Wavelet Neural Networks Wavelet Neural Networks With Applications in Financial Engineering, Chaos, and Classification AntoniosK.Alexandridis SchoolofMathematics,StatisticsandActuarialScience UniversityofKent Canterbury,UnitedKingdom AchilleasD.Zapranis DepartmentofAccountingandFinance UniversityofMacedonia Thessaloniki,Greece Copyright©2014byJohnWiley&Sons,Inc.Allrightsreserved. PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey. PublishedsimultaneouslyinCanada. Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor byanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,(978)750-8400, fax(978)750-4470,oronthewebatwww.copyright.com.RequeststothePublisherforpermission shouldbeaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken, NJ07030,(201)748-6011,fax(201)748-6008,oronlineathttp://www.wiley.com/go/permission. LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitable foryoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublishernor authorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnotlimitedto special,incidental,consequential,orotherdamages. Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactour CustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStatesat (317)572-3993orfax(317)572-4002. Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmay notbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourwebsiteat www.wiley.com. LibraryofCongressCataloging-in-PublicationData: Alexandridis,AntoniosK. Waveletneuralnetworks:withapplicationsinfinancialengineering, chaos,andclassification/AntoniosK.Alexandridis,Schoolof Mathematics,StatisticsandActuarialScience,UniversityofKent, Canterbury,UK,AchilleasD.Zapranis,DepartmentofAccountingand Finance,UniversityofMacedonia,Thessaloniki,Greece. pagescm Includesbibliographicalreferencesandindex. ISBN978-1-118-59252-6(cloth) 1.Wavelets(Mathematics) 2.Neuralnetworks(Computerscience) 3.Financialengineering. I.Zapranis,Achilleas,1965– II.Title. QA403.3.A4282014 006.3′2–dc23 2013047838 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 Toourfamilies Contents Preface xiii 1 MachineLearningandFinancialEngineering 1 FinancialEngineering / 2 FinancialEngineeringandRelatedResearchAreas / 3 FunctionsofFinancialEngineering / 5 ApplicationsofMachineLearninginFinance / 6 FromNeuraltoWaveletNetworks / 8 WaveletAnalysis / 8 ExtendingtheFourierTransform:TheWaveletAnalysis Paradigm / 10 NeuralNetworks / 17 WaveletNeuralNetworks / 19 ApplicationsofWaveletNeuralNetworksinFinancialEngineering, Chaos,andClassification / 21 BuildingWaveletNetworks / 23 VariableSelection / 23 ModelSelection / 24 ModelAdequacyTesting / 25 BookOutline / 25 References / 27 vii viii CONTENTS 2 NeuralNetworks 35 ParallelProcessing / 36 ProcessingUnits / 37 ActivationStatusandActivationRules / 37 ConnectivityModel / 39 Perceptron / 41 TheApproximationTheorem / 42 TheDeltaRule / 42 BackpropagationNeuralNetworks / 44 MultilayerFeedforwardNetworks / 44 TheGeneralizedDeltaRule / 45 BackpropagationinPractice / 49 TrainingwithBackpropagation / 51 NetworkParalysis / 54 LocalMinima / 54 NonuniqueSolutions / 56 ConfigurationReference / 56 Conclusions / 59 References / 59 3 WaveletNeuralNetworks 61 WaveletNeuralNetworksforMultivariateProcessModeling / 62 StructureofaWaveletNeuralNetwork / 62 InitializationoftheParametersoftheWaveletNetwork / 64 TrainingaWaveletNetworkwithBackpropagation / 69 StoppingConditionsforTraining / 72 EvaluatingtheInitializationMethods / 73 Conclusions / 77 References / 78 4 ModelSelection:SelectingtheArchitecture oftheNetwork 81 TheUsualPractice / 82 EarlyStopping / 82 Regularization / 83 Pruning / 84 MinimumPredictionRisk / 86