;Thedefaultisw2=0.
4.
c1iTherstnumericvalueusedtocalculatetheaccelerationcoefcientc1ofthetimevariantPSOalgorithm;thedefaultisc1i=2.
5.
c1fThesecondnumericvalueusedtocalculatetheaccelerationcoefcientc1ofthetimevariantPSOalgorithm;thedefaultisc1f=0.
5.
c2iTherstnumericvalueusedtocalculatetheaccelerationcoefcientc2ofthetimevariantPSOalgorithm;thedefaultisc2i=0.
5.
c2fThesecondnumericvalueusedtocalculatetheaccelerationcoefcientc2ofthetimevariantPSOalgorithm;thedefaultisc2f=2.
5.
bAnumericvalue,usedinthemutationoperator,thatdeterminesthedegreeofdependenceofthemutationontheiterationnumber;thedefaultisb=5.
pr.
mutAnumericvectorofthesamelengthasthenumberofiterationsCcontainingtheprobabilityofmutationforeachtimeinstant.
graphLogical;if'yes',aplotoftheMARSmodelisproduced.
Aplotisproducedonlyifthemodelcontainsmorethanoneexplanatoryvariable.
fn1Therstfunctiontobeoptimised.
Usefn1=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
fn2Thesecondfunctiontobeoptimised.
Usefn2=NULLifthefunctionisun-known(e.
g.
whendesigningexperimentsinappliedproblems).
emma7fn3Thethirdfunctiontobeoptimised.
Usefn3=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
fn4Theforthfunctiontobeoptimised.
Usefn4=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
nrespTheresponsetobeplotted.
Usenresp=1toplottherstresponse.
.
.
DetailsToselectthenewexperimentalpointstobeinvestigated,thefollowingstepsareiterated.
AMARSmodelisttedtothecollecteddatasothatanapproximatedfunctionisobtainedforeachresponse;theseapproximatedfunctionsareusedtopredicttheresponsevaluesatthenon-investigatedexper-imentalpoints.
EachpointintheexperimentalregionE(xspace)isnowassociatedwithavectorofresponsevaluesthathasbeeneithermeasuredorestimated.
Thebest(measuredorestimated)valueofeachresponseisselectedandusedtoidentifythetarget.
Subsequently,foreachexperimentalpointinE,thescalardistancebetweentheresponsevaluesandthetargetiscomputedandthesolu-tionthatisclosesttothetargetisselected.
Ifsuchsolutionhasnotbeentestedyet(seeemmacheck),theexperimentneedstobeperformedanditsresponsevaluesaremeasured.
Thetargetisthenupdated,aswellasthescalardistancesofalltheexperimentalpointsfromthetarget.
Thescalardistancesareusedtoidentifythegoodperformingexperimentalpoints.
Theexperimentalpointwhoseresponsevaluesareclosesttothetargetisreferredtoastheglobalbest.
Similarly,apersonalbestisidentiedforeachparticlebyconsideringtheexperimentalpointsvisitedbythatparticleandselectingthatpointfeaturingtheresponsevaluesthatareclosesttothetarget.
Finally,theparticlesvelocityandpositionareupdatedandanewsetofexperimentalpointsisidentied.
Theparametersw1andw2areusedtocalculatetheinertiaweightwofthePSOalgorithm,namelytheparameterthatcontrolstheinuenceofthepreviousparticlevelocityonthepresentvelocity.
Highvaluesofwfavouraglobalsearch,whereaslowervaluesofwencouragealocalsearch.
InEMMAtheinertiaweightisallowedtodecreaselinearlywithiterationfromw1tow2thusfavouringtheexplorationinitiallyandtheexploitationsubsequently.
Theparametersc1iandc1fareusedtocalculatethecognitiveaccelerationcoefcientc1ofthePSOalgorithm,whereastheparametersc2iandc2fareusedtocalculatethesocialaccelerationcoefcientc2ofthePSOalgorithm.
Highervaluesofc1ensurelargerdeviationoftheparticleinthesearchspace(exploration),whilehighervaluesofc2signifytheconvergencetothecurrentglobalbest(exploitation).
InEMMAc1isallowedtodecreasefromc1itoc1fandc2isallowedtoincreasefromc2itoc2f.
SeeTripathietal.
(2007)formoredetails.
ValueAnobjectofclassemmawiththecomponentslistedbelow:xpopExperimentalpointsinvestigated.
ypopResponsevaluesobservedattheexperimentalpointsinvestigated.
xspaceExperimentalregion.
Itisgivenbyallthepossiblecombinationsofthefactors'levelsandcontainsxpop.
Therownamesuniquelyidentifytheexperimentalpointsandarereportedalsoinxpop.
yspaceResponsevaluesthathavebeeneitherobservedorpredicted.
Observedresponsevaluesarestoredalsoinypop.
PredictedresponsevaluesareobtainedusingaMARSmodelttedtotheavailabledata.
8emmaoptIndicatesifeachsinglefunctioniseitherminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointsselectedinitially(t=0).
naNumberofexperimentalpointsselectedinsubsequentiterations(t>0).
testedIDsofthetestedexperimentalpoints.
timeCurrenttimeinstantoftheEMMAprocedure.
weightRelativeimportanceofeachresponse.
Ifonlyoneresponseisinvestigated,thenweight=1;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
GbIDofthebestexperimentalpointinvestigated(globalbest).
Usexspace[Gb,]tovisualisetheglobalbestanduseyspace[Gb,]tovisualiseitsmeasuredre-sponsevalue(s).
Gbidentiestheexperimentalpointwhoseresponsevaluesareclosesttothetarget;thetargetisasetofdesirableresponsevalueswhichareau-tomaticallyselectedonthebasisofthemeasuredandpredictedresponsevalues.
PbIDofthebestexperimentalpointinvestigatedbyeachparticle(personalbest).
Usexspace[Pb,]toidentifythepersonalbestsanduseyspace[Pb,]tovisu-alisetheirmeasuredresponsevalues.
Amongtheexperimentalpointsassociatedtooneparticle,thePbidentiestheexperimentalpointthatiswhoseresponsevaluesareclosesttothetarget.
Gb.
archArchiveoftheglobalbestsidentied.
Becausetheglobalbestchangesasnewexperimentalpointsareinvestigated,anarchiveismaintained.
Pb.
archArchiveofthepersonalbestsidentied.
Becausethepersonalbestschangeasnewexperimentalpointsareinvestigated,anarchiveismaintained.
vVelocitiesusedtoupdatetheparticlesposition.
Thepositionofaparticleisuniquelydeterminedbythepredictors'values;italsodenestheexperimenttobeperformed.
AteachstepofEMMA,thepositionofaparticleisupdatedbyaddinganumericalvalue(velocity)tothecurrentvalueofeachsinglepredictor.
sam.
xIDsoftheexperimentsthathavebeenselectedinthecurrentiterationoftheprocedure.
Usexspace[sam.
x,]tovisualisetheexperimentstobeperformed.
addLogical.
