Partially Hidden Markov Chain Multivariate Linear Autoregressive model: inference and forecasting—application to machine health prognostics

• Published in: Machine learning Vol. 112; no. 1; pp. 45 - 97
• Main Authors: Dama, Fatoumata, Sinoquet, Christine
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2023; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Artificial Intelligence; Atmospheric models; Autoregressive models; Computer Science; Control; Convergence; Forecasting; Inference; Labels; Life Sciences; Machine Learning; Markov analysis; Markov chains; Mechatronics; Meteorology; Multivariate analysis; Natural Language Processing (NLP); Robotics; Simulation and Modeling; Time series; Turbofan engines; Machine health prognostics; Autoregressive model; Regime-switching model; Time series analysis; Hidden state inference; Forecasting
• ISSN: 0885-6125; 1573-0565; 1573-0565
• DOI: 10.1007/s10994-022-06209-5

Time series subject to regime shifts have attracted much interest in domains such as econometry, finance or meteorology. For discrete-valued regimes, models such as the popular Hidden Markov Chain (HMC) describe time series whose state process is unknown at all time-steps. Sometimes, time series are annotated. Thus, another category of models handles the case with regimes observed at all time-steps. We present a novel model which addresses the intermediate case: (i) state processes associated to such time series are modelled by Partially Hidden Markov Chains (PHMCs); (ii) a multivariate linear autoregressive (MLAR) model drives the dynamics of the time series, within each regime. We describe a variant of the expectation maximization (EM) algorithm devoted to PHMC-MLAR model learning. We propose a hidden state inference procedure and a forecasting function adapted to the semi-supervised framework. We first assess inference and prediction performances, and analyze EM convergence times for PHMC-MLAR, using simulated data. We show the benefits of using partially observed states as well as a fully labelled scheme with unreliable labels, to decrease EM convergence times. We highlight the robustness of PHMC-MLAR to labelling errors in inference and prediction tasks. Finally, using turbofan engine data from a NASA repository, we show that PHMC-MLAR outperforms or largely outperforms other models: PHMC and MSAR (Markov Switching AutoRegressive model) for the feature prediction task, PHMC and five out of six recent state-of-the-art methods for the prediction of machine useful remaining life.