A Bayesian mixture model for Poisson network autoregression

Multivariate count time series arise in a wide range of applications, including the number of COVID-19 cases recorded each week in different counties of the Republic of Ireland. In this example, it is natural to view the counties as nodes in a network, with edges between counties reflecting proximity. One could then model disease spread on a network through a regression model. Often Gaussian errors are assumed for such a model, but for count data this assumption may not be natural. With this motivating example in mind, we develop a model with the following features. We assume that the time series occur on the nodes of a known underlying network where the edges dictate the form of a structural vector autoregression model. In contrast to using a full vector autoregressive model, the network assumption is a means of imposing sparsity. Moreover we aim for a model that is able to accommodate heterogeneous node dynamics, and to cluster nodes that exhibit similar behaviour. To address these aims, we propose a new Bayesian Poisson network autoregression mixture model that we call a PNARM model, which combines ideas from Poisson network autoregression models, grouped network autoregression models, and non-uniform co-clustering priors.