Structure function learning of hierarchical multi-state systems with a change-point: An embedded expectation-maximization algorithm

Hierarchical multi-state systems (HMSSs) are one of the most important structures in engineering practices, and their reliability assessment has received increasing attention in the past decades. However, the degradation behaviors of HMSSs may exhibit two distinct phases during their operations due to a sudden change in their structure functions caused by the shift in functional requirements or working environments. Motivated by this phenomenon, this article studies a new system structure, namely HMSSs with a change-point. A dynamic Bayesian network (DBN) model is applied for reliability assessment of HMSSs with a change-point. A novel embedded Expectation-Maximization (EM) algorithm is developed to learn the unknown parameters, including the change-point and structure functions, of the DBN model using incomplete observation data. In contrast to traditional EM algorithms, an additional optimization procedure for estimating the change-point is embedded into the M-step of the proposed embedded EM algorithm. Two illustrative examples, including a numerical case and a lighting system in a manufacturing workshop, are exemplified to demonstrate the effectiveness of the proposed method. The results indicate that the proposed method can accurately identify the change-point and learn structure functions from incomplete observation sequences.