Maximum Likelihood Estimation for a Hidden Semi-Markov Model with Multivariate Observations


  • This article is part of the Special issue on selected papers from the 7th IMA International Conference on Modelling in Industrial Maintenance and Reliability published in Quality and Reliability Engineering International, Volume 28, Issue 6. The entire Special Issue can be read on Wiley Online Library.

Viliam Makis, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8.



In this paper, a parameter estimation procedure for a condition-based maintenance model under partial observations is presented. The deterioration process of the partially observable system is modeled as a continuous-time homogeneous semi-Markov process. The system can be in a healthy or unhealthy operational state, or in a failure state, and the sojourn time in the operational state follows a phase-type distribution. Only the failure state is observable, whereas operational states are unobservable. Vector observations that are stochastically related to the system state are collected at equidistant sampling times. The objective is to determine maximum likelihood estimates of the model parameters using the Expectation–Maximization (EM) algorithm. We derive explicit formulae for both the pseudo likelihood function and the parameter updates in each iteration of the EM algorithm. A numerical example is developed to illustrate the efficiency of the estimation procedure. Copyright © 2012 John Wiley & Sons, Ltd.