总体最小二乘求解线性回归模型的迭代算法

2014年第9期工程勘察GeotechnicalInvestigation＆Surveying851121．6500932．650093根据线性回归模型的特点，推导了一种解线性回归参数的总体最小二乘算法。并对其单位权中误差的评定进行了探讨，通过理论推导表明按常规的总体最小二乘解法求得的线性回归单位权中误差与实际不符，并对其进行了纠正。通过算例分析且与常规的总体最小二乘解法进行比较，结果表明了算法的正确性和可靠性。总体最小二乘;线性回归;平差模型;迭代解法;奇异值分解TD172+.3AAniterationalgorithmoftotalleastsquaresforlinearregressionmodelsWangQisheng1YangDehong1YangGenxin21．FacultyofLandＲesourceEngineeringKunmingUniversityofScienceandTechnologyKunming650217China2．YunnanLandandＲesourcesVocationalCollegeKunming650217ChinaAbstractAccordingtothecharacteristicsofthelinearregressionmodelanalgorithmoftotalleastsquaresforsolvinglinearregressionparametersisderived．Andtheevaluationoftheerrorofthemeansquareerrorofunitweightisdiscussed．Thetheoreticalderivationshowsthattheerrorestimationofthegeneralleastsquaressolutionfortheconventionallinearregressiondoesnottallywiththeactualmeansquareerrorofunitweightandcorrespondingcorrectionsareapplied．Throughexampleanalysisandcomparingwithconventionaltotalleastsquaresmethodtheresultsshowthecorrectnessandreliabilityofthealgorithm．Keywordstotalleastsquareslinearregressionadjustmentmodeliterationalgorithmsingularvaluedecomposition2013-11-231989－．01。。SVD2～78～10。。5。7。。。、。11.1SVD解法126y=^a0+^a1x1+…+^akxk+ΔV=AX－L186工程勘察GeotechnicalInvestigation＆Surveying2014年第9期ΔyakxkA2VL+L=A+EAXVLvecEA[]～[]00σ20QLQ[]()A2SVD2ALXT－[]1=03126X=ATA－σ2n+1000I[]()n－1ATL44nσ2n+1AL。σ20=σ2n+1/m－n－1mn+1。SVD。SVD。1.2常规迭代解法2Vi=L－AXiTL－AXi/1+XiTXiXi+1=ATA－1ATL+XiVi5X05‖Xi+1－Xi‖＜εσ0=V/m－n。。。22.1算法推导y=^a0+^a1x1+^a2x+…+^anxn66b^0y+b^1x1+b^2x2+…+b^nxn=1b^0=1a^0b^n=^an^a0n=12…7A+EA^x=L88A=y1x11…x1ny2x21…x2n…ymxm1…xmnEA=vy1vx11…vx1nvy2vx21…vx2m…vymvxm1…vxmn^x=b^0b^1b^nL=111v=vecEAvecvecEA。v=vecEA～0σ20ImIn99ImInmnvTv=min10810=vTv－2kTA^x+EA^x－L1111kn+1×1EA^x^xTIn+1vIn+1n+1。11v、x0v－xTIn+1Tk=0ATk+ETAk=01212EA=vec－1xTIn+1Tk1313vec－1vecvec－1xTIn+1Tkm×n+1。128EA^x=^xTIn+1vk=xTIn+1xTIn+1T－1L－A^x141114ATxTIn+1xTIn+1T－1L－A^x=－vec－1xTIn+1TkTk1515^x^x=ATxTIn+1xTIn+1T－1A－1ATxTIn+1xTIn+1T－1L162014年第9期工程勘察GeotechnicalInvestigation＆Surveying871^x0217kki=^xiTIn+1^xiTIn+1T－1L－A^xi^xi+1=ATxiTIn+1xiTIn+1T－1A－1·ATxiTIn+1xiTIn+1T－1L+vec－1xiTIn+1TkiTki1732‖xi+1－xi‖＜ε7b^0y+b^1x1+b^2x2+…+b^nxn=1y=^a0+^a1x1+^a2x+…^anxn^a0=1b^0^an=b^nb^0n=12…182.2单位权中误差评定7σ0=∑mi=1v2yi+∑mni=1j=1v2x()ijm－n+1槡1912vTv=kTxTIn+1xTIn+1Tk20σ0=vTvm－n+槡1=kTxTIxTITkm－n+槡12119。。。y+vy=1+v0^a0+^a1x1+vx1+^a2x2+vx2+…+^anxn+vxn22σ0=∑mi=1v2yi+v20i+v2x1i…+v2xnim－n+1槡2322。珓vy=vy－v0^a0242419。。。。33.1算例165～1012、26、61y=^a0+^a1x。6、1。56.74、0.99。。。3.2算例2MATLAB。z=1.5+x+2yxyz00.005。xy00.031。LS、、2。2。。0.0227。88工程勘察GeotechnicalInvestigation＆Surveying2014年第9期1zxy1－1.4988－1.000－1.01952－1.1979－0.9021－0.89233－0.9019－0.8746－0.82834－0.6012－0.6826－0.73975－0.2899－0.6658－0.57236－0.0113－0.5696－0.500070.3111－0.3976－0.401680.6017－0.3285－0.272790.9050－0.1877－0.1822101.1917－0.0797－0.0895111.49700.02570.0375121.79860.07930.1279132.10210.21350.2072142.39160.30300.2793152.70240.42480.3805162.99390.51610.5358173.30030.62690.5516183.60330.69600.6993193.90160.79560.7415204.20540.93020.9306214.50500.93631.02592a0a1a2LS1.50771.30291.66990.0635TLS1.50721.15751.81900.0280TLS1.50541.00011.97930.0268。。4。。1GOLUBGHVANLCF．AnAnalysisoftheTotalLeastSquaresProblemJ．SIAMJNumer．Anal198017883～893．2．J．201035111351～1354.3．J．2010356711～714．4．J．2011101～4．5．J．2010356708～710.6．J．2012128～10．7．M．2008．8．J．200833121271～1274．9．J．200828577～81．10．J．2010S1檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼檼174～176．84。、—、。。。1．J．1989712483～487．2．—J．200527111～14．3．J．200729143～45．4．J．200510543～45．5GuoDongLiDaisukeYamaguchiMasatakeNagai．AGM11-MarkovchaincombinedmodelwithanapplicationtopredictthenumberofChineseinternationalairlinesJ．TechnologicalForecasting＆SocialChange2007741465～1480．6．J．20002266～8．7．J．201311212～15．8．J．2007324391～395．9．M．2004．