## Spatial Cluster Modelling: An Overview

When analysing spatial data one is often interested in detecting deviations from the expected. For instance, we may be interested in the answers to questions like, “Is there an unusual aggregation of leukemia cases around a nuclear power station ?” or, “Where is it likely that the air pollution level is above the legally allowed limit ?”. In both cases the focus is on ﬁnding regions in (usually two-dimensional) space in which higher than expected counts, or readings, are observed. We shall call such areas clusters and determining their nature forms the focus of this work. This volume brings together a collection of papers on the topic of spatial

cluster modelling and gives descriptions of various approaches which begin to solve the problem of detecting clusters. The papers are statistical in nature but draw on results in other ﬁelds as diverse as astrophysics, medical imaging, ecology and environmental engineering. Two examples of the sort of spatial processes that we shall consider here

are displayed in Figs. 1.1-1.2. Fig. 1.1 is an example of a point process, where each dot is an “event” (in this case the occurrence of a cancer). Here it is of interest to determine whether the cases are more aggregated, or clustered, than expected and whether the clustering relates to the locations of any possible pollution sources. To assess this a background control disease map (which is not shown) is often used to represent the expected variation in the distribution of cases; this is often a function of the relative population density. Fig. 1.2 is an example of a dataset that consists of observations of an

underlying spatial process at a number of locations. The usual aim of an analysis of this type of data is to determine the value of the spatial process at all the locations in the domain of interest, assuming that each measurement is only observed in the presence of a random error component. However, we may also be interested in determining areas where the process is above some predeﬁned limit or even, in some sense, above average.