基于变量投影的结构总体最小二乘算法

测绘领域诸多实际应用中系数矩阵和观测向量具有结构特征,即系数矩阵和观测向量中包含固定量(甚至固定列)和随机量,并且不同位置的随机量线性相关。针对这个问题,从变量误差(errors-in-variables,EIV)函数模型出发,首先,将系数矩阵和观测向量构成的增广矩阵表示为仿射函数形式,并采用变量投影法对函数模型进行重构;然后,利用拉格朗日法推导出了一种结构总体最小二乘(structured total least squares,STLS)估计算法。算例分析结果表明,该算法与已有能够解决系数矩阵和观测向量存在结构特征的加权或结构总体最小二乘算法估计结果一致,说明了该算法的有效性,同时阐明了该算法与已有相关算法的关系。

Structured Total Least Squares Method Based on Variable Projection

Objectives:To solve the problem that the coefficient matrix and the observation vector have structured characteristics for many practical applications in the field of surveying and mapping,that is,the coefficient matrix and the observation vector contain fixed quantities(or even fixed columns)and random quantities,and the random quantities at different positions are linearly related.Methods:Starting from the errors-in-variables(EIV)function model,the augmented matrix composed of the coefficient matrix and the observation vector is expressed as an affine function form,and the function model is reconstructed by the variable projection method.Then,a structured total least squares(STLS)estimation algorithm is derived by the Lagrange method.Results:The example results show that the proposed algorithm is consistent with the existing weighted or structured total least squares estimation algorithms that can solve structured problem in the coefficient matrix and the observation vector.Compared with the weighted total least squares(WTLS)estimation algorithms,the proposed algorithm only needs to establish a positive definite weight matrix of independent random variables and reduces the number of estimates.Compared with other STLS estimation algorithms,the proposed algorithm takes into account the overall structure of the coefficient matrix and the observation vector.Conclusions:This shows the effectiveness of the proposed algorithm.The STLS estimation and the WTLS estimation ensure the statistical optimality from the perspective of the function model and the stochastic model,respectively.