基于进化加权神经网络模糊ANARX模型的非平稳非线性时间序列自适应预测(IJITCS-V8-N10-

I.J.InformationTechnologyandComputerScience,2016,10,1-10PublishedOnlineOctober2016inMECS()DOI:10.5815/ijitcs.2016.10.01Copyright©2016MECSI.J.InformationTechnologyandComputerScience,2016,10,1-10AdaptiveForecastingofNon-StationaryNonlinearTimeSeriesBasedontheEvolvingWeightedNeuro-Neo-Fuzzy-ANARX-ModelZhengbingHuSchoolofEducationalInformationTechnology,CentralChinaNormalUniversity,Wuhan,ChinaE-mail:[e-mail]lUniversityofRadioElectronics,Kharkiv,UkraineE-mail:[e-mail]ersityofRadioElectronics,Kharkiv,UkraineE-mail:[e-mail],[e-mail]—Anevolvingweightedneuro-neo-fuzzy-ANARXmodelanditslearningproceduresareintroducedinthearticle.Thissystemisbasicallyusedfortimeseriesforecasting.It’sbasedonneo-fuzzyelements.Thissystemmaybeconsideredasapoolofelementsthatprocessdatainaparallelmanner.Theproposedevolvingsystemmayprovideonlineprocessingdatastreams.IndexTerms—ComputationalIntelligence,timeseriesprediction,neuro-neo-fuzzySystem,MachineLearning,ANARX,DataStream.I.INTRODUCTIONMathematicalforecastingofdatasequences(timeseries)isnowadayswellstudiedandthereisalargenumberofpublicationsonthistopic.Therearemanymethodsforsolvingthistask:regression,correlation,spectralanalysis,exponentialsmoothing,etc.,andmoreadvancedintellectualsystemsthatrequiresometimesrathercomplicatedmathematicalmethodsandahighqualificationofauser.Theproblembecomesmorecomplicatedwhenanalyzedtimeseriesarebothnon-stationaryandnonlinearandcontainunknownbehaviortrendsaswellasquasiperiodic,stochasticandchaoticcomponents.ThebestresultshavebeenshownbynonlinearforecastingmodelsbasedonmathematicalmethodsofComputationalIntelligence[1-3]suchasneuro-fuzzysystems[4-5]duetotheirapproximatingandextrapolatingproperties,learningabilities,results’transparencyandinterpretability.Themodelstobeespeciallynotedaretheso-calledNARX-models[6]whichhavetheformˆ1,...,,1,...,yxykfykyknxkxkn(1)whereˆykisanestimateofforecastedtimeseriesatdiscretetime1,2,...k;fstandsforacertainnonlineartransformationimplementedbyaneuro-fuzzysystem,xkisanobservedexogenousfactorthatdefinesabehaviorofyk.ItcanbenoticedthatpopularBox–JenkinsAR-,ARX-,ARMAX-modelsaswellasnonlinearNARMA-modelscanbedescribedbytheexpression(1).Thesemodelshavebeenwidelystudied;therearemanyarchitecturesandlearningalgorithmsthatimplementthesemodels,butitisassumedthatmodels’ordersynandxnaregivenapriori.Theseordersarepreviouslyunknowninacaseofstructuralnon-stationarityforanalyzedtimeseries,andtheyalsohavetobeadjustedduringalearningprocedure.Inthiscase,itmakessensetouseevolvingconnectionistsystems[7-10]thatadjustnotonlytheirsynapticweightsandactivation-membershipfunctions,butalsotheirarchitectures.Therearemanyalgorithmsthatimplementtheselearningmethodsbothinabatchmodeandinasequentialmode.Theproblembecomesmorecomplicatedifdataarefedtothesystemwithahighfrequencyintheformofadatastream[11].Here,themostpopularevolvingsystemsturnouttobetoocumbersomeforlearningandinformationprocessinginanonlinemode.Asanalternative,arathersimpleandeffectivearchitecturecanbeconsidered.It’stheso-calledANARX-model(AdditiveNARX)thathastheform[12,13]121ˆ1,12,2......,,nnllykfykxkfykxkfyknxknfyklxkl(2)2AdaptiveForecastingofNon-StationaryNonlinearTimeSeriesBasedontheEvolvingWeightedNeuro-Neo-Fuzzy-ANARX-ModelCopyright©2016MECSI.J.InformationTechnologyandComputerScience,2016,10,1-10(heremax,yxnnn),anoriginaltaskoftheforecastingsystem’ssynthesisisdecomposedintomanylocaltasksofparametricidentificationfornodemodelswithtwoinputvariablesykl,xkl,1,2,...,,...ln.Authors[12,13]usedelementaryRosenblattperceptronswithsigmoidalactivationfunctionsassuchnodes.TheANARX-modelprovidedahighforecastingquality,butgenerallyspeakingitrequiresalargenumberofnodesinitsarchitecture[14].Somesynthesisproblemsofforecastingneuro-fuzzy[15]andneo-fuzzy[15-17]systemsbasedontheANARX-modelsareconsideredinthiswork.Thesesystemsavoidtheabovementioneddrawbacks.Sinceweconsideracaseofstochasticnonlineardynamicsignalsinthisarticle,thebasicnoveltyhastodowithdefiningamodel’sdelayorderinanonlinemode.Theremainderofthispaperisorganizedasfollows:Section2describesanarchitectureofaneuro-fuzzy-ANARX-model.Section3describesanarchitectureofaneo-fuzzy-ANARX-model.Section4presentsaweightedneuro-neo-fuzzy-ANARX-model.Section5presentsseveralreal-worldapplicationstobesolvedwiththehelpoftheproposedsystem.Conclusionsaregiveninthefinalsection.II.ANEURO-FUZZY-ANARX-MODELAnarchitectureoftheANARX-modelisshowninFig.1.Itisformedbytwolinesoftimedelayelements1z11zykykandnnodes[]lNwhicharesimultaneouslylearned.Thesenodesaretunedindependentlyfromeachother.Byaddingnewnodesorremovingunnecessaryones,itdoesn’thaveanyinfluenceonotherneurons,i.e.theevolvingprocessforthissystemisimplementedbychanginganumberofthenodes.Fig.1.TheANARX-model.Fig.2.Aneuro-fuzzynodeoftheANARX-model.It’srecommendedtousean