If'0'indicatesthatanadditionalexperimentalpointneedstobeinves-tigated;if'1'indicatesthatanadditionalexperimentalpointisnotrequired.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
FriedmanJ.
H.
(1991)'Multivariateadaptiveregressionsplines'(withdiscussion),TheAnnalsofStatistics19,1:141.
emmacheck9TripathiP.
K.
,BandyopadhyayS.
,PalS.
K.
(2007)'Multi-objectiveparticleswarmoptimizationwithtimevariantinertiaandaccelerationcoefcients'InformationSciences,177,5033:5049.
Examples##1responsevariable##in.
namex1","x2")nlevx1","x2")nlevx"),nd=10,na=5,weight,C,w1=0.
7,w2=0.
4,c1i=2.
5,c1f=0.
5,c2i=0.
5,c2f=2.
5,b=5,pr.
mut,graph="yes",fn1=ackley,fn2=peaks,nresp=2)emmacheckCheckingtheneedforadditionalexperimentsDescriptionThefunctionevaluatesifoneadditionalexperimentalpointisrequired.
Ifthisisthecase,thefunctionprovideswithdetailsabouttheadditionalexperimenttobeperformed.
Usageemmacheck(x,graph,fn1=NULL,fn2=NULL,fn3=NULL,fn4=NULL,nresp)10emmacheckArgumentsxAnobjectofclassemmatn.
graphLogical;if"yes",aplotoftheMARSmodelisproduced.
Notethataplotisproducedonlyifthemodelcontainsmorethanoneexplanatoryvariable.
fn1Therstfunctiontobeoptimised.
Usefn1=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
fn2Thesecondfunctiontobeoptimised.
Usefn2=NULLifthefunctionisun-known(e.
g.
whendesigningexperimentsinappliedproblems).
fn3Thethirdfunctiontobeoptimised.
Usefn3=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
fn4Thefourthfunctiontobeoptimised.
Usefn4=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
nrespTheresponsetobeplotted.
Usenresp=1toplottherstresponse.
.
.
DetailsOncetheexperimentsidentiedbyemmaareimplemented,theobservedresponsevalues,thepre-dictedresponsevalues,thetargetandthescalardistancesfromthetargetareupdated.
Thesolutionwiththeresponsevaluesclosesttothetargetisthusidentied.
Ifsuchasolutionhasnotbeentestedyet,emmacheckselectsitasanadditionalexperimentalpointthatneedstobeinvestigated.
ValueAnobjectofclassemmatnwiththecomponentslistedbelow:xpopExperimentalpointsinvestigated.
ypopResponsevaluesobservedattheexperimentalpointsinvestigated.
xspaceExperimentalregion.
Itisgivenbyallthepossiblecombinationsofthefactors'levelsandcontainsxpop.
Therownamesuniquelyidentifytheexperimentalpointsandarereportedalsoinxpop.
yspaceResponsevaluesthathavebeeneitherobservedorpredicted.
Observedresponsevaluesarestoredalsoinypop.
PredictedresponsevaluesareobtainedusingaMARSmodelttedtotheavailabledata.
optIndicatesifeachsinglefunctioniseitherminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointsselectedinitially(t=0).
naNumberofexperimentalpointsselectedinsubsequentiterations(t>0).
GbIDofthebestexperimentalpointinvestigated.
Usexspace[Gb,]tovisualisethebestexperimentalpointanduseyspace[Gb,]tovisualisethemeasuredre-sponsevalue(s).
Gbidentiestheexperimentalpointwhoseresponsevaluesareclosesttothetarget.
Thetargetisasetofdesirableresponsevalueswhichareautomaticallyselectedonthebasisofthemeasuredandpredictedresponseval-ues.
addLogical.
If'0'indicatesthatanadditionalexperimentalpointneedstobeinves-tigated;if'1'indicatesthatanadditionalexperimentalpointisnotrequired.
emmacheck11testIDsofthetestedexperimentalpoints.
timeCurrenttimeinstantoftheEMMAprocedure.
weightImportanceofeachresponse.
Ifonlyoneresponseisinvestigated,thenweight=1;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
FriedmanJ.
H.
(1991)'Multivariateadaptiveregressionsplines'(withdiscussion),TheAnnalsofStatistics19,1:141.
Examples##definetheproblemvariablesin.
namex1","x2")nlevxperimentalruns(initialization)tnxperimentalrunsduringsubsequentstepsofthe##EMMAprocedurefor(tin1:(C-1)){tnxperimentalruns(initialization)tnxperimentsin\code{tn$xpop}andmeasuretheresponse##values,thenloadthemeasuredresponsevaluesin\code{tn$ypop}tn$ypopxpop)##identifytheexperimentalrunsduringsubsequentstepsofthe##EMMAprocedurefor(tin1:(C-1)){tnxpop)tnxpop)}emmat0DeningtheinitialdesignDescriptionThefunctioninitializestheEMMAprocedure.
Itgeneratestheexperimentalspaceandselectstheinitialsetofexperimentalpoints,namelytheinitialsetofexperimentstobeperformed.
Randomsamplingisusedforthatpurpose.
Usageemmat0(in.
name,nlev,lower,upper,out.
name,nd,fn1=NULL,fn2=NULL,fn3=NULL,fn4=NULL)Argumentsin.
nameAvectorcontainingthenamesoftheinputvariables(factors).
nlevAnumericvectorofthesamelengthasin.
name,containingthenumberoffactorlevels.
lowerAnumericvectorofthesamelengthasin.
name,containingthelowervaluesofthefactors.
upperAnumericvectorofthesamelengthasin.
name,containingtheuppervaluesofthefactors.
out.
nameAvectorcontainingthename(s)oftheoutput/responsevariable(s).
ndNumberofexperimentalpointstobeselectedwhent=0.
emmat013fn1Therstfunctiontobeoptimised;usefn1=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn2Therstfunctiontobeoptimised;usefn2=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn3Thethirdfunctiontobeoptimised;usefn3=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn4Thefourthfunctiontobeoptimised;usefn4=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
DetailsAtthemomentthefunctiondoesnotimplementtheuseofconstraintsforthefactors.
Unfeasibleexperimentsareeasilyexcludedbymanipulatingthematrixxspaceinanobjectofclassemmat0.
ValueAnobjectofclassemmat0withthecomponentslistedbelow:xpopExperimentalpointsinvestigated.
ypopResponsevaluesobservedattheexperimentalpointsinvestigated.
xspaceExperimentalregion.
yspaceResponsevaluesthathavebeeneitherobservedorpredicted.
Observedresponsevaluesarestoredalsoinypop.
PredictedresponsevaluesareobtainedusingaMARSmodelttedtotheavailabledata.
optIndicatesifeachsinglefunctioniseitherminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointsselectedinitially(t=0).
naNumberofexperimentalpointsselectedinsubsequentiterations(t>0).
testedIDofthetestedexperimentalpoints.
timeCurrenttimeinstantoftheEMMAprocedure.
optIndicatesifeachsingleobjectivefunctioniseitherminimized('mn')ormaxi-mized('mx').
