Ferreira JTAS., Denison DGT., Holmes CC.
This chapter serves as an introduction to the use of partition models to estimate a spatial process z(x) over some p-dimensional regionof interest X. Partition models can be useful modelling tools as, unlike standard spatial models (e.g. kriging) they allow the correlation structure between points to vary over the space of interest. Typically, the correlation between points is assumed to be a fixed function which is most likely to be parameterised by a few variables that can be estimated from the data (see, for example, Diggle et al. (1998)). Partition models avoid the need for pre-examination of the data to find a suitable correlation function to use. This removes the bias necessarily introduced by picking the correlation function and estimating its parameters using the same set of data. Spatial clusters are, by their nature, regions which are not representative of the entire space of interest. Therefore it seems inappropriate to assume a stationary covariance structure over X. The partition model relaxes this assumption by breaking up the space into regions where the data are assumed to be generated independently from locally parameterised models. This can naturally place in a single region those points relating to an unusual cluster, and these points do not necessarily have to influence the response function in nearby locations. Further, by assuming independence between the regions the response function at the cluster centre tends not to be oversmoothed. We now describe the partition model and its implementation in a Bayesian framework, via Markov chain Monte Carlo (MCMC) methods. We first give the model used for Gaussian response data but also discuss how the framework can be extended to count data (e.g. for disease mapping applications). Further, when analysing count data we show how, when covariate information is available, we can incorporate this into the analysis. This method is shown to be useful for both linear and nonlinear modelling of the covariate efects.