Bayesian Hierarchical Modelling of Noisy Gamma Processes: Model Formulation, Identifiability, Model Fitting, and Extensions to Unit-to-Unit Variability

TL;DR

This paper develops a Bayesian hierarchical framework for noisy gamma process models, addressing identifiability issues, extending to multiple units, and enabling predictive inference for degradation and failure times.

Contribution

It introduces a reparameterized Bayesian hierarchical model for noisy gamma processes, resolving identifiability problems and extending to unit-to-unit variability with practical model selection methods.

Findings

Identifiability issues can be mitigated with stronger priors or additional data.

Extensions effectively model unit-to-unit variability in degradation data.

Bayesian model selection via cross-validation improves predictive performance.

Abstract

The gamma process is a natural model for monotonic degradation processes. In practice, it is desirable to extend the single gamma process to incorporate measurement error and to construct models for the degradation of several nominally identical units. In this paper, we show how these extensions are easily facilitated through the Bayesian hierarchical modelling framework. Following the precepts of the Bayesian statistical workflow, we show the principled construction of a noisy gamma process model. We also reparameterise the gamma process to simplify the specification of priors and make it obvious how the single gamma process model can be extended to include unit-to-unit variability or covariates. We first fit the noisy gamma process model to a single simulated degradation trace. In doing so, we find an identifiability problem between the volatility of the gamma process and the measurement error when there are only a few noisy degradation observations. However, this lack of identifiability can be resolved by including extra information in the analysis through a stronger prior or extra data that informs one of the non-identifiable parameters, or by borrowing information from multiple units. We then explore extensions of the model to account for unit-to-unit variability and demonstrate them using a crack-propagation data set with added measurement error. Lastly, we perform model selection in a fully Bayesian framework by using cross-validation to approximate the expected log probability density of new observation. We also show how failure time distributions with uncertainty intervals can be calculated for new units or units that are currently under test but are yet to fail.