总体最小二乘的扰动分析-王乐洋

33120132JOURNALOFGEODESYANDGEODYNAMICSVol．33No．1Feb．20131671-5942201301-0040-05*1213300132()344000、。P207ARESEARCHONPERTURBATIONANALYSISOFTHETOTALLEASTSQUARESWangLeyang121FacultyofGeomaticsEastChinaInstituteofTechnologyNanchang3300132JiangxiProvinceKeyLaboratoryforDigitalLandFuzhou()344000AbstractOnthebasisoftheconditionnumberdefinitionofcoefficientmatrixandfromtheviewpointofnu-mericalanalysistheperturbationanalysisequationsofleastsquaresLSanddataleastsquaresDLSarede-ducedinfollowingcaseswhentheperturbationisonlycontainedintheobservationdatatheperturbationsarecon-tainedintheobservationdataandcoefficientmatrixandtheperturbationisonlycontainedinthecoefficientma-trix．Onthebasisoftheclosed-formexpressionofthetotalleastsquaresTLSsolutionandwiththetoolofthesingularvaluedecompositionSVDtheperturbationanalysisequationofTLSisdeduced．ThroughtheanalysistheperturbationanalysisequationofTLSistheunifiedformoftheperturbationanalysisequationofLSandDLS．KeywordstotalleastsquaresTLSleastsquaresLSdataleastsquaresDLSperturbationanalysiscon-ditionnumber1totalleastsquaresTLSGol-ubVanLoan119802-5。、、2*[e-mail]16、7。2Golub-VanLoan8。8。、、。、。22．1AX≈b1A∈Rm×nm＞nX∈Rn×1b∈Rm×1。ALSNX=L2N=ATAL=ATbbδb1LSX^lsbδbδXXNX+δX=L+δL3δL=ATδb。X^ls=X+δX4NX=L≠0NδX=δL5‖δX‖≤‖N－1‖‖δL‖61‖X‖≤‖N‖‖L‖7‖δX‖‖X‖≤‖N‖‖N－1‖‖δL‖‖L‖8N910condN=‖N‖p‖N－1‖p9‖·‖ppp1、2∞。8bLSX^ls‖δL‖/‖L‖‖N‖‖N－1‖condNcondNcondN1condN。2．21AbLSdataleastsquaresDLS112。NδN1DLSX^dlsNδNδ珚XX。AX=bA+δAX+δ珚X=bAδ珚X+δAδ珚X+δAX=0ATATAδ珚X+ATδAδ珚X+ATδAX=0A+δATA+δAX+δ珚X=A+δATbN+δNX+δ珚X+ATδAX+ATδAδ珚X+δATAδ珚X=LN+δNX+δ珚X+δATAδ珚X－ATAδ珚X=LδATAδ珚X－ATAδ珚X≈0N+δNX+δ珚XL10δN=δATδAδAA。X^dls=X+δ珚X11N+δNδ珚X=－δNX1210‖B‖＜1I±B‖I±B－1‖≤11－‖B‖131433‖·‖。‖N－1‖‖δN‖＜1N+δN=NI+N－1δN9‖N+δN－1‖≤‖N－1‖1－‖N－1‖‖δN‖1412δ珚X=－N+δN－1δNX15‖δ珚X‖‖X‖≤‖N－1‖‖N‖‖δN‖N1－‖N－1‖‖N‖‖δN‖‖N‖1616condN‖N‖‖N－1‖N‖δN‖/‖N‖‖δ珚X‖/‖X‖‖δN‖/‖N‖‖N‖‖N－1‖。2．31AbLS2NδNbδbδNδbδ珚X'X。AX=bA+δAX+δ珚X'=b+δbAδ珚X'+δAδ珚X'+δAX=δbATATAδ珚X'+ATδAδ珚X'+ATδAX=ATδbA+δATA+δAX+δ珚X'=A+δATb+δbN+δNX+δ珚X'+ATδAX+ATδAδ珚X'+δATAδ珚X'=L+ATδb+δLN+δNX+δ珚X'+δATAδ珚X'－ATAδ珚X=L+δLδATAδ珚X'－ATAδ珚X'≈0N+δNX+δ珚X'L+δL17δL=δATδb。X^'ls=X+δ珚X'18172δ珚X'=N－1δL－N－1δNX－N－1δNδ珚X'1919‖δ珚X'‖≤‖N－1‖‖δL‖+‖N－1‖‖δN‖‖X‖+‖N－1‖‖δN‖‖δ珚X'‖201－‖N－1‖‖δN‖‖δ珚X'‖≤‖N－1‖‖δL‖+‖δN‖‖X‖21δN‖N－1‖‖δN‖＜1‖δ珚X'‖≤‖N－1‖1－‖N－1‖‖δN‖‖δL‖+‖δN‖‖X‖222‖L‖≤‖N‖‖X‖231‖X‖≤‖N‖‖L‖242224‖δ珚X'‖‖X‖≤‖N－1‖1－‖N－1‖‖δN‖‖δL‖L‖N‖+‖δN()‖25‖δ珚X‖‖X‖≤‖N‖‖N－1‖1－‖N‖‖N－1‖‖δN‖N‖δL‖‖L‖+‖δN‖‖N()‖2626AbδAδbLS‖δL‖/‖L‖‖δN‖/‖N‖condN‖N‖‖N－1()‖‖δL‖/‖L‖‖δN‖/‖N‖LS‖δL‖/‖L‖‖δN‖/‖N‖condN‖N‖‖N－1()‖LS。