从信息论角度理解间接平差

Understanding Adjustment from an Information Theoretic Perspective

在协方差矩阵、协因数阵、权阵等概念的基础上,引入了Fisher信息矩阵(简称信息矩阵),介绍了信息矩阵的一些重要性质,强调了总体信息矩阵与样本信息矩阵两种概念的区别;推导了多元正态分布情况下的信息矩阵,揭示了总体/样本信息矩阵与总体/样本协方差矩阵以及协因数阵/权阵的关系,指出权阵为归一化信息矩阵;在信息矩阵的基础上引入信息向量的概念,推导了信息域间接平差方法,该方法对信息矩阵与信息向量进行估计,在结果层面,该方法与估计原参数向量与协方差矩阵的普通间接平差方法等价,但形式更简单、结构更明确,为理解间接平差提供了一种新的视角,而且新方法在模型不可解场合、序贯/递归平差的初始化方面等具有特殊优势;给出了用于动态状态空间模型滤波的信息域动态平差算法,即为与Kalman滤波算法等价的信息滤波算法。

Objects:It is tried to understand the adjustment from an information theoretic viewpoint.Methods:Besides the concepts of covariance matrix,cofactor matrix and weight matrix,which are often introduced in“Surveying Adjustment”courses,we introduce the concept of Fisher information matrix(or simply information matrix)in this paper.Results:Several important properties of the information matrix are shown;the population and the sample information matrices are distinguished.The information matrix under the assumption of multivariate normal distribution is derived;the links among the population/sample information matrix,the population/sample covariance matrix and the cofactor/weight matrix are explained;and it is revealed that the weight matrix can be viewed as a normalized information matrix.Based on the information matrix,the concept of information vector is introduced;the adjustment in information domain is derived in which the information matrix and the information vector are calculated rather than the parameter vector and the covariance matrix calculated in standard adjustment.Though equivalent with the standard adjustment in terms of the results,the information-domain adjustment has simpler format and clearer structure;this provides a new perspective for understanding the traditional adjustment;more importantly the information-domain method has special merits in solving under-determined model and also in initializing the recursive adjustment method.Finally,the dynamic adjustment in information domain is derived for filtering a dynamic state space model;it is nothing but the information filter whih is equivalent to the standard Kalman filter.Conclusions:Information theory provides new insights for comprehensively and thoroughly understanding the surveying adjustment theory.