奇异谱分析课件 I.pdf

《奇异谱分析课件 I.pdf》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、NewinsightsintoseparabilityinSingularSpectrumAnalysisNinaGolyandinaSt.PetersburgStateUniversityMathematicalDepartmentChinaworkshopon‘SingularSpectrumAnalysisanditsapplications’17-20May2012,Beijing,China1/27NinaGolyandinaNewinsightsintoseparabilityOutlineBa

2、sicSingularSpectrumAnalysis(SSA).DiscussionoftheSVDstepforseparabilityProjectionpursuitICA-SSASeparabilitycut-offpoints“SingularSpectrumAnalysisfortimeseries”Golyandina,Zhigljavsky,20122/27NinaGolyandinaNewinsightsintoseparabilityBasicSSAalgorithmTimeseries

3、X=(x1,...,xN)Parameters:windowlengthL;1

4、1,...,d}=IjP√X=X+...+X,whereX=λUVTI1IcImmmm∈I2DiagonalaveragingMatrixX99KHankelmatrixXe↔rec.seriesX(I)IIOutput:X=X(I1)+...+X(Ic).3/27NinaGolyandinaNewinsightsintoseparabilitySVDPd√SingularValueDecompositionX=λUVT:mmmm=1ThebestapproximationprocedureTheuniqu

5、edecomposition,whichissymmetricwithrespecttorowsandcolumns.Inparticular,itistheuniquedecomposition,whichconstructsorthonormalbasisinboththecolumnandrowspaces:{Um}and{Vm}.Therefore,itisverynaturalfortheSSA,wheretherowsandcolumnsofthetrajectorymatrixaretaken

6、fromthesametimeseries.Thisisthedifferencewiththemethodsofmultivariateanalysis(e.g.PCA),wherecasesandvariablesdifferbytheirsense.4/27NinaGolyandinaNewinsightsintoseparabilitySeparabilitybySSAwiththeSVDX=X(1)+X(2)P√SVD:X=X+...+X=λUVTI1IcmmmDefinitionTheseparabi

7、lityiscalledweakifthereexistsanSVDofthetrajectorymatrixXsuchthatwecansplittheSVDmatrixtermsintotwodifferentgroups,sothatthesumsoftermswithinthegroupsgiveX(1)andX(2).DefinitionTheseparabilityiscalledstrong,ifthisistrueforanySVDofthetrajectorymatrix.Example(1)

8、(2)xn=A1sin(2πω1+φ1),xn=A2sin(2πω2+φ2),ω16=ω2.Thereisweakseparability(maybe,approximate).Theproblem:A1=A2=⇒nostrongseparability!5/27NinaGolyandinaNewinsightsintoseparabilityLackofstrongseparabilityProperdecom