Bayesian spatio-temporal modelling of rainfall through non-homogenous hidden Markov models

Abstract

Multi-site statistical models for daily rainfall should account for spatial and temporal dependence amongst measurements and also allow for the event of no rain. Recent research into climate change and variability has sparked interest in the relationship between rainfall and climate, stimulating the development of statistical models that relate large-scale atmospheric variables to local precipitation. Although modelling daily rainfall presents a challenging and topical problem, there have been few attempts taking a subjective Bayesian approach. This thesis is concerned with developing hidden Markov models (HMMs) for the spatio-temporal analysis of rainfall data, within a Bayesian framework. In these models, daily rainfall patterns are driven by a finite number of unobserved states, interpreted as weather states, that evolve in time as a first order Markov chain. The weather states explain space time structure in the data so that reasonably simple models can be adopted within states. Throughout this thesis, the models and procedures are illustrated using data from a small dense network of six sites situated in Yorkshire, UK. First we study a simple (homogeneous) HMM in which rainfall occurrences and amounts, given occurrences, are conditionally independent in space and time, given the weather state, and have Bernoulli and gamma distributions, respectively. We compare methods for approximating the posterior distribution for the number of weather states. This simple model does not incorporate atmospheric information and appears not to capture the observed spatio-temporal structure. We therefore investigate two non-homogeneous hidden Markov models (NHMMs) in which we allow the transition probabilities between weather states to depend on time-varying atmospheric variables and successively relax the conditional independence assumptions. The first NHMM retains the simple conditional model for non-zero rainfall amounts but allows occurrences to form a Markov chain of autologistic models, given the weather state. The second introduces latent multivariate normal random variables to form a hierarchical NHMM in which neither rainfall occurrences nor non-zero amounts are conditionally spatially or temporally independent, given the weather state. Throughout this thesis, we emphasise the elicitation of prior distributions that convey genuine initial beliefs. For each hidden Markov model studied we demonstrate techniques to assist in this task.