A Comparison of Hidden Markov and Semi-Markov Modeling for a Deterioration System subject to Vibration Monitoring

We compare a hidden Markov and Erlang semi-Markov modeling of a partially observable deteriorating system operating under a varying load and subject to multi-sensor vibration monitoring. The evolution of the unknown state process is described by a hidden, two state semi-Markov process with an Erlang sojourn time distribution in the healthy state. The unknown model parameters are estimated using the EM algorithm. We derive explicit formulae for the parameter re-estimation in the EM algorithm, which leads to a fast estimation procedure. An optimal Bayesian maintenance policy is developed minimizing the long run expected average cost per unit time and a formula for the mean residual time in the healthy state is derived as a function of the posterior probability statistic. The results show that a simple hidden Erlang model performs better than a hidden Markov model, which is encouraging for considering a more general hidden semi-Markov modeling in future research.