Abstract
In mapping studies, extreme risk areas may arise in proximity to one another in a smooth spatial surface. They may also arise as isolated โhotspotsโ or โlowspotsโ, which are quite distinct from those of neighbouring sites. In this paper, we develop spatial methods which encompass both types of extreme risks. The former is modelled by a spatially smooth surface using a conditional autoregressive model; the latter is addressed with the addition of a discrete clustering component, which offers the flexibility of accommodating extreme isolated risks and is not limited by spatial smoothness. The autoregressive component incorporates the spatially correlated risk as a baseline surface, acknowledging that environmental activity, often spatially correlated, influences risk responses. The discrete component identifies hotspots/lowspots of activity beyond the spatially correlated baseline risk surface. Both types of extreme risk are important, but isolated extremes may provide insight into areas with potential of being a centre for future spatially correlated extreme risks. Hence these may be particularly important in terms of surveillance. A Bayesian approach to inference is employed and graphical techniques for isolating extremes are illustrated. Model assessment is conducted via crossโvalidation posterior predictive checks. Three examples demonstrate the utility of the methods and case studies show the procedures to be useful for pinpointing extreme risks. In addition, sensitivity to priors is investigated. Copyright ยฉ 2007 John Wiley & Sons, Ltd.