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
14emmatnExamples##1responsevariable####definetheproblemvariablesin.
namex1","x2")nlevxperimentalruns(initialization)##simulatedproblem(withknownobjectivefunction)tnxperimentsin\code{tn$xpop}andmeasurethe##responsevalues,thenloadin\code{tn$ypop}themeasured##responsevalues#tn$ypopx1","x2")nlevxperimentalpointsDescriptionGiventhesetofexperimentalpointsinvestigatedinpreviousstepsoftheEMMAprocedureandtheirmeasuredresponsevalues,emmatnreturnsanewsetofexperimentalpointstobeinvestigated(andthusnewexperimentstobeperformed).
emmatn15Usageemmatn(t,x,na,opt,weight,C,w1,w2,c1i,c1f,c2i,c2f,b,pr.
mut,graph,fn1=NULL,fn2=NULL,fn3=NULL,fn4=NULL,nresp)ArgumentstAnumericvalueindicatingthecurrenttimeinstantoftheEMMAprocedure.
xAnobjectofclassemmat0oremmatn.
Useemmat0ift=1;useemmatnift>1.
naAnumericvalueindicatingthenumberofexperimentalpointstobeselectedwhent>0.
optAcharactervectorofthesamelengthasthenumberofresponses,indicatingforeachresponsefunction,iftheresponsemustbeminimized('mn')ormaximized('mx').
weightAnumericalvectorofthesamelengthasthenumberofresponses,reectingtherelevanceofeachresponse.
Useweight=1ifonlyoneresponseisinvestigated;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
CAnumericvalueindicatingthemaximumnumberofiterations.
ThedefaultisC=20.
w1TherstnumericvalueusedtocalculatetheinertiaweightparameterofthetimevariantPSOalgorithm.
Thedefaultisw1=0.
7.
w2ThesecondnumericvalueusedtocalculatetheinertiaweightparameterofthetimevariantPSOalgorithm.
Thedefaultisw2=0.
4.
c1iTherstnumericvalueusedtocalculatetheaccelerationcoefcientc1ofthetimevariantPSOalgorithm.
Thedefaultisc1i=2.
5.
c1fThesecondnumericvalueusedtocalculatetheaccelerationcoefcientc1ofthetimevariantPSOalgorithm.
Thedefaultisc1f=0.
5.
c2iTherstnumericvalueusedtocalculatetheaccelerationcoefcientc2ofthetimevariantPSOalgorithm.
Thedefaultisc2i=0.
5.
c2fThesecondnumericvalueusedtocalculatetheaccelerationcoefcientc2ofthetimevariantPSOalgorithm.
Thedefaultisc2f=2.
5.
bAnumericvalue,usedinthemutationoperator,thatdeterminesthedegreeofdependenceofthemutationontheiterationnumber.
Thedefaultisb=5.
pr.
mutAnumericvectorofthesamelengthasthenumberofiterationsCcontainingtheprobabilityofmutationforeachtimeinstant.
graphLogical;if'yes',aplotoftheMARSmodelisproduced.
Aplotisproducedonlyifthemodelcontainsmorethanoneexplanatoryvariable.
fn1Therstfunctiontobeoptimised;usefn1=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn2Therstfunctiontobeoptimised;usefn2=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
16emmatnfn3Thethirdfunctiontobeoptimised;usefn3=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn4Thefourthfunctiontobeoptimised;usefn4=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
nrespTheresponsetobeplotted.
Usenresp=1toplottherstresponse.
.
.
DetailsTheparametersw1andw2areusedtocalculatetheinertiaweightwofthePSOalgorithm,namelytheparameterthatcontrolstheinuenceofthepreviousparticlevelocityonthepresentvelocity.
Highvaluesofwfavouraglobalsearch,whereaslowervaluesofwencouragealocalsearch.
InEMMAtheinertiaweightisallowedtodecreaselinearlywithiterationfromw1tow2thusfavouringtheexplorationinitiallyandtheexploitationsubsequently.
Theparametersc1iandc1fareusedtocalculatethecognitiveaccelerationcoefcientc1ofthePSOalgorithm,whereastheparametersc2iandc2fareusedtocalculatethesocialaccelerationcoefcientc2ofthePSOalgorithm.
Highervaluesofc1ensurelargerdeviationoftheparticleinthesearchspace(exploration),whilehighervaluesofc2signifytheconvergencetothecurrentglobalbest(exploitation).
InEMMAc1isallowedtodecreasefromc1itoc1fandc2isallowedtoincreasefromc2itoc2f.
SeeTripathietal.
(2007)formoredetails.
ValueAnobjectofclassemmawiththecomponentslistedbelow:xpopExperimentalpointsinvestigated.
ypopResponsevaluesobservedattheexperimentalpointsinvestigated.
xspaceExperimentalregion.
Itisgivenbyallthepossiblecombinationsofthefactors'levelsandcontainsxpop.
Therownamesuniquelyidentifytheexperimentalpointsandarereportedalsoinxpop.
yspaceResponsevaluesthathavebeeneitherobservedorpredicted.
Observedresponsevaluesarestoredalsoinypop.
PredictedresponsevaluesareobtainedusingaMARSmodelttedtotheavailabledata(xpop,ypop).
optIndicatesifeachsinglefunctioniseitherminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointsselectedinitially(t=0).
naNumberofexperimentalpointsselectedinsubsequentiterations(t>0).
testedIDsofthetestedexperimentalpoints.
timeCurrenttimeinstantoftheEMMAprocedure.
weightRelativeimportanceofeachresponse.
Ifonlyoneresponseisinvestigated,thenweight=1;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
GbIDofthebestexperimentalpointinvestigated(globalbest).
Gbidentiestheexperimentalpointwhoseresponsevaluesareclosesttothedesirableresponsevalues(target);thetargetisautomaticallyselectedonthebasisofthemeasuredandpredictedresponsevalues.
Usexspace[Gb,]tovisualisetheglobalbestanduseyspace[Gb,]tovisualiseitsmeasuredresponsevalue(s).
emmatn17PbIDofthebestexperimentalpointinvestigatedbyeachparticle(personalbest).
Usexspace[Pb,]toidentifythepersonalbestsanduseyspace[Pb,]tovisu-alisetheirmeasuredresponsevalues.
Amongtheexperimentalpointsassociatedtooneparticle,thePbidentiestheexperimentalpointthatiswhoseresponsevaluesareclosesttothetarget.
Gb.
archArchiveoftheglobalbestsidentied.
Becausetheglobalbestchangesasnewexperimentalpointsareinvestigated,anarchiveismaintained.
Pb.
archArchiveofthepersonalbestsidentied.
Becausethepersonalbestschangeasnewexperimentalpointsareinvestigated,anarchiveismaintained.
vVelocitiesusedtoupdatetheparticlesposition.
Thepositionofaparticleisuniquelydeterminedbythepredictors'values;italsodenestheexperimenttobeperformed.
AteachstepofEMMA,thepositionofaparticleisupdatedbyaddinganumericalvalue(velocity)tothecurrentvalueofeachsinglepredictor.
sam.
xIDsoftheexperimentsthathavebeenselectedinthecurrentiterationoftheprocedure.
