Abstract

The root cause diagnostics of product quality defects in multistage manufacturing processes often requires a joint identification of crucial stages and process variables. To meet this requirement, this article proposes a novel penalized matrix regression methodology for two-dimensional variable selection. The method regresses a scalar response variable against a matrix-based predictor using a generalized linear model. The unknown regression coefficient matrix is decomposed as a product of two factor matrices. The rows of the first factor matrix and the columns of the second factor matrix are simultaneously penalized to inspire sparsity. To estimate the parameters, we develop a Block Coordinate Proximal Descent (BCPD) optimization algorithm, which cyclically solves two convex sub-optimization problems. We have proved that the BCPD algorithm always converges to a critical point with any initialization. In addition, we have also proved that each of the sub-optimization problems has a closed-form solution if the response variable follows a distribution whose (negative) log-likelihood function has a Lipschitz continuous gradient. A simulation study and a dataset from a real-world application are used to validate the effectiveness of the proposed method.

摘要

多阶段制造过程中产品质量缺陷的根本原因诊断通常需要联合识别关键阶段和过程变量。为了满足这一要求，本文提出了一种用于二维变量选择的新型惩罚矩阵回归方法。该方法使用广义线性模型将标量响应变量与基于矩阵的预测器进行回归。未知回归系数矩阵被分解为两个因子矩阵的乘积。第一个因子矩阵的行和第二个因子矩阵的列同时受到惩罚以激发稀疏性。为了估计参数，我们开发了一种块坐标近端下降（BCPD）优化算法，它循环地解决了两个凸子优化问题。我们已经证明了 BCPD 算法在任何初始化时总是收敛到一个临界点。此外，我们还证明了如果响应变量遵循其（负）对数似然函数具有 Lipschitz 连续梯度的分布，则每个子优化问题都有一个封闭形式的解。模拟研究和来自实际应用程序的数据集用于验证所提出方法的有效性。