| 冯志强1
，张鸿燕2*.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[J].海南师范大学学报自科版,2022,35(3):250-259 |
| 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 LinearModel via Minimizing l1-Norm of Residual Vector and Basis Pursuit |
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| DOI：10.12051/j.issn.1674-4942.2022.03.003 |
| 中文关键词: 多元线性模型 参数估计 剩余向量 稀疏优化 基追踪方法 |
| 英文关键词: multivariate linear model parameter estimation residual vector sparse optimization basis-pursuit method |
| 基金项目:海南省自然科学基金项目（2019RC199）；国家自然科学基金项目（62167003） |
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| 中文摘要: |
| 本文提出了多元线性模型参数估计的最小l1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l1残差向量
将l1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l1
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l1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。 |
| 英文摘要: |
| In this paper we present a robust algorithm for minimizing the l1-norm of the residual vector for the multiple lin⁃
ear 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 l1-norm minimization problem is con⁃
verted to a non-differentiable optimization problem with constraints via minimizing the residual vector measured by l1-
norm. Thirdly, a sparse-optimization procedure is adopted for solving the residual vector of the interest with minimal l1-
norm based on basis-pursuit method. Finally, the compatible linear equations are solved to obtain the final objective solu⁃
tion. The Moore-Penrose inverse matrix is introduced to deal with the exception when the rank of coefficient matrix is defi⁃
cient, 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. |
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