Residual lifetime prediction for heterogeneous degradation data by Bayesian semi-parametric method

TL;DR

This work tackles residual lifetime prediction under heterogeneous degradation by introducing a Bayesian semi-parametric degradation model in which unit-specific random effects follow a Dirichlet process mixture of normals ($F \sim DP(\gamma, G_0)$). Inference combines Gibbs sampling for the hierarchical model with transformation-based MCMC (TMCMC) to sample from the derived residual lifetime distributions, yielding $F_{T_{new}|\bm{Y}}(t)$ on $(0,\infty)$. A linear degradation path $\eta(t; \boldsymbol{\alpha}, \boldsymbol{\beta}) = \alpha + \beta t$ is embedded within a DP prior to flexibly capture heterogeneity, and the method is validated against simulations with gamma, Weibull, and normal mixtures as well as a Fatigue-C crack size dataset. Results show that the semi-parametric approach better adapts to multi-modal random effects and provides accurate unit-specific residual lifetimes, particularly for slowly degrading units, while remaining competitive for faster degraders. The framework offers a practical, flexible tool for condition-based maintenance and can extend to medical prognosis contexts where heterogeneous degradation processes are present.

Abstract

Degradation data are considered for assessing reliability in highly reliable systems. The usual assumption is that degradation units come from a homogeneous population. But in presence of high variability in the manufacturing process, this assumption is not true in general; that is different sub-populations are involved in the study. Predicting residual lifetime of a functioning unit is a major challenge in the degradation modeling especially in heterogeneous environment. To account for heterogeneous degradation data, we have proposed a Bayesian semi-parametric approach to relax the conventional modeling assumptions. We model the degradation path using Dirichlet process mixture of normal distributions. Based on the samples obtained from posterior distribution of model parameters we obtain residual lifetime distribution for individual unit. Transformation based MCMC technique is used for simulating values from the derived residual lifetime distribution for prediction of residual lifetime. A simulation study is undertaken to check performance of the proposed semi-parametric model compared with parametric model. Fatigue Crack Size data is analyzed to illustrate the proposed methodology.

Figures (20)

• Figure 1: Degradation paths generated with $n=10$ and $m=31$.
• Figure 2: Development of crack sizes over time Lu
• Figure 3: For each row, the left side plot is for histogram for Residual lifetimes, middle one is for trace plot and right side denotes autocorrelation plot of the samples produced for Unit 1-3 and $n=10$, $m=31$ by semi-parametric method.
• Figure 4: For each row, the left side plot is for histogram for Residual lifetimes, middle one is for trace plot and right side denotes autocorrelation plot of the samples produced for Unit 1-3 and $n=10$, $m=31$ by parametric method.
• Figure 5: For each row, the left side plot is for histogram for Residual lifetimes, middle one is for trace plot and right side denotes autocorrelation plot of the samples produced for Unit 1-3 and $n=10$, $m=31$ by semi-parametric method.