Penalised hidden semi-Markov models with flexible sojourn-time distributions

Abstract In the class of hidden semi-Markov models with non-parametric sojourn-time distribution, we present a framework that penalises the latter with respect to its departure from a parametric base kernel. The penalised approach explicitly bridges parametric and non-parametric assumptions for the sojourn-time distributions, also in terms of the effective number of parameters. Inference is obtained via an expectation–maximisation algorithm. For Ridge-type penalties, we reduce the M step to a univariate optimisation problem, thereby greatly improving the computational burden. The penalty parameter is chosen using a computationally efficient Akaike-type information criterion. We illustrate our method with a simulation study, and a real data application to a large number of time series measuring traffic flow in several spots of five European cities. In the real data example, penalised hidden semi-Markov models are often preferred to hidden Markov models, and to hidden semi-Markov models with parametric and non-parametric sojourn-time specifications.