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从信息论角度理解间接平差

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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.
Understanding Adjustment from an Information Theoretic Perspective
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 in-troduced 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 infor-mation 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 infor-mation matrix,the concept of information vector is introduced;the adjustment in information domain is de-rived 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 ad-justment in terms of the results,the information-domain adjustment has simpler format and clearer struc-ture;this provides a new perspective for understanding the traditional adjustment;more importantly the in-formation-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 Kal-man filter.Conclusions:Information theory provides new insights for comprehensively and thoroughly un-derstanding the surveying adjustment theory.

indirect/parameter adjustmentFisher information matrixmultivariate normal distributioninformation vectorinformation-domain adjustmentinformation filtering

常国宾、张书毕、刘志平

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中国矿业大学环境与测绘学院,江苏 徐州,221116

间接/参数平差 Fisher信息矩阵 多元正态分布 信息向量 信息域平差 信息滤波

江苏省高等教育教改研究中国矿业大学教学研究

2019JSJG2602021YB035

2024

10.13203/j.whugis20210383

武汉大学学报(信息科学版)
武汉大学

武汉大学学报(信息科学版)

CSTPCD北大核心
影响因子：1.072
ISSN：1671-8860
年,卷(期)：2024.49(2)
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