基于残差向量l1范数最小化与基追踪的多元线性模型参数估计方法  

A Novel Approach for Estimating Parameters of Multivariate Linear Model via Minimizing l1-Norm of Residual Vector and Basis Pursuit

作　　者：冯志强 张鸿燕 FENG Zhiqiang;ZHANG Hongyan(School of Mathematics and Statistics,Hainan Normal University,Haikou 571158,China;School of Information Science and Technology,Hainan Normal University,Haikou 571158,China)

机构地区：[1]海南师范大学数学与统计学院,海南海口571158 [2]海南师范大学信息科学技术学院,海南海口571158

出　　处：《海南师范大学学报（自然科学版）》2022年第3期250-259,267,共11页Journal of Hainan Normal University（Natural Science）

基　　金：海南省自然科学基金项目(2019RC199);国家自然科学基金项目(62167003)。

摘　　要：本文提出了多元线性模型参数估计的最小l范数解的一种新方法。该方法分为4步:首先将参数估计问题描述为由观测数据确定的超定线性方程组的形式;然后利用最小l残差向量将l范数最小化问题转化为一个有约束的不可微最优化问题;接下来利用基追踪方法求得最小l残差向量的稀疏解;最后求解相容线性方程组得到原方程组的最小l范数解。对于系数矩阵存在秩亏的情况,采用Moore-Penrose广义逆进行了有效的处理,这极大地扩展了算法的适应性。数值算例表明该方法具有良好的鲁棒性与很高的数值精度,并且容许较高的临界外点比例。In this paper we present a robust algorithm for minimizing the l-norm of the residual vector for the multiple linear model. The algorithm consists of four major steps. Firstly, the multiple linear model is specified by the overdetermined linear system with the observation data.Secondly, the equivalent transformation of the l-norm minimization problem is converted to a non-differentiable optimization problem with constraints via minimizing the residual vector measured by l-norm. Thirdly, a sparse-optimization procedure is adopted for solving the residual vector of the interest with minimal l-norm based on basis-pursuit method. Finally, the compatible linear equations are solved to obtain the final objective solution. The Moore-Penrose inverse matrix is introduced to deal with the exception when the rank of coefficient matrix is deficient, which enlarges the adaptability of the algorithm.The accuracy and robustness of the algorithm proposed are verified and evaluated with numerical examples in case of a high level of critical proportion of outliers.

关 键 词：多元线性模型 参数估计 剩余向量 稀疏优化 基追踪方法 

分 类 号：O212.1[理学—概率论与数理统计]