Usexspace[sam.
x,]tovisualisetheexperimentstobeperformed.
addLogical.
If'0'indicatesthatanadditionalexperimentalpointneedstobeinves-tigated;if'1'indicatesthatanadditionalexperimentalpointisnotrequired.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
FriedmanJ.
H.
(1991)'Multivariateadaptiveregressionsplines'(withdiscussion),TheAnnalsofStatistics19,1:141.
TripathiP.
K.
,BandyopadhyayS.
,PalS.
K.
(2007)'Multi-objectiveparticleswarmoptimizationwithtimevariantinertiaandaccelerationcoefcients'InformationSciences,177,5033:5049.
Examples##Notrun:##1responsevariable##in.
namex1","x2")nlevxpop)for(tin1:(C-1)){tnxpop)tnxpop)}##End(Notrun)##2responsevariables##in.
namex1","x2")nlevx"),weight,pr.
mut=pr.
mut,graph="yes",fn1=ackley,fn2=peaks)tnx)ArgumentsxAmatrixcontainingthevaluesoftheinputvariables.
ValueVectorofthesamelengthasxgivingthevaluesofthebenchmarkfunction.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanExamplesx1x2x1xxpand.
grid(x1,x2)zx(peaks(x),nrow=length(x1))nrzx1,x2,z,col=color[facetcol],theta=0,phi=10,expand=1,xlab="x1",ylab="x2",zlab="f(x1,x2)",ticktype="detailed")20plot.
emmaplot.
emma3DsimulationplotDescriptionForaproblemwith1responseand2inputvariables(factors)plotsa3Dgraphandshowshowthesimulationevolves.
Usage##S3methodforclassemmaplot(x,n=50,fn,C=10,.
.
.
)ArgumentsxAnobjectofclass'emma'.
nThenumberoffactors'levelstobeplotted.
fnTheoptimizationfunction.
CThenumberoftimeinstantsusedintheEMMAprocedure.
.
.
.
Otherargumentsnotused.
Value.
.
.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanExamplesin.
namex1","x2")nlevx",nd=10,na=5,weight,C,w1=0.
7,w2=0.
4,c1i=2.
5,c1f=0.
5,c2i=0.
5,c2f=2.
5,b=5,pr.
mut,graph="no",fn1=peaks)plot(emma.
peaks,fn=peaks,n=50,C=20)IndexTopic\textasciitildeDesignedExperimentsdistance,4emma,5emma-package,2emmacheck,9emmat0,12emmatn,14Topic\textasciitildeOptimizationackley,3distance,4emma,5emmacheck,9emmat0,12emmatn,14peaks,19plot.
emma,20ackley,3distance,4EMMA(emma-package),2emma,5,7,10,16emma-package,2emmacheck,7,9emmat0,5,12,13,15emmatn,5,10,14,15peaks,19plot.
emma,2021
1-0Date2011-10-22AuthorLauraVillanova,KateSmith-MilesandRobJHyndman.
MaintainerLauraVillanovaDependsR(>=2.
9.
2),earth,clusterSimImportsmethodsDescriptionTheevolutionarymodel-basedmultiresponseapproach(EMMA)isanovelmethodologytoprocessoptimisationandproductimprovement.
Theapproachissuitabletocontextsinwhichtheexperimentalcostand/ortimelimitthenumberofimplementabletrials.
LicenseGPL(>=2)LazyLoadyesRepositoryCRANDate/Publication2011-10-2617:59:36NeedsCompilationnoRtopicsdocumented:emma-package2ackley3distance4emma5emmacheck9emmat012emmatn14peaks19plot.
emma2012emma-packageIndex21emma-packageDesigningexperimentsforprocessoptimizationDescriptionTheevolutionarymodel-basedmultiresponseapproach(EMMA)isaprocedureforprocessopti-mizationandproductimprovement.
Itisparticularlysuitedtoprocessesfeaturingirregularexper-imentalregionduetoconstraintsontheinputvariables(factors),multipleresponsesnotaccomo-datedbypolynomialmodels,andexpensiveortime-consumingexperiments.
EMMAiterativellyselectsnewexperimentalpointsthatincreasinglyconcentrateonthemostpromisingregionsoftheexperimentalspace.
Theselectionofthenewexperimentalpointsisperformedonthebasisoftheresultsachievedduringprevioustrials.
Amultivariateadaptiveregressionsplines(MARS)modelandaparticleswarmoptimization(PSO)algorithmareusedtodrivethesearchoftheoptimum.
DetailsPackage:emmaType:PackageVersion:1.
0Date:2011-02-22License:GPL(>=2)LazyLoad:yesAuthor(s)LauraVillanova,KateSmith-MilesandRobJHyndmanMaintainer:LauraVillanovaReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
Examplesin.
namex1","x2")nlevx",nd=10,na=5,weight,C,w1=0.
7,w2=0.
4,c1i=2.
5,c1f=0.
5,c2i=0.
5,c2f=2.
5,b=5,pr.
mut,graph="no",fn1=peaks)plot(emma.
peaks,fn=peaks,n=50,C=20)ackleyAckleytestproblemDescriptionGeneratestheAckleybenchmarkfunction.
TheAckleyfunctionisacommonlyusedtestproblemforglobaloptimizationprocedures.
Usageackley(x)ArgumentsxAmatrixcontainingthevaluesoftheinputvariables.
ValueVectorofthesamelengthasxgivingthevaluesoftheAckleyfunction.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferenceshttp://www-optima.
amp.
i.
kyoto-u.
ac.
jp/member/student/hedar/Hedar_files/TestGO_files/Page295.
htm4distanceExamplesx1x2x1xxpand.
grid(x1,x2)zx(ackley(x),nrow=length(x1))nrzx1,x2,z,col=color[facetcol],theta=0,phi=10,expand=1,xlab="x1",ylab="x2",zlab="f(x1,x2)",ticktype="detailed")distanceDistancemeasurefromthetargetDescriptionComputesascalardistancebetweenthetarget(asetofdesirablevaluesfortheresponses)andtheresponsesvaluesthathavebeeneitherobservedorestimatedforeachpointintheexperimentalspace.
Suchadistanceisusedtoidentifyadditionalexperimentalpointstobeinvestigated.
Usagedistance(xpop,xspace,yspace,weight,opt)ArgumentsxpopAdataframecontainingthefactorvaluesfortheexperimentalpointsinvesti-gated;therownamesuniquelyidentifyeachexperimentalpoint(ID).
xspaceAdataframecontainingthefactorvaluesfortheexperimentalpointsdeningtheentireexperimentalregion;therownamesuniquelyidentifyeachexperi-mentalpoint(ID).
yspaceAdataframecontainingtheresponsevalues(eitherobservedorestimated)forthepointsintheexperimentalregion.
weightAnumericalvector,ofthesamelengthasthenumberofresponses,containingtheweightsassignedtotheeachresponse;thesumoftheweightsmustbeequalto1.
optAcharactervector,ofthesamelengthasthenumberofresponses,deningifeachresponseneedstobeminimizedormaximized.
