A convex two-dimensional variable selection method for the root-cause diagnostics of product defects

Many multistage manufacturing processes consist of multiple identical stages. The root cause diagnostic of the product quality defects of these processes often involves the simultaneous identification of crucial stages and process variables that are related to product anomalies. In the literature, this is typically achieved by using penalized matrix regression that regresses the index of product defect against a matrix whose rows and columns respectively represent the stages and process variables. However, most existing models have some limitations that compromise their applicability and/or performance. For example, some models have an assumption on the rank of the coefficient, which often cannot be satisfied; some others formulate a nonconvex optimization criterion that easily results in a local optimum. Also, most models only provide diagnostics results with group-wise (i.e., stage- and variable-wise) sparsity. To address these challenges, this article proposes a novel convex two-dimensional variable selection method that can inspire both group-wise and element-wise sparsity. This is accomplished by proposing a new generalized matrix regression model and simultaneously penalizing the rows, columns, and elements of the regression coefficient matrix using an ℓ2, ℓ2, and ℓ1 norm, respectively. Simulated and real-world data are used to validate the effectiveness of the proposed method.