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Understanding the current health condition of complex systems and their temporal evolution is an important step in prognostics and health management (PHM). However, when managing a fleet of complex systems, variations arising from manufacturing, environmental factors, mission profiles, and maintenance practices result in diverse health index (HI) trajectories. Therefore, in PHM, it is essential not only to identify common fleet-wide trends but also to account for individual asset-level variations when inferring HIs. While several data-driven approaches exist for inferring individual asset-level HIs from unsupervised run-to-failure degradation data, little research has been devoted to deriving analytical probabilistic representations of HIs that encompass both fleet-level trends and individual asset-level fluctuations. This paper aims to bridge this gap by addressing the research question of how to obtain an analytical representation of probability distributions for the time to reach intermediate degradation levels, using run-to-failure data or incomplete degradation trajectories from a fleet of complex systems. In this work, it is assumed that suitable, asset-specific HI curves have been inferred through a fusion of deep learning techniques and prior expert knowledge of degradation physics . Given this context, we derive an analytical probabilistic description of the health index (HI) that reflects both fleet-wide trends and asset-specific conditions in the cases of Gamma or Weibull time-to-failure (TTF) distributions. Our approach involves defining HIs with a power law function, enabling the modeling of TTF and time to reach intermediate degradation levels. Moreover, we also detail the procedure for estimating the power law exponent from field data through regression analysis and conduct a sensitivity analysis regarding this exponent. To illustrate our methodology, we present two case studies based on the N-CMAPPS dataset of turbofan engines and Li-ion batteries, validating the aforementioned assumptions and illustrating our methodology steps
Dersin et al. (Thu,) studied this question.