Theallowedvaluesare'mn'(minimize)and'mx'(maximize).
emma5DetailsThefunctionnormalizestheresponsevalueswithrespecttotheestimatedlimitsoftheresponsespace,sothattheresponsevaluesliebetween0and1.
Subsequently,thefunctionidentiesthetargetandcomputesascalardistancebetweenthetargetandtheresponsevalues.
ValuefitThescalardistancesbetweenthetargetandtheresponse(s)valuesfortheexper-imentalpointsinxpop.
obj.
nnScalardistancefromthetargetforthebestexperimentalpointidentiedbyEMMA.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesFriedmanJ.
H.
(1991)'Multivariateadaptiveregressionsplines'(withdiscussion),TheAnnalsofStatistics19,1:141.
VillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165emmaEvolutionaryModel-basedMultiresponseApproachDescriptionEMMAdesignstheexperimentsusingaprocedurebasedontheParticleSwarmOptimization(PSO)algorithm.
Firstly,EMMAselectsasetofinitialexperimentalpoints(seeemmat0)thatdenetheinitialpositionoftheparticles;subsequently,foragivennumberofiterations,theparticlesaremovedandnewexperimentalpointsareselected(seeemmatn).
Usageemma(in.
name,nlev,lower,upper,out.
name,opt="mn",nd=10,na=5,weight,C=20,w1=0.
7,w2=0.
4,c1i=2.
5,c1f=0.
5,c2i=0.
5,c2f=2.
5,b=5,pr.
mut,graph,fn1=NULL,fn2=NULL,fn3=NULL,fn4=NULL,nresp)6emmaArgumentsin.
nameAvectorcontainingthenamesoftheinputvariables(factors).
nlevAnumericvectorofthesamelengthasin.
name,containingthenumberoffactorlevels.
lowerAnumericvectorofthesamelengthasin.
name,containingthelowervaluesofthefactors.
upperAnumericvectorofthesamelengthasin.
name,containingtheuppervaluesofthefactors.
out.
nameAvectorcontainingthename(s)oftheoutput/responsevariable(s).
optAcharactervectorofthesamelengthasthenumberofresponses,indicatingforeachresponsefunction,iftheresponsemustbeminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointstobeselectedwhent=0.
naAnumericvalueindicatingthenumberofexperimentalpointstobeselectedwhent>0.
weightAnumericalvectorofthesamelengthasthenumberofresponses,reectingtherelevanceofeachresponse.
Useweight=1ifonlyoneresponseisinvestigated;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
CAnumericvalueindicatingthemaximumnumberofiterations.
w1TherstnumericvalueusedtocalculatetheinertiaweightparameterofthetimevariantPSOalgorithm;thedefaultisw1=0.
7.
w2ThesecondnumericvalueusedtocalculatetheinertiaweightparameterofthetimevariantPSOalgorithm;Thedefaultisw2=0.
4.
c1iTherstnumericvalueusedtocalculatetheaccelerationcoefcientc1ofthetimevariantPSOalgorithm;thedefaultisc1i=2.
5.
c1fThesecondnumericvalueusedtocalculatetheaccelerationcoefcientc1ofthetimevariantPSOalgorithm;thedefaultisc1f=0.
5.
c2iTherstnumericvalueusedtocalculatetheaccelerationcoefcientc2ofthetimevariantPSOalgorithm;thedefaultisc2i=0.
5.
c2fThesecondnumericvalueusedtocalculatetheaccelerationcoefcientc2ofthetimevariantPSOalgorithm;thedefaultisc2f=2.
5.
bAnumericvalue,usedinthemutationoperator,thatdeterminesthedegreeofdependenceofthemutationontheiterationnumber;thedefaultisb=5.
pr.
mutAnumericvectorofthesamelengthasthenumberofiterationsCcontainingtheprobabilityofmutationforeachtimeinstant.
graphLogical;if'yes',aplotoftheMARSmodelisproduced.
Aplotisproducedonlyifthemodelcontainsmorethanoneexplanatoryvariable.
fn1Therstfunctiontobeoptimised.
Usefn1=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
fn2Thesecondfunctiontobeoptimised.
Usefn2=NULLifthefunctionisun-known(e.
g.
whendesigningexperimentsinappliedproblems).
emma7fn3Thethirdfunctiontobeoptimised.
Usefn3=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
fn4Theforthfunctiontobeoptimised.
Usefn4=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
nrespTheresponsetobeplotted.
Usenresp=1toplottherstresponse.
.
.
DetailsToselectthenewexperimentalpointstobeinvestigated,thefollowingstepsareiterated.
AMARSmodelisttedtothecollecteddatasothatanapproximatedfunctionisobtainedforeachresponse;theseapproximatedfunctionsareusedtopredicttheresponsevaluesatthenon-investigatedexper-imentalpoints.
EachpointintheexperimentalregionE(xspace)isnowassociatedwithavectorofresponsevaluesthathasbeeneithermeasuredorestimated.
Thebest(measuredorestimated)valueofeachresponseisselectedandusedtoidentifythetarget.
Subsequently,foreachexperimentalpointinE,thescalardistancebetweentheresponsevaluesandthetargetiscomputedandthesolu-tionthatisclosesttothetargetisselected.
Ifsuchsolutionhasnotbeentestedyet(seeemmacheck),theexperimentneedstobeperformedanditsresponsevaluesaremeasured.
Thetargetisthenupdated,aswellasthescalardistancesofalltheexperimentalpointsfromthetarget.
Thescalardistancesareusedtoidentifythegoodperformingexperimentalpoints.
Theexperimentalpointwhoseresponsevaluesareclosesttothetargetisreferredtoastheglobalbest.
Similarly,apersonalbestisidentiedforeachparticlebyconsideringtheexperimentalpointsvisitedbythatparticleandselectingthatpointfeaturingtheresponsevaluesthatareclosesttothetarget.
Finally,theparticlesvelocityandpositionareupdatedandanewsetofexperimentalpointsisidentied.
Theparametersw1andw2areusedtocalculatetheinertiaweightwofthePSOalgorithm,namelytheparameterthatcontrolstheinuenceofthepreviousparticlevelocityonthepresentvelocity.
Highvaluesofwfavouraglobalsearch,whereaslowervaluesofwencouragealocalsearch.
InEMMAtheinertiaweightisallowedtodecreaselinearlywithiterationfromw1tow2thusfavouringtheexplorationinitiallyandtheexploitationsubsequently.
Theparametersc1iandc1fareusedtocalculatethecognitiveaccelerationcoefcientc1ofthePSOalgorithm,whereastheparametersc2iandc2fareusedtocalculatethesocialaccelerationcoefcientc2ofthePSOalgorithm.
Highervaluesofc1ensurelargerdeviationoftheparticleinthesearchspace(exploration),whilehighervaluesofc2signifytheconvergencetothecurrentglobalbest(exploitation).
