A Bayesian Tweedie exponential dispersion process with a change-point for two-phase degradation data

Abstract

Degradation analysis is an efficient way to evaluate a product's reliability in various applications by capturing its degradation characteristics quickly and accurately. In the degradation analysis, the accuracy of the reliability inference depends highly on the selected model for fitting observed degradation measurements. Covering popular stochastic process models as special cases, the Tweedie exponential dispersion process (TEDP) model has potential in describing more extensive degradation phenomena in engineering applications. In this work, we propose a TEDP with a change-point (CPTEDP) to model two-phase degradation pattern of products, where unit-specific two-piece drifts are considered to explain heterogeneous degradation patterns of testing units. We introduce a three-stage hierarchical Bayesian inference on the parameters of the CPTEDP model without the need for approximating its likelihood function that has no closed-form expression using Markov chain Monte Carlo (MCMC). To overcome difficulties posed by the continuity constraint of degradation processes on modeling and computation under the hierarchical Bayesian framework, we adopt the pseudo likelihood function instead of the real likelihood for computational efficiency. After fitting the three-stage hierarchical Bayesian CPTEDP model by adopting the Gibbs algorithm, we derive the approximations of reliability measures (e.g., mean time to failure (MTTF), remaining useful life (RUL), mean residual life (MRL)) of explicit analytical forms by substituting posterior estimates obtained from the Gibbs algorithm. Analytical results from a case study and a variety of simulations demonstrate that the hierarchical Bayesian CPTEDP model can provide more suitable fits to degradation data with two-phase patterns, improving the accuracy of reliability estimation for products.