《误差理论与测量平差基础教学课件》第五讲

第二章误差理论与最小二乘原理,ErrorTheoryandTheLeastSquaresPrinciple,第四讲参数估计与最小二乘原理（复习）,思考题,1、衡量估计量性质的标准有哪些，它们分别有什么含义？,无偏性：估值的数学期望等于真值。,有效性：方差最小的无偏估值。,一致性：随着观测值个数的增大估值以概率收敛于真值。,思考题,第四讲参数估计与最小二乘原理（复习）,3、什么情况下最小二乘估计与最大或然估计是一致的。,当观测值服从正态分布时。,似然函数：,第四讲参数估计与最小二乘原理（复习）,第四讲参数估计与最小二乘原理（复习）,4、协方差阵与权阵之间的关系。,权矩阵是与协方差矩阵的逆矩阵成比例的矩阵。,第四讲参数估计与最小二乘原理（复习）,（1）形式上的不同。（2）都是均方差的估计值。（3）真误差难得，残差易得，第二个公式应用面更广。（4）后一式称Bessel公式,第五讲方差及协方差矩阵的传播PropagationofVariance-CovarianceMatrix,1、RaiseAQuestion,2、VarianceandStandardErrorofFunctionsofRandomvariable,3、PropagationofThevectorsVariance-CovarianceMatrix,4、RelationofCovarianceMatrixBetweenVectors,No.5PropagationofVariance-CovarianceMatrix,补充知识,1协方差（covariance）,(协方差的估值，仍然叫协方差),估值：,协方差是两种真误差所以可能取值乘积的理论平均值,协方差本身可以小于零,补充知识,2.向量的协方差矩阵（covariancematrix）,补充知识,3.向量间的协方差矩阵（covariancematrix）,设：,补充知识,4.向量的微分,设：,令：,补充知识,4.向量的微分,JacobianMatrix：isalsosometimescalledCoefficientMatrix(since,inthecaseofalinearmathematicalmodel,thepartialderivativesarejustthecoefficientoftheObs.),补充知识,5.观测值真误差与其函数真误差的关系,设：,补充知识,5.观测值真误差与其函数真误差的关系,设：,泰勒级数展开：,补充知识,5.观测值真误差与其函数真误差的关系,设：,No.5PropagationofVariance-CovarianceMatrix,1、RaiseAQuestion,1)ThequestionishowtoestimatetheerrorsinZasfunctionsofrandomvariable,No.5PropagationofVariance-CovarianceMatrix,1、RaiseAQuestion,1)ThequestionishowtoestimatetheerrorsinZasfunctionsofrandomvariable,No.5PropagationofVariance-CovarianceMatrix,1、RaiseAQuestion,1)ThequestionishowtoestimatetheerrorsinZasfunctionsofrandomvariable,No.5PropagationofVariance-CovarianceMatrix,1、RaiseAQuestion,Reducetothefollowquestion,观测值的方差与观测值函数的方差之间的关系式，称为误差传播律。,No.5PropagationofVariance-CovarianceMatrix,2)ErrorsPropagation:istheprocessofevaluatingtheerrorsinestimatedquantities(Z)asfunctionsoftheerrorinthemeasurements(L),1、RaiseAQuestion,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,设函数,由全微分得真误差之间得关系,式中,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,设,得到,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,根据方差定义,代之以中误差形式,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,StepofSolution:,ConstructtheMathematicalmodel;Ifthemodelisno-linear,linearizingitfirstly;ApplyingtheLawofPropagationofErrors;Substitutingstandarderrorforvariance.,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,几种特殊情况：,1观测值不相关时,2线性函数,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,几种特殊情况：,3倍数函数,4和差函数,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,中误差传播时,1观测值不相关时,2线性函数,3倍数函数,4和差函数,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,Example1:,在1：500的图上，量的两点间距离d23.4mm，d的量测中误差md0.2mm，求这两点间实地距离S及其中误差ms。,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,Example2:,设X为独立观测值L1，L2，L3的函数，x=1/7L1+2/7L2+4/7L3，已知L1，L2，L3的中误差分别为m1=3mm，m2=2mm，m3=1mm，求函数X的中误差mx。,No.5PropagationofVariance-CovarianceMatrix,2、VarianceandStandardErrorofFunctionsofRandomvariable,Example3:,设在测站A上，已知BACa，设a无误差，观测值b1和b2的中误差m1=m2=1”.4，协方差m12=1（秒2），求角x的中误差mx。,No.5PropagationofVariance-CovarianceMatrix,3、PropagationofThevectorsVariance-CovarianceMatrix,Nowwehavemorethanoneunknown,sayavectorZoftunknowns,thatarerelatedtothenrandomvariable.Inthiscase,wewilldiscussthelinearmodelfirstly.,No.5PropagationofVariance-CovarianceMatrix,3、PropagationofThevectorsVariance-CovarianceMatrix,函数形式,对应真误差得关系,依定义,No.5PropagationofVariance-CovarianceMatrix,3、PropagationofThevectorsVariance-CovarianceMatrix,非线性形式,线性化,于是,No.5PropagationofVariance-CovarianceMatrix,3、PropagationofThevectorsVariance-CovarianceMatrix,对应中误差的形式,No.5PropagationofVariance-CovarianceMatrix,3、PropagationofThevectorsVariance-CovarianceMatrix,例：设观测值L1,L2,L3，又知道a1，a2分别为a1=L2-l1a2=l3-l2并且求ma1，ma2，ma1a2,No.5PropagationofVariance-CovarianceMatrix,3、PropagationofThevectorsVariance-CovarianceMatrix,Summaryofsteps:,Given:X,itsvariancematrix,modelRequired:unknownvectorsvariancematrix1)Formthedirectmathematicalmodel2)EvaluatetheelementsofJacobianMatrix3)Applythelawofcovariance.,No.5PropagationofVariance-CovarianceMatrix,4、RelationofCovarianceMatrixBetweenVectors,GeneralLawofPropagationofVar