总体最小二乘的迭代解法

3511201011#GeomaticsandInformationScienceofWuhanUniversityVol.35No.11Nov.2010:2010-09-15:(40874010);(2007GZC0474,2008GQC0001,2008GZS0041);(080104);(TJES0802);(DLLJ200506):1671-8860(2010)11-1351-04:A总体最小二乘的迭代解法鲁铁定1,2周世健1,3(1,56,344000)(2,129,430079)(3,,330029):针对总体最小二乘解算问题,应用测量平差中的间接平差原理推导了总体最小二乘的迭代逼近解算公式,通过与奇异值分解法进行比较,得出两种解算方法具有等价性实验数据分析验证了算法的有效性:总体最小二乘;奇异值分解;迭代算法;测量平差:P207.2-,,,[1][1][2],,GolubLanLoan[3],[4-7],[8],,,,[7]1,:(A+$A)X=L+$L(1)[7]:$Lvec($A)~00,2L002AªIn=R20In00ImªIn(2),vec(#)(1):(A+EA)X=L+e(3):A+EAL+e#X-1=AL+E#X-1=0(4),E=EAe[4]:tr(E#ET)=tr(EAETA+eeT)=vec(EA)Tvec(EA)+eTe=min(5)AL[4]:AL=U1U2m+1n-(m+1)#20#VT=U1#2#VT(6),U1=U11U12m1(7)#2010112#VT=210022m1V11V12V21V22m1Tm1(8)Eckart-Young-Mirsky[9],X[10]:X^=-V12#V-122=-1vm+1,m+1V12(9)viALTAL,[10]:ATAATLLTALTLX^-1=R2m+1X^-1[7]:^eT#^e+(vecE^A)T#vecE^A=tr(^e#^eT+E^A#E^TA)=tr(E^#E^T)=R2m+1(10)R20(TLS)=R2m+1/(n-m)(11)D(X^)UR20(N-R2m+1Im)-1N(N-R2m+1Im)-1(12),N=ATA22.1(3)[7]:L=AX-e+EAX=AX+[-I,XTªI]#eeAT(13)evec(EA)~00,R20In00ImªIn(14),eA=vec(EA)(13):€e=AX-L(15)€e=[In-XTªIn]eeA=e-EAX(16)[11]:Q€e=[In-XTªIn]In00ImªInIn-XªIn=In+XTXªIn=(1+XTX)In(17)(5):€eTQ-1€e€e=min(18):5e^-TQ-1€ee^-5X^=2e^-TQ-1€eA-2e^-Te^-X^T(1+X^TX^)2=0(19)(17),Q€eX,,(19):ATQ-1€ee^--X^e^-Te^-(1+X^TX^)2=0(20)(15)(16)(20):ATAX^-ATL=X^e^-Te^-/(1+X^TX^)=X^(AX^-L)T(AX^-L)/(1+X^TX^)=X^^M(21),^MB=(L-AX^)T(L-AX^)/(1+X^TX^),(21)[7]:X^=(ATA)-1(ATL+X^^v)(22)(18)Q-1€eX,,:ATQ-1€eAX^-ATQ-1€eL=0(23)(15)(23),:X^=(ATQ-1€eA)-1ATQ-1€eL=[AT(1+XTX)InA]-1AT(1+XTX)L=(ATA)-1ATL(24),(24),[11-13],,,,[8]Q€e,,,Q€e(22):R20(TLS)=^v/(n-m)(25)D(X^)UR20(N-^MIm)-1N(N-^vIm)-1(26)2.2(22),:1)^M=0,X^(1)=N-1c,N=ATA,c=ATL2)^M(i)=(L-AX^(i))T(L-AX^(i))/(1+(X^(i))TX^(i))3)X^(i+1)=N-1(c+X^(i)^M(i))4)X^(i+1)-X^(i)E,2.3(15):e^-=[In,-X^TªIn]^e^eA=^e-E^AX^=AX^-L(27),(27),:In-X^TªIIn-X^ªIK=e^-=AX^-L13523511::K=(1+X^TX^)-1€e=(1+X^TX^)-1(AX^-L):^e^eA=In-X^ªIK=(1+X^TX^)-1(AX^-L)-X^ªIn(1+X^TX^)-1(AX^-L)(28)(28):^e=(1+X^TX^)-1(AX^-L)(29)^eA=(-X^ªIn)(1+X^TX^)-1(AX^-L)(30)Kronecker,(30):E^A=-(1+X^TX^)-1(AX^-L)X^T(31)^M[7]:^M=(L-AX^)T(L-AX^)/(1+X^TX^)=[LT(L-AX^)+X^T(X^^M)]/(1+X^TX^)=LTL-(ATL)TX^(32)(22)(32),[7]:ATAATLLTALTLX^-1=^MX^-1,^M=Mmin\0(33)[10],[14],^M=^eT^e+^eTA^eA=min,GolubLanLoan(10)(32),^M=^eT^e+^eTA^eA=R2m+1,3[15]5-2,25,y=a+bxx=c+dy1,1Tab.1ValueoftheObservationsYXYX110.9835.3149.5739.1211.1329.71510.9446.8312.5130.8169.5848.548.4058.81710.0959.359.2761.4188.1170.068.7371.3196.8370.076.3674.4208.8874.588.5076.6217.6872.197.8270.7228.4758.1109.1457.5238.8644.6118.2446.42410.3833.41212.1928.92511.0828.61311.8828.1方案1xyy=a+bx¹:y=13.6284-0.0799xºSVD:V,:^a^b=-1vm+1,m+1V12=14.1952-0.0897y=14.1952-0.0897x»:M(1)=0.09737,^a^bT=14.1951-0.0897T,y=14.1952-0.0897x方案2yxx=c+dy1,1,:¹:y=15.3022-0.1117x;ºSVD:y=14.1951-0.0897x;»:y=14.1948-0.0897x12,,,,,,,4,,,,[7],,,,Q€e,Q€eX[1],,.