奇异谱分析课件 II.pdf

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1、OnthechoiceofparametersinSingularSpectrumAnalysisandrelatedsubspace-basedmethodsNinaGolyandinaSt.PetersburgStateUniversityMathematicalDepartmentUK-Chinaworkshoponsingularspectrumanalysisanditsapplications20September2010,CardiffUniversity,UK1/22NinaGolyandinaOnthechoiceofparametersinSSAOutlineConside

2、rSingularSpectrumAnalysis(SSA)andsubspace-basedmethodsinsignalprocessing.DescribecommonandspecificfeaturesofthesemethodsConsiderdifferentkindsofproblems:signalreconstruction,forecastingandparameterestimation.ProvidegeneralrecommendationsonthechoiceofparametersDemonstratethattheerrorbehaviordependsont

3、hetypeofresiduals,deterministicorstochastic,andwhetherthenoiseiswhiteorred.DiscussconvergencerateTheanalysisisbasedonknowntheoreticalresultsandextensivecomputersimulations.“OnthechoiceofparametersinSingularSpectrumAnalysisandrelatedsubspace-basedmethods”Journal:“StatisticsandItsInterface”,SpecialIs

4、sue:TheoryandPracticeinSingularSpectrumAnalysisofTimeSeries.20102/22NinaGolyandinaOnthechoiceofparametersinSSABasicSSAalgorithmTimeseriesFN=(f0,...,fN−1)Parameters:windowlengthL;1

5、pfL−1fL...fN−12SVDPd√√TTTXX⇒(λm,Um)⇒X=λmUmVm,Vm=XUm/λm.m=1Reconstruction1GroupingP√PTTX=XI1+...+XIc,whereXI=λmUmVm=UmUmXm∈Im∈I2Diagonalaveraging(I)MatrixXI99KHankelmatrixXeI↔rec.seriesF(I1)(Ic)Output:F=F+...+F.Results:eigenvalues(λi),eigenvectorsEOF(Ui),factorvectors/principal(Ij)components(Vi),

6、reconstructedcomponentsRC(F).3/22NinaGolyandinaOnthechoiceofparametersinSSASeriesstructureFN=(f0,...,fN−1)FN=SN+RN,whereSNisasignal,RNisaresidual.SN=(s0,...,sN−1)isgovernedbyalinearrecurrentformula(LRF)Xrsi+r=aksi+r−k.k=1Ifrisminimal,itiscalledtherankofthesignalSN.PnGeneralformofthesignalP:sn=kPmk(

7、n)ρkcos(2πωkn+φk)∈Rnorsn=kPmk(n)µk∈C,whereµk∈C.Problems:ExtractionofthesignalSignalforecastingParameterestimation4/22NinaGolyandinaOnthechoiceofparametersinSSASignalextractionbySSAFN=SN+RN,FN=(f0,...,fN−1)L