误差理论与测量平差基础教学课件第五讲06

1、第二章 误差理论与最小二乘原理,Error Theory and The Least Squares Principle,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

2、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 Propagation of Variance-Covariance Matrix,1、Raise A Question,2、Variance and Standard Error of Functions of Random variable,3、Propagation of The vectors Variance-Covariance Matrix,4

3、、Relation of Covariance Matrix Between Vectors,No.5 Propagation of Variance-Covariance Matrix,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（covariance matrix）,补充知识,3.向量间的协方差矩阵（covariance matrix）,设：,补充知识,4.向量的微分,设：,令：,补充知识,4.向量的微分,Jacobian Matrix：is also

4、 sometimes called Coefficient Matrix(since,in the case of a linear mathematical model,the partial derivatives are just the coefficient of the Obs.),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.5 Propagation of Variance-Covariance Matrix,1、 Raise A Quest

5、ion,1) The question is how to estimate the errors in Z as functions of random variable,No.5 Propagation of Variance-Covariance Matrix,1、 Raise A Question,1) The question is how to estimate the errors in Z as functions of random variable,No.5 Propagation of Variance-Covariance Matrix,1、 Raise A Quest

6、ion,1) The question is how to estimate the errors in Z as functions of random variable,No.5 Propagation of Variance-Covariance Matrix,Reduce to the follow question,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。,求函数的方差,已知函数关系式,以及观测值的方差协方差,No.5 Propagation of Variance-Covariance Matrix,2) Errors Propagation: is the p

7、rocess of evaluating the errors in estimated quantities (Z) as functions of the error in the measurements (L),1、 Raise A Question,No.5 Propagation of Variance-Covariance Matrix,2.Variance and standard error of functions of random variable,设随机变量的函数,由全微分得真误差之间得关系,式中,式中：,为随机变量,令：,No.5 Propagation of Va

8、riance-Covariance Matrix,2.Variance and standard error of functions of random variable,随机变量的函数,上式取平方,中误差形式,式中：,根据,两端取期望,No.5 Propagation of Variance-Covariance Matrix,Step of Solution:,1）Construct the Mathematical model; 2）If the model is no-linear, linearizing it firstly; 3）Applying the Law of Prop

9、agation of Errors; 4）Substituting standard error for variance.,2.Variance and standard error of functions of random variable,No.5 Propagation of Variance-Covariance Matrix,Some special cases：,1观测值不相关时,2线性函数,No.5 Propagation of Variance-Covariance Matrix,Some special cases：,1观测值不相关时,2线性函数,3倍数函数,4和差函数

10、,No.5 Propagation of Variance-Covariance Matrix,中误差传播时,2.Variance and standard error of functions of random variable,1观测值不相关时,2线性函数,3倍数函数,4和差函数,No.5 Propagation of Variance-Covariance Matrix,Example1:,在视距测量中，测得标尺读书差d0.565m，中误差md 0.2mm，当视距常数为100时，求这两点间距离S及其中误差ms。,2.Variance and standard error of func

11、tions of random variable,Example1:,在视距测量中，测得标尺读书差d0.565m，中误差md 0.2mm，当视距常数为100时，求这两点间距离S及其中误差ms。,解：,函数关系式：,距离观测值,距离观测值的中误差,Example2:,例2：已知水平角是两个方向观测值的差，两方向观测值独立等精度，设方向观测值的中误差 ，求角度的中误差。,解：,函数关系式：,应用和差函数的方差公式,得,Example3:,例3：用线段比较法求航高时，设地面点 间的距离为 ，其中误差为 ，量得 两点在相片上的象点 间的距离为 ，其中误差为 ，摄影主焦距为 （常数），试求由公式 计算航

12、高的中误差。,解：,函数关系式：,真误差关系式,得,注意：本例的两个距离值是独立的,解：,得：,No.5 Propagation of Variance-Covariance Matrix,3、Propagation of the vectors variance-covariance matrix,Now we have more than one unknown, say a vector Y of m unknowns,that are related to the n random variable X.In this case,we will discuss the linear m

13、odel firstly.,函数形式,对应真误差得关系,依方差定义,非线性函数：,求向量的微分,得到真误差的线性形式,传播形式,No.5 Propagation of Variance-Covariance Matrix,对应中误差的形式,3、Propagation of the vectors variance-covariance matrix,3、Propagation of the vectors variance-covariance matrix,解：,函数关系式：,例：设在测站O等精度观测了三个方向a1，a2，a3，中误差均为ma，试求协方差矩阵 和按zx-y算出的z角中误差。,

14、又知道：,根据误差传播定律,Z的函数可写为,根据误差传播定律,No.5 Propagation of Variance-Covariance Matrix,3、Propagation of The vectors Variance-Covariance Matrix,Summary of steps :,Given: X , its variance matrix,model Required: unknown vectors variance matrix 1) Form the direct mathematical model 2) Evaluate the elements of Jacobian Matrix 3) Apply the law of covariance.,No.5 Propagation of Variance-Covariance Matrix,4、Relation of Covariance Matrix Between Vectors,General Law of Propagation of Variance-Covariance Matrix,设,No.5 Propagation of Varianc