Modeling Product Degradation with Heterogeneity: A General Random-Effects Wiener Process Approach

Degradation of many products in practical applications is often subject to unit-to-unit heterogeneity. Such heterogeneity can be attributed to heterogeneous quality of the raw materials and the fluctuation of the manufacturing process, as well as the heterogeneous usage conditions and environments. The heterogeneity leads to the scattering of the degradation rates and diffusion intensities in the population. To model this phenomenon, this study proposes a general random-effects Wiener process model that accounts for the unit-to-unit heterogeneity in the degradation drift and the volatility simultaneously. In particular, the drift of the Wiener process is characterized by a normal distribution and the diffusion parameter is characterized by an independent inverse Gaussian distribution. The proposed model is flexible for characterization of heterogeneous degradation, and permits an analytically tractable model inference. An EM algorithm incorporating the variational Bayesian method is developed to estimate the model parameters, and a parametric bootstrap approach is proposed to construct confidence intervals. The performance of the proposed model and the estimation algorithm is validated by Monte Carlo simulations. The degradation data of an infrared LED device and the wearing data of the magnetic head of a hard disk driver are studied based on the proposed model. With comprehensive comparative studies, the good performance of the proposed model in fitting the real degradation data is validated.