A degradation model indexed by two scales

Degradation is a common and important phenomenon for industrial products, which is the basis of preventive maintenance planning. The degradation process manifests as the gradual degradation of certain performance characteristics, which can be indexed through calendar time or cumulative usage level. Most existing studies model the degradation as a process indexed by a single scale, while the degradation process is actually related to both the calendar time and usage process. In particular, due to the heterogeneity of customers or operating scenarios, the accumulation of usage, i.e. the usage process itself, is often of uncertainty. In this case, indexing the degradation process solely by time or usage is not sufficient. This study proposes a degradation process model indexed by two scales, i.e., the time and the usage. The proposed model is the sum of two correlated components, where each component representing the degradation along a single scale can be formulated as a Wiener process. We develop the maximum likelihood estimation procedure for the proposed model and verify its performance through Monte Carlo simulations.