InEMMAc1isallowedtodecreasefromc1itoc1fandc2isallowedtoincreasefromc2itoc2f.
SeeTripathietal.
(2007)formoredetails.
ValueAnobjectofclassemmawiththecomponentslistedbelow:xpopExperimentalpointsinvestigated.
ypopResponsevaluesobservedattheexperimentalpointsinvestigated.
xspaceExperimentalregion.
Itisgivenbyallthepossiblecombinationsofthefactors'levelsandcontainsxpop.
Therownamesuniquelyidentifytheexperimentalpointsandarereportedalsoinxpop.
yspaceResponsevaluesthathavebeeneitherobservedorpredicted.
Observedresponsevaluesarestoredalsoinypop.
PredictedresponsevaluesareobtainedusingaMARSmodelttedtotheavailabledata.
8emmaoptIndicatesifeachsinglefunctioniseitherminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointsselectedinitially(t=0).
naNumberofexperimentalpointsselectedinsubsequentiterations(t>0).
testedIDsofthetestedexperimentalpoints.
timeCurrenttimeinstantoftheEMMAprocedure.
weightRelativeimportanceofeachresponse.
Ifonlyoneresponseisinvestigated,thenweight=1;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
GbIDofthebestexperimentalpointinvestigated(globalbest).
Usexspace[Gb,]tovisualisetheglobalbestanduseyspace[Gb,]tovisualiseitsmeasuredre-sponsevalue(s).
Gbidentiestheexperimentalpointwhoseresponsevaluesareclosesttothetarget;thetargetisasetofdesirableresponsevalueswhichareau-tomaticallyselectedonthebasisofthemeasuredandpredictedresponsevalues.
PbIDofthebestexperimentalpointinvestigatedbyeachparticle(personalbest).
Usexspace[Pb,]toidentifythepersonalbestsanduseyspace[Pb,]tovisu-alisetheirmeasuredresponsevalues.
Amongtheexperimentalpointsassociatedtooneparticle,thePbidentiestheexperimentalpointthatiswhoseresponsevaluesareclosesttothetarget.
Gb.
archArchiveoftheglobalbestsidentied.
Becausetheglobalbestchangesasnewexperimentalpointsareinvestigated,anarchiveismaintained.
Pb.
archArchiveofthepersonalbestsidentied.
Becausethepersonalbestschangeasnewexperimentalpointsareinvestigated,anarchiveismaintained.
vVelocitiesusedtoupdatetheparticlesposition.
Thepositionofaparticleisuniquelydeterminedbythepredictors'values;italsodenestheexperimenttobeperformed.
AteachstepofEMMA,thepositionofaparticleisupdatedbyaddinganumericalvalue(velocity)tothecurrentvalueofeachsinglepredictor.
sam.
xIDsoftheexperimentsthathavebeenselectedinthecurrentiterationoftheprocedure.
Usexspace[sam.
x,]tovisualisetheexperimentstobeperformed.
addLogical.
If'0'indicatesthatanadditionalexperimentalpointneedstobeinves-tigated;if'1'indicatesthatanadditionalexperimentalpointisnotrequired.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
FriedmanJ.
H.
(1991)'Multivariateadaptiveregressionsplines'(withdiscussion),TheAnnalsofStatistics19,1:141.
emmacheck9TripathiP.
K.
,BandyopadhyayS.
,PalS.
K.
(2007)'Multi-objectiveparticleswarmoptimizationwithtimevariantinertiaandaccelerationcoefcients'InformationSciences,177,5033:5049.
Examples##1responsevariable##in.
namex1","x2")nlevx1","x2")nlevx"),nd=10,na=5,weight,C,w1=0.
7,w2=0.
4,c1i=2.
5,c1f=0.
5,c2i=0.
5,c2f=2.
5,b=5,pr.
mut,graph="yes",fn1=ackley,fn2=peaks,nresp=2)emmacheckCheckingtheneedforadditionalexperimentsDescriptionThefunctionevaluatesifoneadditionalexperimentalpointisrequired.
Ifthisisthecase,thefunctionprovideswithdetailsabouttheadditionalexperimenttobeperformed.
Usageemmacheck(x,graph,fn1=NULL,fn2=NULL,fn3=NULL,fn4=NULL,nresp)10emmacheckArgumentsxAnobjectofclassemmatn.
graphLogical;if"yes",aplotoftheMARSmodelisproduced.
Notethataplotisproducedonlyifthemodelcontainsmorethanoneexplanatoryvariable.
fn1Therstfunctiontobeoptimised.
Usefn1=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
fn2Thesecondfunctiontobeoptimised.
Usefn2=NULLifthefunctionisun-known(e.
g.
whendesigningexperimentsinappliedproblems).
fn3Thethirdfunctiontobeoptimised.
Usefn3=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
fn4Thefourthfunctiontobeoptimised.
Usefn4=NULLifthefunctionisunknown(e.
g.
whendesigningexperimentsinappliedproblems).
nrespTheresponsetobeplotted.
Usenresp=1toplottherstresponse.
.
.
DetailsOncetheexperimentsidentiedbyemmaareimplemented,theobservedresponsevalues,thepre-dictedresponsevalues,thetargetandthescalardistancesfromthetargetareupdated.
Thesolutionwiththeresponsevaluesclosesttothetargetisthusidentied.
Ifsuchasolutionhasnotbeentestedyet,emmacheckselectsitasanadditionalexperimentalpointthatneedstobeinvestigated.
ValueAnobjectofclassemmatnwiththecomponentslistedbelow:xpopExperimentalpointsinvestigated.
ypopResponsevaluesobservedattheexperimentalpointsinvestigated.
xspaceExperimentalregion.
Itisgivenbyallthepossiblecombinationsofthefactors'levelsandcontainsxpop.
Therownamesuniquelyidentifytheexperimentalpointsandarereportedalsoinxpop.
yspaceResponsevaluesthathavebeeneitherobservedorpredicted.
Observedresponsevaluesarestoredalsoinypop.
PredictedresponsevaluesareobtainedusingaMARSmodelttedtotheavailabledata.
optIndicatesifeachsinglefunctioniseitherminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointsselectedinitially(t=0).
naNumberofexperimentalpointsselectedinsubsequentiterations(t>0).
GbIDofthebestexperimentalpointinvestigated.
Usexspace[Gb,]tovisualisethebestexperimentalpointanduseyspace[Gb,]tovisualisethemeasuredre-sponsevalue(s).
Gbidentiestheexperimentalpointwhoseresponsevaluesareclosesttothetarget.
Thetargetisasetofdesirableresponsevalueswhichareautomaticallyselectedonthebasisofthemeasuredandpredictedresponseval-ues.
addLogical.
If'0'indicatesthatanadditionalexperimentalpointneedstobeinves-tigated;if'1'indicatesthatanadditionalexperimentalpointisnotrequired.
emmacheck11testIDsofthetestedexperimentalpoints.
timeCurrenttimeinstantoftheEMMAprocedure.
weightImportanceofeachresponse.
Ifonlyoneresponseisinvestigated,thenweight=1;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
FriedmanJ.
