简化的实数、复数和四元数神经模糊学习算法(IJISA-V10-N5-1)

I.J.IntelligentSystemsandApplications,2018,5,1-13PublishedOnlineMay2018inMECS()DOI:10.5815/ijisa.2018.05.01Copyright©2018MECSI.J.IntelligentSystemsandApplications,2018,5,1-13SimplifiedReal-,Complex-,andQuaternion-ValuedNeuro-FuzzyLearningAlgorithmsRyusukeHataDept.ofHumanandArtificialIntelligenceSystems,GraduateSchoolofEngineering,UniversityofFukui,JapanE-mail:[e-mail]neering,KhulnaUniversityofEngineeringandTechnology,BangladeshE-mail:[e-mail]neering,BangladeshUniversityofEngineeringandTechnology,BangladeshE-mail:[e-mail]lligenceSystems,GraduateSchoolofEngineering,UniversityofFukui,JapanE-mail:[e-mail]:09November2017;Accepted:29January2018;Published:08May2018Abstract—Theconventionalreal-valuedneuro-fuzzymethod(RNF)isbasedonclassicfuzzysystemswithantecedentmembershipfunctionsandconsequentsingletons.RulesinRNFaremadebyallthecombinationsofmembershipfunctions;thus,thenumberofrulesaswellastotalparametersincreaserapidlywiththenumberofinputs.Althoughnetworkparametersarerelativelylessintherecentlydevelopedcomplex-valuedneuro-fuzzy(CVNF)andquaternionneuro-fuzzy(QNF),parametersincreasewithnumberofinputs.Thisstudyinvestigatessimplifiedfuzzyrulesthatconstrainrapidincrementofruleswithinputs;andproposedsimplifiedRNF(SRNF),simplifiedCVNF(SCVNF)andsimplifiedQNF(SQNF)employingtheproposedsimplifiedfuzzyrulesinconventionalmethods.Theproposedsimplifiedneuro-fuzzylearningmethodsdifferfromtheconventionalmethodsintheirfuzzyrulestructures.Themethodstunefuzzyrulesbasedonthegradientdescentmethod.Thenumberofrulesinthesemethodsareequaltothenumberofdivisionsofinputspace;andhencetheyrequiresignificantlylessnumberofparameterstobetuned.Theproposedmethodsaretestedonfunctionapproximationsandclassificationproblems.Theyexhibitmuchlessexecutiontimethantheconventionalcounterpartswithequivalentaccuracy.Duetolessnumberofparameters,theproposedmethodscanbeutilizedfortheproblems(e.g.,real-timecontroloflargesystems)wheretheconventionalmethodsaredifficulttoapplyduetotimeconstrain.IndexTerms—Fuzzyinference,neuro-fuzzy,complex-valuedneuralnetwork,quaternionneuralnetwork,functionapproximation,classification.I.INTRODUCTIONNeuro-fuzzymethodsrefertocombinationsofartificialneuralnetworksandfuzzymodels[1-3].Artificialneuralnetworksarethecomputationalmodelsofneuronalcellbehaviorsinthebrain,andhavehighlearningabilityaswellasparallelprocessingability.Thesepropertiesallowsystemstoperformwellinenvironmentsthataredifficulttoformulate.Fuzzylogicisbasedoninferencerulesandallowsthesystemstousehuman-like“fuzziness.”Inparticular,fuzzyinferencesystemsbasedonif–thenrulesprovidehighrobustnessandhuman-likeinference[4,5].However,itisusuallyhardforhumanbeingtodesignproperfuzzyrulesresultingtheconsumptionofaconsiderabletimetotunefuzzyrules.Neuro-fuzzymethodshavinglearningalgorithmsofartificialneuralnetworksinthefuzzyinferencesystemscansolvetheseproblems.Conceivingcomplementarystrengthsofneuralandfuzzysystems,neuro-fuzzeshavebeenappliedtohandlenumerousreal-lifeproblemsincludingcontrol,functionapproximations,classifications,etc.[1-3,6-8].Avarietyofsystemstructuresandlearningalgorithmsareavailableforneuro-fuzzymethods[9–26].Learningoftheclassicalneuro-fuzzysystemsisbasedonthegradientdescentmethod[9].Itismodifiedtoavoidnon-firingorweakfiring[10,11]andtoimprovelearningefficiency[12].Geneticalgorithmsareappliedtoaneuro-fuzzywithradial-basis-function-basedmembershipfortheautomaticgenerationoffuzzyrules[13].Anadaptiveneuro-fuzzysystemforbuildingandoptimizingfuzzymodelshasbeenproposed[14].Avarietyofneuro-fuzzymethodsarealsoproposedrecently[15–21].Theapplicationsofneuro-fuzzymethodsincludefeatureselection[19,21],2SimplifiedReal-,Complex-,andQuaternion-ValuedNeuro-FuzzyLearningAlgorithmsCopyright©2018MECSI.J.IntelligentSystemsandApplications,2018,5,1-13classification[15–20],andimageprocessing[17].Furthermore,neuro-fuzzymethodsusingcomplex-valuedinputsandoutputshavebeenproposedandappliedtoimageprocessingandtime-seriesprediction[22,23].Inaddition,recentstudiesofneuro-fuzzymethodsheavingcomplex-valuedorquaternion-valuedinputsandreal-valuedoutputsexhibitbetterlearningabilitythantheconventionalmethods[24-28].Theconventionalreal-valuedneuro-fuzzymethod(RNF)isbasedonclassicfuzzysystemswithantecedentmembershipfunctionsandconsequentsingletons.Whentrainingdatawithinput-outputmappingaregiventothenetwork,themembershipfunctionsandsingletonsaretunedbybackpropagationalgorithm.However,thenumberoffuzzyrulesrapidlyincreaseswiththeincrementofthenumberofinputs.IntheRNF,thenumberoffuzzyrulesiscalculatedbythenumberofinputstothepowerofthenumberofdivisionsofinputspace;hence,thelearningtimeincreaseswithinputs.RNFextensionsincomplexandquaterniondomainsreducethenetworkparametersofagivenproblemwithalessnumberofinputs.Complex-valuedneuro-fuzzymethod(CVNF)hascomplex-valuedfuzzyruleswhoseinputs,members