秩亏EIV模型平差理论与方法研究

Errors-in-variables (EIV) models have been substantially investigated theoretically and practically since the publication of the pioneering works by Adcock in 1877 and Pearson in 1901, in particular, also the landmark work by Golub and van Loan in 1980. Almost all theoretical and practical works on EIV models assume that they are of full rank. Nothing has ever been done about EIV models with rank defect up to the present. The only theoretical result we have now is the non-uniqueness condition and the solution approach of minimum norm total least squares given by Golub and van Loan (1980) and van Huffel and Vandewalle (1991). The major purposes of this research project are summarized as follows: (i) given a full EIV model with all the elements of the coefficient matrix being assumed to be random and stochastically independent, we will prove that the eigenvalues of the corresponding normal matrix of such an EIV model are almost surely distinguishable with probability one, indicating that the non-uniqueness condition of total least squares (TLS) solutions, as reported in Golub and van Loan (1980) and van Huffel and Vandewalle (1991), cannot be true with probability one. Actually, we even do not find any formal definition of EIV models with rank defect. As a result, we will also provide a formal definition of EIV models with rank defect; (ii) even if a full EIV model is assumed to be theoretically of rank defect, we will prove that the rank deficiency of such an EIV model cannot be identifiable from measurements with probability one; (iii) if an EIV model is identified to be of rank defect, it must be a partial EIV model; and finally, (iv) we will propose methods to construct a unique solution to an EIV model with rank defect. More precisely, we will extend the minimum norm total least squares method by Golub and van Loan (1980) and van Huffel and Vandewalle (1991) to cover with all the unknown parameters, namely, the model parameters and the corrections to the stochastically independent elements of the coefficient matrix.

EIV模型的总体最小二乘法自Adcock，Pearson与Golub/van Loan的开创性论文发表以来,得到了极大的发展和广泛应用。但是,秩亏EIV模型的研究几乎一片空白。秩亏EIV模型研究的唯一成果仅有Golub/van Loan和van Huffel/Vandewalle提出的EIV模型解的不唯一性条件及其最小范数最小二乘解。本项目拟解决以下主要理论问题：1)给定一个全EIV模型，假定系数阵元素随机且统计独立，我们将证明Golub/van Loan与van Huffel/Vandewalle提出的解不唯一性条件以概率1几乎肯定不成立。2)即使全EIV模型理论上是秩亏的，我们将证明理论上秩亏的全EIV模型以概率1不能通过观测值判别出来。3)一个秩亏的EIV模型必定是部分EIV模型。我们也将扩展秩亏EIV模型最小范数最小二乘解的范数以覆盖包含模型参数及系数矩阵改正数在内的所有未知参数。

尽管EIV模型（即高斯马尔可夫模型中的系数矩阵元素全部或者部分具有随机性）模型自从被提出来以后，一直有学者持续关注并深入研究，但是EIV模型的秩亏问题一直没有恰当的解决，多年来项目组成员在前人研究成果的基础上对EIV 模型的秩亏问题开展了大量的理论研究工作，最终得出成果如下：.（1）给定一个满秩EIV模型，其系数矩阵的元素是连续随机变量，且系数矩阵元素的方差-协方差矩阵正定，则这样的EIV模型概率为1不可能秩亏，另外一方面也表明由Golub and van Loan [1980] and van Huffel and Vandewalle [1991,1988]提出的整体最小二乘TLS解（Total Least Squares solutions）的非唯一性条件概率为1不成立；.（2）即使一个全部EIV模型（Full Errors-In-Variables models）是理论上秩亏，其秩亏不可根据观测值定义的概率为1，因为系数矩阵的随机性，一个理论上秩亏的全部EIV模型几乎一直被概率为1地错误定义为一个可估模型；.（3）如果一个EIV 模型秩亏，则其必定是一个部分EIV模型（Partial Errors-In-Variables models），不过部分EIV模型不一定秩亏，即部分EIV 模型是秩亏的必要非充分条件，此外，如果一个部分EIV模型秩亏，我们则不可能从数值上获得其精确秩。.相关的这些理论结果具有实际的工程应用价值，但是如果没有相关工程人员的关注，这些根据随机观测值而建立的秩亏EIV模型的解可能全部是不正确的，原因是针对这样的EIV模型，可能存在无数个数学上等价的解，项目组经过广泛长期的深入研究，通过额外增加经验约束/信息，推荐了两种解算部分秩亏EIV模型的唯一解方法。