Penalized Mixed-Effects Decomposition (PMD) for Functional Data

Abstract

The code is for the Penalized Mixed-effects Decomposition (PMD) Method. The PMD method can decompose the profiles (curves/ functional data) into four components: fixed effects, normal effects, defective effects ,and noise. The PMD method can be used for the multichannel profile detection of in-line Raman spectra, and it can also be used for feature extraction of other signals (profiles/functional data). Reference: Yue, X., Yan, H., Park, J.G., Liang, Z., Shi, J., “A Wavelet-based Penalized Mixed Effects Decomposition for Multichannel Profile Monitoring Based on In-line Raman Spectroscopy”, IEEE Transactions on Automation Science and Engineering, (in press). Abstract of the paper: Modeling and analysis of profiles, especially high-dimension nonlinear profiles, is an important and challenging topic in statistical process control. Conventional mixed-effects models have several limitations in solving the multichannel profile detection problems for in-line Raman spectroscopy, such as the inability to separate defective information from random effects, computational inefficiency, and inability to handle high dimensional extracted coefficients. In this paper, a new wavelet-based penalized mixed-effects decomposition (PMD) method is proposed to solve the multichannel profile detection problem in Raman spectroscopy. The proposed PMD exploits a regularized high-dimensional regression with linear constraints to decompose the profiles into four parts: fixed effects, normal effects, defective effects, and signal-dependent noise. An optimization algorithm based on the accelerated proximal gradient (APG) is developed to do parameter estimation efficiently for the proposed model. Finally, the separated fixed effects coefficients, normal effects coefficients, and defective effects coefficients can be used to extract the quality features of fabrication consistency, within-sample uniformity, and defect information, respectively. Using a surrogated data analysis and a case study, we evaluated the performance of the proposed PMD method and demonstrated a better detection power with less computational time. Note to Practitioners— This paper was motivated by the need of implementing multichannel profile detection for Raman spectra to realize in-line process monitoring and quality control of continuous manufacturing of carbon nanotube buckypaper. Existing approaches, like the mixed-effects model or the smooth sparse decomposition method, cannot separate defective information in random effects effectively. This paper develops a penalized mixed-effects decomposition which decomposes Raman spectra into four components: fixed effects, normal effects, defective effects, and signal-dependent noise respectively. The first three components can be applied to monitor the fabrication consistency, degree of uniformity, and defect information of buckypaper, respectively. With this new approach, several quality features can be monitored simultaneously and the algorithm based on accelerated proximal gradient method can satisfy the computation speed requirement of in-line monitoring. This research provides a solid foundation for in-line process monitoring and quality control for scalable nanomanufacturing of carbon nanotube buckypaper. Furthermore, the developed methodology can be applied in the decomposition of other signal systems with fixed effects, normal and defective effects.