Hierarchical sparse functional principal component analysis for multistage multivariate profile data

Modern manufacturing systems typically involve multiple production stages, the real-time status of which can be tracked continuously using sensor networks that generate a large number of profiles associated with all process variables at all stages. The analysis of the collective behavior of the multistage multivariate profile data is essential for understanding the variance patterns of the entire manufacturing process. For this purpose, two major challenges regarding the high data dimensionality and low model interpretability have to be well addressed. This article proposes integrating Multivariate Functional Principal Component Analysis (MFPCA) with a three-level structured sparsity idea to develop a novel Hierarchical Sparse MFPCA (HSMFPCA), in which the stage-wise, profile-wise and element-wise sparsity are jointly investigated to clearly identify the informative stages and variables in each eigenvector. In this way, the derived principal components would be more interpretable. The proposed HSMFPCA employs the regression-type reformulation of the PCA and the reparameterization of the entries of eigenvectors, and enjoys an efficient optimization algorithm in high-dimensional settings. The extensive simulations and a real example study verify the superiority of the proposed HSMFPCA with respect to the estimation accuracy and interpretation clarity of the derived eigenvectors.