H.
(1991)'Multivariateadaptiveregressionsplines'(withdiscussion),TheAnnalsofStatistics19,1:141.
Examples##definetheproblemvariablesin.
namex1","x2")nlevxperimentalruns(initialization)tnxperimentalrunsduringsubsequentstepsofthe##EMMAprocedurefor(tin1:(C-1)){tnxperimentalruns(initialization)tnxperimentsin\code{tn$xpop}andmeasuretheresponse##values,thenloadthemeasuredresponsevaluesin\code{tn$ypop}tn$ypopxpop)##identifytheexperimentalrunsduringsubsequentstepsofthe##EMMAprocedurefor(tin1:(C-1)){tnxpop)tnxpop)}emmat0DeningtheinitialdesignDescriptionThefunctioninitializestheEMMAprocedure.
Itgeneratestheexperimentalspaceandselectstheinitialsetofexperimentalpoints,namelytheinitialsetofexperimentstobeperformed.
Randomsamplingisusedforthatpurpose.
Usageemmat0(in.
name,nlev,lower,upper,out.
name,nd,fn1=NULL,fn2=NULL,fn3=NULL,fn4=NULL)Argumentsin.
nameAvectorcontainingthenamesoftheinputvariables(factors).
nlevAnumericvectorofthesamelengthasin.
name,containingthenumberoffactorlevels.
lowerAnumericvectorofthesamelengthasin.
name,containingthelowervaluesofthefactors.
upperAnumericvectorofthesamelengthasin.
name,containingtheuppervaluesofthefactors.
out.
nameAvectorcontainingthename(s)oftheoutput/responsevariable(s).
ndNumberofexperimentalpointstobeselectedwhent=0.
emmat013fn1Therstfunctiontobeoptimised;usefn1=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn2Therstfunctiontobeoptimised;usefn2=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn3Thethirdfunctiontobeoptimised;usefn3=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn4Thefourthfunctiontobeoptimised;usefn4=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
DetailsAtthemomentthefunctiondoesnotimplementtheuseofconstraintsforthefactors.
Unfeasibleexperimentsareeasilyexcludedbymanipulatingthematrixxspaceinanobjectofclassemmat0.
ValueAnobjectofclassemmat0withthecomponentslistedbelow:xpopExperimentalpointsinvestigated.
ypopResponsevaluesobservedattheexperimentalpointsinvestigated.
xspaceExperimentalregion.
yspaceResponsevaluesthathavebeeneitherobservedorpredicted.
Observedresponsevaluesarestoredalsoinypop.
PredictedresponsevaluesareobtainedusingaMARSmodelttedtotheavailabledata.
optIndicatesifeachsinglefunctioniseitherminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointsselectedinitially(t=0).
naNumberofexperimentalpointsselectedinsubsequentiterations(t>0).
testedIDofthetestedexperimentalpoints.
timeCurrenttimeinstantoftheEMMAprocedure.
optIndicatesifeachsingleobjectivefunctioniseitherminimized('mn')ormaxi-mized('mx').
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
14emmatnExamples##1responsevariable####definetheproblemvariablesin.
namex1","x2")nlevxperimentalruns(initialization)##simulatedproblem(withknownobjectivefunction)tnxperimentsin\code{tn$xpop}andmeasurethe##responsevalues,thenloadin\code{tn$ypop}themeasured##responsevalues#tn$ypopx1","x2")nlevxperimentalpointsDescriptionGiventhesetofexperimentalpointsinvestigatedinpreviousstepsoftheEMMAprocedureandtheirmeasuredresponsevalues,emmatnreturnsanewsetofexperimentalpointstobeinvestigated(andthusnewexperimentstobeperformed).
emmatn15Usageemmatn(t,x,na,opt,weight,C,w1,w2,c1i,c1f,c2i,c2f,b,pr.
mut,graph,fn1=NULL,fn2=NULL,fn3=NULL,fn4=NULL,nresp)ArgumentstAnumericvalueindicatingthecurrenttimeinstantoftheEMMAprocedure.
xAnobjectofclassemmat0oremmatn.
Useemmat0ift=1;useemmatnift>1.
naAnumericvalueindicatingthenumberofexperimentalpointstobeselectedwhent>0.
optAcharactervectorofthesamelengthasthenumberofresponses,indicatingforeachresponsefunction,iftheresponsemustbeminimized('mn')ormaximized('mx').
weightAnumericalvectorofthesamelengthasthenumberofresponses,reectingtherelevanceofeachresponse.
Useweight=1ifonlyoneresponseisinvestigated;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
CAnumericvalueindicatingthemaximumnumberofiterations.
ThedefaultisC=20.
w1TherstnumericvalueusedtocalculatetheinertiaweightparameterofthetimevariantPSOalgorithm.
Thedefaultisw1=0.
7.
w2ThesecondnumericvalueusedtocalculatetheinertiaweightparameterofthetimevariantPSOalgorithm.
Thedefaultisw2=0.
4.
c1iTherstnumericvalueusedtocalculatetheaccelerationcoefcientc1ofthetimevariantPSOalgorithm.
Thedefaultisc1i=2.
5.
c1fThesecondnumericvalueusedtocalculatetheaccelerationcoefcientc1ofthetimevariantPSOalgorithm.
Thedefaultisc1f=0.
5.
c2iTherstnumericvalueusedtocalculatetheaccelerationcoefcientc2ofthetimevariantPSOalgorithm.
Thedefaultisc2i=0.
5.
c2fThesecondnumericvalueusedtocalculatetheaccelerationcoefcientc2ofthetimevariantPSOalgorithm.
Thedefaultisc2f=2.
5.
bAnumericvalue,usedinthemutationoperator,thatdeterminesthedegreeofdependenceofthemutationontheiterationnumber.
Thedefaultisb=5.
pr.
mutAnumericvectorofthesamelengthasthenumberofiterationsCcontainingtheprobabilityofmutationforeachtimeinstant.
graphLogical;if'yes',aplotoftheMARSmodelisproduced.
Aplotisproducedonlyifthemodelcontainsmorethanoneexplanatoryvariable.
fn1Therstfunctiontobeoptimised;usefn1=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn2Therstfunctiontobeoptimised;usefn2=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
16emmatnfn3Thethirdfunctiontobeoptimised;usefn3=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
fn4Thefourthfunctiontobeoptimised;usefn4=NULLiftheobjectivefunctionisunknown,likeinappliedproblems.
nrespTheresponsetobeplotted.
Usenresp=1toplottherstresponse.
.
.
DetailsTheparametersw1andw2areusedtocalculatetheinertiaweightwofthePSOalgorithm,namelytheparameterthatcontrolstheinuenceofthepreviousparticlevelocityonthepresentvelocity.
Highvaluesofwfavouraglobalsearch,whereaslowervaluesofwencouragealocalsearch.
InEMMAtheinertiaweightisallowedtodecreaselinearlywithiterationfromw1tow2thusfavouringtheexplorationinitiallyandtheexploitationsubsequently.
