一种加权整体最小二乘估计的高效算法

An Efficient Algorithm for Weighted Total Least Squares Method

摘要:

加权整体最小二乘法（WTLS）是估计errors-in-variables （EIV）模型参数严密的方法，当面临大数据集时，其计算效率有限。针对EIV模型中设计矩阵呈现出的结构性特征，在最小二乘准则的约束条件下，通过仅给设计矩阵的随机列赋予权重，推证了适用于EIV模型参数估计的部分加权整体最小二乘法（PWTLS）。PWTLS无需借助拉格朗日辅助法，能够精确估计EIV模型参数；另外，该算法缩减了矩阵的维数，同时在迭代过程中避免了估计设计矩阵的随机误差，从而减小了矩阵运算量，提升了计算效率。最后以真实数据和模拟数据为例与其他7种同类算法进行对比，结果表明， PWTLS取得了与同类算法相同的精度，但计算效率显著提高，验证了算法的可行性。

Abstract:

The weighted total least-squares （WTLS） adjustment is a rigorous method for estimating parameters in the errors-in-variables （EIV） model. However， the WTLS are not proper for larger data problem in terms of computational efficiency. Aimed at the structural characteristics of the design matrix in the EIV model，a partially weighted total least-squares （PWTLS） algorithm is proposed based on weighted least-squares （WLS） adjustment by weighting the random column of the design matrix. The PWTLS can obtain an exact solution of the EIV model without applying Lagrange multipliers in a straightforward manner. In addition， the PWTLS reduces the dimensions of the cofactor matrix and does not estimate the random error of the design matrix， as this would greatly improve the computational efficiency. Finally， real and simulated examples are used to demonstrate the accuracy and computational performance of the proposed algorithms . The results show that the PWTLS can obtain the same accuracy as the existing seven improved algorithms， but the computational efficiency is significantly improved .