Regularized Periodic Gaussian Process for Nonparametric Sparse Feature Extraction From Noisy Periodic Signals

This study proposes a nonparametric sparse feature extraction approach based on a periodic Gaussian process (PGP) for highly nonlinear sparse periodic signals, which may not be effectively modeled by conventional linear models based on user-specified dictionaries. The PGP model is reformulated as a mixed-effects model. Hence a regularization term is allowed to be imposed on the random effect of the PGP model, called regularized PGP (RPGP) in this study, for sparse feature extraction. Unlike conventional sparse models, the proposed RPGP can simultaneously model fixed and random effects (global trend and local sparsity) of the signals. A computationally scalable algorithm based on the alternating direction method of multipliers (ADMM) is tailored for RPGP to iteratively optimize the fixed and random effects. The efficient computation of RPGP is achieved by a customized circulant-based acceleration technique that utilizes fast Fourier transform on circulant matrices. The performance of RPGP is evaluated through a simulation study on synthetic signals and a case study on real vibration signals. Note to Practitioners—This work is motivated by the problem of sparse feature extraction from highly nonlinear periodic signals with a complex global trend. The key issues involved in this problem include: 1) how to model the highly nonlinear periodic signals; 2) how to separate the periodic signals and the global trend from noisy signals; 3) how to extract the sparse periodic feature. Existing approaches for sparse feature extraction assume that the signals have a constant trend and can be effectively modeled by certain dictionaries. This work proposes a novel approach that leverages a regularized PGP to simultaneously extract the sparse periodic feature and the global trend. Moreover, the nonparametric nature of the PGP model enables automatic feature extraction, eliminating the need to manually select a dictionary for sparse periodic feature extraction. The proposed approach involves four main steps: 1) collecting periodic signals from embedded sensors; 2) modeling the signals with PGP models following the fixed effects framework; 3) training the model parameters using the proposed optimization procedure; 4) extracting the global trend and sparse periodic feature based on the model parameters. This paper demonstrated the application of rolling bearing vibration signals, while the effectiveness of the proposed model is not limited to this topic. Other periodic signals with similar characteristics can also be analyzed by applying the proposed model. In the future, this approach will be extended to handle sparse feature extraction from pseudo-periodic signals in a more complex noise environment.