Modern industrial systems are often subject to multiple failure modes, and their conditions are monitored by multiple sensors, generating multiple time-series signals. Additionally, time-to-failure data are commonly available. Accurately predicting a system's remaining useful life (RUL) requires effectively leveraging multi-sensor time-series data alongside multi-mode failure event data. In most existing models, failure modes and RUL prediction are performed independently, ignoring the inherent relationship between these two tasks. Some models integrate multiple failure modes and event prediction using black-box machine learning approaches, which lack statistical rigor and cannot characterize the inherent uncertainty in the model and data. This paper introduces a unified approach to jointly model the multi-sensor time-series data and failure time concerning multiple failure modes. This proposed model integrate a Cox proportional hazards model, a Convolved Multi-output Gaussian Process, and multinomial failure mode distributions in a hierarchical Bayesian framework with corresponding priors, enabling accurate prediction with robust uncertainty quantification. Posterior distributions are effectively obtained by Variational Bayes, and prediction is performed with Monte Carlo sampling. The advantages of the proposed model is validated through extensive numerical and case studies with jet-engine dataset.

Building a predictive model that rapidly adapts to real-time condition monitoring (CM) signals is critical for engineering systems/units. Unfortunately, many current methods suffer from a trade-off between representation power and agility in online settings. For instance, parametric methods that assume an underlying functional form for CM signals facilitate efficient online prediction updates. However, this simplification leads to vulnerability to model specifications and an inability to capture complex signals. On the other hand, approaches based on over-parameterized or non-parametric models can excel at explaining complex nonlinear signals, but real-time updates for such models pose a challenging task. In this paper, we propose a neural process-based approach that addresses this trade-off. It encodes available observations within a CM signal into a representation space and then reconstructs the signal's history and evolution for prediction. Once trained, the model can encode an arbitrary number of observations without requiring retraining, enabling on-the-spot real-time predictions along with quantified uncertainty and can be readily updated as more online data is gathered. Furthermore, our model is designed to incorporate qualitative information (i.e., labels) from individual units. This integration not only enhances individualized predictions for each unit but also enables joint inference for both signals and their associated labels. Numerical studies on both synthetic and real-world data in reliability engineering highlight the advantageous features of our model in real-time adaptation, enhanced signal prediction with uncertainty quantification, and joint prediction for labels and signals.