Using hidden Markov models to model spatial dependence in a network

Summary This study considers spatial dependence in the number of injury crashes reported on a road network. The aggregated crash counts are considered realisations of a Poisson random variable; thus, we model both over‐dispersion and serial correlation using the Poisson hidden Markov model (PHMM). PHMMs have typically been used for modelling temporal dependence, but they have rarely been used to model spatial dependence. Our interest, however, is specifically in relation to an underlying point process which is constrained to occur on a network. We illustrate the use of the PHMM with police‐reported data on injury road collisions on selected motorways in the United Kingdom over a 5‐year period (2010–2014). We use officially recorded estimates of traffic volume as an exposure variable. The aim is to identify highway segments which might have distinctly high crash rates. To do this, we first select an optimal model in terms of the number of latent states. As we use a Bayesian approach, we can assign a posterior classification probability to each segment in terms of membership of the various underlying risk states. Model fitting is conducted using the Markov chain Monte Carlo approach; we develop a new modified version of the Akaike Information Criterion and Bayesian Information Criterion, approximated from a Bayesian framework, to select the best model in terms of number of states.