31AbTLS2N－σ2n+1IX=L27N=ATAL=ATbσn+1AbAb=U∑VT28U=U1U2U1=u1…unU2=un+1…umui∈Rm×1UTU=ImV=V11V12V21V[]22n1=v1…vn+1vi∈Rn+1×1VTV=In+1241Σ=Σ100Σ[]2=diagσ1…σn+1∈Rm×n+1Σ1=diagσ1…σn∈Rn×nΣ2=σn+10…0T∈Rm－n×1σ1≥…≥σn+1≥0。NδNbδb1TLSX^tlsδNδbδ珛XXATδAX、ATδAδX、δATAX、δATAδX、2σn+1δσn+1X、2σn+1δσn+1δX、ATδbδATbN+δN－σ2n+1+δσn+12IX+δ珛XL+δL29δL=δATδbδσn+1δAδb。X^tls=X+δ珛X302927Nδ珛X=δL－δNX+δσn+12X+σ2n+1δ珛X－δNδ珛X+δσn+12δ珛X3131N－1δ珛X=N－1δL－N－1δNX+N－1δσn+12X+N－1σ2n+1δ珛X－N－1δNδ珛X+N－1δσn+12δ珛X3232‖δ珛X‖≤‖N－1‖‖δL‖+‖N－1‖‖δN‖‖X‖+‖N－1‖‖δN‖‖δ珛X‖+σ2n+1‖N－1‖‖δ珛X‖+δσn+12‖N－1‖‖X‖+δσn+12‖N－1‖‖δ珛X‖331－‖N－1‖‖δN‖－σ2n+1‖N－1‖－δσn+12‖N－1‖‖δ珛X‖≤‖N－1‖‖δL‖+‖δN‖‖X‖+δσn+12‖X‖342-9‖A‖2=λAT槡A=λ槡N35λATA=λNATAN。A210A=U'∑'V'T36U'=U'1U'2U'1=u'1…u'nU'2=u'n+1…u'mu'j∈Rm×1U'TU'=ImV=v'1…v'nv'i∈Rn×1V'TV'=InΣ'=diagσ'1…σ'n∈Rm×nσ'1≥…≥σ'n≥0。AATAN36‖N‖2=σ'2137212σ1≥σ'1≥…≥σn≥σ'n≥σn+1380＜σ2n+1‖N－1‖=σ2n+1σ'21＜139δσn+12‖N－1‖=δσn+12σ'21≈040‖N－1‖‖δN‖=δσ'12σ'21≈0411－‖N－1‖‖δN‖－σ2n+1‖N－1‖－δσn+12‖N－1‖=1－δσ'12σ'21－σ2n+1σ'21－δσn+12σ'21≈σ'21－σ2n+1σ'21＞042341－‖N－1‖‖δN‖－σ2n+1‖N－1‖－δσn+12‖N－1‖‖δ珛X‖≤‖N－1‖1－‖N－1‖‖δN‖－σ2n+1‖N－1‖－δσn+12‖N－1‖‖δL‖+‖δN‖‖X‖+δσn+12‖X‖4343‖X‖‖δ珛X‖‖X‖≤‖N－1‖1－‖N－1‖‖δN‖－σ2n+1‖N－1‖－δσn+12‖N－1‖‖δL‖‖X‖+‖δN‖+δσn+1()24427‖L‖≤‖N‖+σ2n+1‖X‖451‖X‖≤‖N‖+σ2n+1‖L‖464446‖δ珛X‖‖X‖≤‖N‖‖N－1‖1－‖N‖‖N－1‖‖δN‖+σ2n+1+δσn+12‖N‖‖δL‖‖L‖+‖δN‖‖N‖+1‖N‖‖L‖σ2n+1‖δL‖+δσn+12‖L‖4747AbTLS‖δL‖/‖L‖、‖δN‖/‖N‖34331/‖N‖‖L‖σ2n+1‖δL‖+δσn+12‖L‖condN‖N‖‖N－1‖‖δL‖/‖L‖‖δN‖/‖N‖TLS‖δL‖/‖L‖、‖δN‖/‖N‖1/‖N‖‖L‖σ2n+1‖δL‖‖+δσn+12‖‖L‖condN‖N‖‖N－1‖TLS。47LS1LS‖δN‖=0σ2n+1=0δσn+1=04782LSDLS‖δL‖=0σ2n+1=0δσn+1=047163LSσ2n+1=0δσn+1=04726。4、。、。1GolubGHandandVanLoanCF．AnanalysisofthetotalleastsquaresproblemJ．SIAMJNumerAnal．198017883-893．2VanHuffelSandVandewalleJ．ThetotalleastsquaresproblemComputationalaspectsandanalysisM．SIAMPhiladelphia1991．3VanHuffelSEd．．Recentadvancesintotalleastsquarestechniquesanderrors-in-variablesmodelingM．SIAMPhiladelphia1997．4VanHuffelSandLemmerlingPEds．．Totalleastsquaresanderrors-in-variablesmodelingAnalysisalgorithmsandapplicationsM．DordrechtKluwerAcademicPublishers2002．5．J．2012548－5257．WangLeyang．ResearchonpropertiesoftotalleastsquaresestimationJ．JouralofGes-desyandGeodynamics2012548－52576．J．A199493304-311WeiMushengandChenGuoliang．SolutionsetsandpropertyforweightedtotalleastsquaresproblemJ．AppliedMathemat-ics-AJournalofChineseUniversities199493304-311．7．TLSLSJ．2003254479-492LiuYonghuiandWeiMusheng．OnthecomparisionofthetotalleastsquaresandtheleastsquaresproblemsJ．MathematicaNumericaSinica2003254479-492．8ZhouLiangminetal．Perturbationanalysisandconditionnumb