Abstract
Motivated by a real data problem, an anisotropic and inhomogeneous spatio-temporal Hidden Markov model (HMM) with an unknown number of states is made up on a cylindrical lattice. A Bayesian inference procedure, based on a reversible jump Markov chain Monte Carlo algorithm, is proposed to estimate both the dimension and the unknown parameters of the model. The real data problem is the modelling in time and in space of the concentrations of three dissolved inorganic nitrogens recorded monthly by the Scottish Environmental Protection Agency in the 56 major Scottish rivers. The 56 gauging stations can be linked to create a circle and so the spatio-temporal data set can be displayed on a cylinder. The states of the hidden Markov process allows the classification of the observations in a small set of groups. The different states can represent increasing levels of pollution. In the Bayesian model presented here, the hidden Markov process is an anisotropic and inhomogeneous Potts model. The Potts model is widely used in statistical mechanics to model the spins of elementary particles that are placed on a lattice. Here the hidden Potts model is assumed to be anisotropic (i.e., variant under rotations) and inhomogeneous (i.e., variant under translations). Anisotropy is due to the presence of two different parameters describing the link between neighbouring pixels: one for the temporal relation and the other for the spatial relation. Inhomogeneity is established by assuming that the spatial relation is a function of the physical distance between two neighbouring sites.