[J].#,1353#2010112008,33(12):1271-1274[2],,.[J].,2008,l28(5):77-81[3]GolubGH,LanLoanFC.AnAnalysisoftheTotalLeastSquaresProblem[J].SIAMJournalonNumericalAnalysis,1980,17(6):883-893[4]SchaffrinB,FelusYA.OntheMultivariateTotalLeast-squaresApproachtoEmpiricalCoordinateTransformations[J].ThreeAlgorithmsJGeod,2008,82:373-383[5]SchaffrinB,FelusAY.MultivariateTotalLeast-squaresAdjustmentforEmpiricalAffineTransformations[C].The6thHotineMarussiSymposiumforTheoreticalandComputationalGeodesy,Springer,Berlin,2007[6]SchaffrinB,LeeIP,FelusYA,etal.TotalLeast-squares(TLS)forGeodeticStraight-lineandPlaneAdjustment[J].BollGeodSciAffini,2006,65(3):141-168[7]SchaffrinB.ANoteonConstrainedTotalLeast-squaresEstimation[J].LinearAlgebraAppl,2006,417(1):245-258[8],,.[J].#,2008,33(5):504-507[9]EckartC,YoungG.TheApproximationofOneMatrixbyAnotherofLowerRank[J].Psychometrika,1936,1(3):211-218[10]VanHuffelS,VandewalleJ.TheTotalLeast-squaresProblemComputationalAspectsandAnalysis[M].Philadelphia:SocietyforIndustrialandAppliedMathematics,1991[11],.[J].,2008,26(1):109-111[12],,,.GIS[M].:,1999[13],,.[J].#,2010,35(2):181-184[14]StrangG.LinearAlgebraandItsApplications[M].3rded.SanDiego:HarcourtBraceJovanovich,1988[15],.[M].:,2005:,,,E-mail:[e-mail]rTotalLeastSquaresEstimationLUTieding1,2ZHOUShijian1,3(1SchoolofGeoscienceandSurveyingEngineering,EastChinaInstituteofTechnology,56XuefuRoad,Fuzhou344000,China)(2SchoolofGeodesyandGeomatics,WuhanUniversity,129LuoyuRoad,Wuhan430079,China)(3JiangxiAcademyofSciences,ShangfangRoad,Nanchang,330029,China)Abstract:Totalleastsquares(TLS)approachaimsatestimatingamatrixofparametersfromalinearmodelwhenthereareerrorsinboththeobservationvectorLandthedatamatrixA.TheauthorsderivedaniterativealgorithmtosolvetheTLSproblembyusingtheprincipleofindirectadjustment.Comparedwiththemethodbasedonsingular-valuedecomposition,theiterativealgorithmcoincideswiththeSVDalgorithm.Thecalculatedexamplehasprovedthattheiterativealgorithmisvalidityandrationality.Keywords:TLS;singular-valuedecomposition;iterativealgorithm;surveyingadjustmentAboutthefirstauthor:LUTieding,associateprofessor,Ph.Dcandidate,majorsinsurveyingdataprocessingandgeodesy.E-mail:[e-mail]1354