On partially supervised learning and inference in dynamic Bayesian networks for prognostics with uncertain factual evidence : Illustration with Markov switching models

This paper describes an Autoregressive Partially-hidden Markov model (ARPHMM) for fault detection and prognostics of equipments based on sensors’ data. It is a particular dynamic Bayesian network that allows to represent the dynamics of a system by means of a Hidden Markov Model (HMM) and an autoregressive (AR) process. The Markov chain assumes that the system is switching back and forth between internal states while the AR process ensures a temporal coherence on sensor measurements. A sound learning procedure of standard ARHMM based on maximum likelihood allows to iteratively estimate all parameters simultaneously. This paper suggests a modification of the learning procedure considering that one may have prior knowledge about the structure which becomes partially hidden. The integration of the prior is based on the Theory of Weighted Distributions which is compatible with the Expectation-Maximization algorithm in the sense that the convergence properties are still satisfied. We show how to apply this model to estimate the remaining useful life based on health indicators. The autoregressive parameters can indeed be used for prediction while the latent structure can be used to get information about the degradation level. The interest of the proposed method for prognostics and health assessment is demonstrated on CMAPSS datasets. 1. AR-MARKOV MODELLING FOR PROGNOSTICS Autoregressive (AR) models have been shown to be appropriate for time-series modelling in various fields such as econometric (Ng & Vogelsang, 2002) and climate forecasting Pablo Juesas et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 United States License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. (Storch & Zwiers, 1999). In condition monitoring, it is generally used for fault detection by establishing an AR model using healthy conditions under various loads and using this model on the observed data recorded in in-service conditions in order to trend the residual signals. Such approaches does not make use of analytical description of faults or collection of typical fault patterns (Farrar & Worden, 2013; Serdio et al., 2013). Such a method has been used for gear monitoring by (W. Wang & Wong, 2002). (Yan, Ko, & Lee, 2004) used a similar approach together with a logit model to determine the probability of failure of an elevator door motion system. (Thanagasundram, Spurgeon, & Schlindwein, 2008) suggested to use a pole representation of an AR process to detect bearings faults. (Saha, Goebel, & Christophersen, 2009) compared ARIMA (AR with integrated moving average) with two other models for battery prognostics. In structural safety, AR models were used by (He & De Roeck, 1997; Huang, 2001) to identify structural dynamic characteristics of system subjected to ambient excitations. (Ling, Shantz, Mahadevan, & Sankararaman, 2011) suggested to use an ARMA to characterize and reconstruct fatigue loads for prognostics application on mechanical components. The authors proposed to adapt the parameters of this model by Bayesian updating to accommodate variability in loading and data sparsity. (Lehman, Nemati, & Mark, 2015) were interested in the modelling of switching autoregressive dynamics from multivariate vital sign time series in order to stratify mortality risks of intensive care units patients receiving particular treatments. In this biomedical application, the authors made use of a combination of an AR process and a Hidden Markov Model (HMM) called ARHMM. Such a model is able to cope with multistate non-stationary systems which are generally en-