贝叶斯方法：对已知和未知白噪声的周期图和时间轴估计的替代(IJMSC-V4-N2-3)

I.J.MathematicalSciencesandComputing,2018,2,22-33PublishedOnlineApril2018inMECS()DOI:10.5815/ijmsc.2018.02.03Availableonlineat:AnAlternativetoPeriodogramandTimeAxesEstimationforKnownandUnknownWhiteNoiseOlanrewajuRasakiOlawaleaaDepartmentofStatistics,UniversityofIbadan,Ibadan,200284,Nigeria.Received:14February2018;Accepted:08March2018;Published:08April2018AbstractThisstudydescribestheBayesianapproachasanalternativeapproachforestimatingtimeaxesparametersandtheperiodogram(powerspectrum)associatedwithsinusoidalmodelwhenthewhitenoise(sigma)isknownorunknown.TheconventionalmethodofestimatingthetimeaxesparametersandtheperiodogramhasbeenviatheSchustermethodthatreliessolelyonMaximumLikelihoodEstimation(MLE).TheBayesianalternativeapproachproposedinthiswork,ontheotherhand,adoptedtheMaximumaPosteriori(MAP)viatheMarkovChainMonteCarlo(MCMC)inordertocheckmatetheproblemofre-parameterizationandover-parameterizationassociatedwithMLEintheconventionalpractice.Theratesofheartbeatvariabilityatexactlyanhourandtwohoursafterbirthofonethousandeighthundred(1800)newlybornbabiesinastatehospitalwererecordedandsubjectedtoboththeBayesianapproachandSchusterapproachforinferences.Theperiodogramestimates,exactlyanhourandtwohoursofafterbirth,wereestimatedtobe0.7395and0.7549,respectively-anditwasdeducedthatratesofheartbeat(frequency)variabilitymoderatedandstabilizedthepulseamongthebabiesaftertwohoursofbirth.Inaddition,MAPmeanestimatesoftheparametersapproximatelyequalstothetruemeanofestimateswhenrounduptocurbtheproblemofre-parameterizationandover-parameterizationthatdoaffectSchustermethodviaMLE.IndexTerms:Bayesian,MaximumAPosteriori(MAP),MarkovChainMonteCarlo(MCMC),MaximumLikelihoodEstimation(MLE),andPeriodograms.©2018PublishedbyMECSPublisher.Selectionand/orpeerreviewunderresponsibilityoftheResearchAssociationofModernEducationandComputerScience1.IntroductionTimeseriesmodelinganddataanalysisareconventionallyrelatedtoBayesiandataanalysiswithitsgeneral*Correspondingauthor.Tel:[phone]E-mailaddress:[e-mail]:AnAlternativetoPeriodogramandTimeAxesEstimationforKnownand23UnknownWhiteNoiseapproachtomodelingmethodsanditsprinciples.Itisaknown,andalreadyestablishedfactthatstochastictimeseriesmodelsevolverounddeterministic(whichisattributedtofrequencychangeorFourierdecompositioninvoicesignals,vibrations,Electrocardiogram(ECG)etc.)timeseriesforparametersembeddedinsinusoidalmodeltobeproperstudiedandinterpreted.Thetypicalandwell-knownfrequencyistheperiodogram;accordingto[1],periodogramwhichisotherwiseknownasclassicalFourierPowerSpectrumiscloselyrelatedtothePosteriorProbabilityDensityFunction(PDF)functionofaBayesiansettingoverthefrequencyparameterofaSinusoidalmodel.ThisimpliesthataPosteriorPDFfunction(/,)PUMisneededforagivenmodel“M”withitvaluesofparametersthatbestdescribesthedata“U”.NomenclaturePeriodogramParametervectororspaceA&BTimeaxesySingletimeseriesvariableUSetofeventswithvariableofconstanttimevaryingvariationMModel()itWhitenoiseprocessNoise(sigma)()ftSinusoidalmodel()pJeffrey’sprior,/,PUMBayesianperiodogram2.RelatedWorkContributionsby[2]and[3]cannotbementionedwhenitcomestotheSingularSpectrumAnalysis(SSA)approachoftimeaxesviaoscillatingcomponentoftheunknownperiodogramandtheuseofnon-parametricpriorapproachonspectraldensitytoestablishedpseudo-posteriordistributionforashort-memoryGaussiantimeseriesundersomeconditionsonthepriorforfrequencytimeseriesmodelrespectively.Awell-providedmethodforcalculatingsignalingtimeofthecommunitymodelvialatesignalingcostforthedatafusionusingtheDynamicTransformationModel(DTM)by[4]hasbeenthelinkbetweentwoprocessesinsignalingandtimeaxesindexes;thesignalingtimewasestimatedbasedonthedatatransmissiontimeandprocessingdelaybasedonthetwoimmediatefilterlevelsviadesignedalgorithm.[5]gaveaclearpictureofhowspectraltimeseriesofmultispectralandperiodogramrecognitionschemesinthecontextsofimageacquisition,irissegmentation,textureanalysis,andmatchingandperformanceevaluationwhile[6]thoroughlydealtwithFourieranalysisongraphswithbothpositiveandnegativeedges;[6]investigatedtheimpactsofintroducingnegativeedgesandexaminepatternsinthespectralspaceofthegraphs’adjacencymatrix.Theirtheoreticalresults[5]and[6]showedthatcommunitiesinak-balancedsignedgrapharedistinguishableinthespectralspaceofitssignedadjacencymatrixevenifconnectionsbetweencommunitiesaredensewithanillustrationempiricalevaluationonbothsyntheticdataandreallifedata.[7]AlsomaintainedthattheWignerquasi-distributionplaysanalternativeroleinbothtime-frequencyanalysisandquantummechanicsthewhitenoiseinsteadoftheconventionalGaussiandistributionbeenuseinboththe24BayesianApproach:AnAlternativetoPeriodogramandTimeAxesEstimationforKnownandUnknownWhiteNoiseclassicalandsuggestedBayesianapproach.Theyallmaintainedthegroundofestimatingtheperiodogramandtimeaxesparametersviatheclassicalapproach.ThisresearchgivesaninsightofestimatingtheparametersviaBayesianapproachwithorwithoutthepriorknowledgeofthenois