Theparametersc1iandc1fareusedtocalculatethecognitiveaccelerationcoefcientc1ofthePSOalgorithm,whereastheparametersc2iandc2fareusedtocalculatethesocialaccelerationcoefcientc2ofthePSOalgorithm.
Highervaluesofc1ensurelargerdeviationoftheparticleinthesearchspace(exploration),whilehighervaluesofc2signifytheconvergencetothecurrentglobalbest(exploitation).
InEMMAc1isallowedtodecreasefromc1itoc1fandc2isallowedtoincreasefromc2itoc2f.
SeeTripathietal.
(2007)formoredetails.
ValueAnobjectofclassemmawiththecomponentslistedbelow:xpopExperimentalpointsinvestigated.
ypopResponsevaluesobservedattheexperimentalpointsinvestigated.
xspaceExperimentalregion.
Itisgivenbyallthepossiblecombinationsofthefactors'levelsandcontainsxpop.
Therownamesuniquelyidentifytheexperimentalpointsandarereportedalsoinxpop.
yspaceResponsevaluesthathavebeeneitherobservedorpredicted.
Observedresponsevaluesarestoredalsoinypop.
PredictedresponsevaluesareobtainedusingaMARSmodelttedtotheavailabledata(xpop,ypop).
optIndicatesifeachsinglefunctioniseitherminimized('mn')ormaximized('mx').
ndNumberofexperimentalpointsselectedinitially(t=0).
naNumberofexperimentalpointsselectedinsubsequentiterations(t>0).
testedIDsofthetestedexperimentalpoints.
timeCurrenttimeinstantoftheEMMAprocedure.
weightRelativeimportanceofeachresponse.
Ifonlyoneresponseisinvestigated,thenweight=1;ifmultipleresponsesareinvestigated,thesumofthevaluesinweightmustbe1.
GbIDofthebestexperimentalpointinvestigated(globalbest).
Gbidentiestheexperimentalpointwhoseresponsevaluesareclosesttothedesirableresponsevalues(target);thetargetisautomaticallyselectedonthebasisofthemeasuredandpredictedresponsevalues.
Usexspace[Gb,]tovisualisetheglobalbestanduseyspace[Gb,]tovisualiseitsmeasuredresponsevalue(s).
emmatn17PbIDofthebestexperimentalpointinvestigatedbyeachparticle(personalbest).
Usexspace[Pb,]toidentifythepersonalbestsanduseyspace[Pb,]tovisu-alisetheirmeasuredresponsevalues.
Amongtheexperimentalpointsassociatedtooneparticle,thePbidentiestheexperimentalpointthatiswhoseresponsevaluesareclosesttothetarget.
Gb.
archArchiveoftheglobalbestsidentied.
Becausetheglobalbestchangesasnewexperimentalpointsareinvestigated,anarchiveismaintained.
Pb.
archArchiveofthepersonalbestsidentied.
Becausethepersonalbestschangeasnewexperimentalpointsareinvestigated,anarchiveismaintained.
vVelocitiesusedtoupdatetheparticlesposition.
Thepositionofaparticleisuniquelydeterminedbythepredictors'values;italsodenestheexperimenttobeperformed.
AteachstepofEMMA,thepositionofaparticleisupdatedbyaddinganumericalvalue(velocity)tothecurrentvalueofeachsinglepredictor.
sam.
xIDsoftheexperimentsthathavebeenselectedinthecurrentiterationoftheprocedure.
Usexspace[sam.
x,]tovisualisetheexperimentstobeperformed.
addLogical.
If'0'indicatesthatanadditionalexperimentalpointneedstobeinves-tigated;if'1'indicatesthatanadditionalexperimentalpointisnotrequired.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanReferencesVillanovaL.
,FalcaroP.
,CartaD.
,PoliI.
,HyndmanR.
,Smith-MilesK.
(2010)'FunctionalizationofMicroarrayDevices:ProcessOptimizationUsingaMultiobjectivePSOandMultiresponseMARSModelling',IEEECEC2010,DOI:10.
1109/CEC.
2010.
5586165CartaD.
,VillanovaL.
,CostacurtaS.
,PatelliA.
,PoliI.
,Vezzu'S.
,ScopeceP.
,LisiF.
,Smith-MilesK.
,HyndmanR.
J.
,HillA.
J.
,FalcaroP.
(2011)'MethodforOptimizingCoatingPropertiesBasedonanEvolutionaryAlgorithmApproach',AnalyticalChemistry83(16),6373-6380.
FriedmanJ.
H.
(1991)'Multivariateadaptiveregressionsplines'(withdiscussion),TheAnnalsofStatistics19,1:141.
TripathiP.
K.
,BandyopadhyayS.
,PalS.
K.
(2007)'Multi-objectiveparticleswarmoptimizationwithtimevariantinertiaandaccelerationcoefcients'InformationSciences,177,5033:5049.
Examples##Notrun:##1responsevariable##in.
namex1","x2")nlevxpop)for(tin1:(C-1)){tnxpop)tnxpop)}##End(Notrun)##2responsevariables##in.
namex1","x2")nlevx"),weight,pr.
mut=pr.
mut,graph="yes",fn1=ackley,fn2=peaks)tnx)ArgumentsxAmatrixcontainingthevaluesoftheinputvariables.
ValueVectorofthesamelengthasxgivingthevaluesofthebenchmarkfunction.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanExamplesx1x2x1xxpand.
grid(x1,x2)zx(peaks(x),nrow=length(x1))nrzx1,x2,z,col=color[facetcol],theta=0,phi=10,expand=1,xlab="x1",ylab="x2",zlab="f(x1,x2)",ticktype="detailed")20plot.
emmaplot.
emma3DsimulationplotDescriptionForaproblemwith1responseand2inputvariables(factors)plotsa3Dgraphandshowshowthesimulationevolves.
Usage##S3methodforclassemmaplot(x,n=50,fn,C=10,.
.
.
)ArgumentsxAnobjectofclass'emma'.
nThenumberoffactors'levelstobeplotted.
fnTheoptimizationfunction.
CThenumberoftimeinstantsusedintheEMMAprocedure.
.
.
.
Otherargumentsnotused.
Value.
.
.
Author(s)LauraVillanova,KateSmith-MilesandRobJHyndmanExamplesin.
namex1","x2")nlevx",nd=10,na=5,weight,C,w1=0.
7,w2=0.
4,c1i=2.
5,c1f=0.
5,c2i=0.
5,c2f=2.
5,b=5,pr.
mut,graph="no",fn1=peaks)plot(emma.
peaks,fn=peaks,n=50,C=20)IndexTopic\textasciitildeDesignedExperimentsdistance,4emma,5emma-package,2emmacheck,9emmat0,12emmatn,14Topic\textasciitildeOptimizationackley,3distance,4emma,5emmacheck,9emmat0,12emmatn,14peaks,19plot.
emma,20ackley,3distance,4EMMA(emma-package),2emma,5,7,10,16emma-package,2emmacheck,7,9emmat0,5,12,13,15emmatn,5,10,14,15peaks,19plot.
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