Modeling left-truncated degradation data using random drift-diffusion Wiener processes

For products whose performance characteristic (PC) gradually degrades with time, one usually observes its degradation levels repeatedly to predict its remaining useful life (RUL). Due to the limited storage space of the server and the low resolution of a measurement instrument, we seldom record the low-magnitude degradation values at the early degradation stage in applications. Such observation setting introduces left-truncated degradation data, in which the data collection starts later than the unit's installation. This brings sampling biases and complicates the degradation data analysis. Moreover, due to the uncontrollable factors in applications, the degradation drift and the degradation diffusion may differ among various units. Motivated by an application of high-speed train bearings, we propose a Wiener process model for the left-truncated degradation data and jointly consider the drift-diffusion random effects. Closed-form formulas are available in the expectation-maximization (EM) algorithm for estimating the model parameters. We derive the RUL distribution in closed form. We also extend the proposed model to the multivariate degradation process. The parameters are estimated with the help of the Monte Carlo EM (MCEM) algorithm. An additional laser application illustrates the performance of the proposed model in RUL prediction, which may help to design a predictive maintenance strategy