Applications of MCMC in Fisheries Science

Abstract

Spatial data contain information about both the attribute of interest and its location. Examples canbe found ina largenumberofdisciplines, includingecology, geology, epidemiology, geography, image analysis, meteorology, forestry, and geosciences. The location may be a set of coordinates, such as the latitude and longitude associatedwith an observed pollutant level, or it may be a small region such as a county associated with an observed disease rate. FollowingCressie (1993), we categorize spatial data into three distinct types: (i) geostatistical or point-level data, as in the pollutant levels observed at several monitors across a region; (ii) lattice or “areal” (regionally aggregated) data, for example, US disease rates provided by county; and (iii) point process data, where the locations themselves are random variables and of interest, as in the set of locations where a rare animal species was observed. Point processes where random variables associatedwith the random locations are also of interest are referred to as marked point processes. In this chapter, we only consider spatial data that fall into categories (i) and (ii). We will use the following notation throughout. Denote a real-valued spatial process in ddimensions by {Z(s) : s ∈ D ⊂ Rd}, where s is the location of the process Z(s) and s varies over the index set D, resulting in a multivariate random process. For point-level data D is a continuous, fixed set, while for lattice or areal data D is discrete and fixed. For spatial point processes, D is stochastic and usually continuous. The distinctions among the above categories may not always be apparent in any given context, so determining a category is part of the modeling process. The purpose of this chapter is to discuss the use of Gaussian randomfields formodeling avariety of point-level and areal spatial data, and to point out the flexibility inmodel choices afforded byMarkov chain Monte Carlo (MCMC) algorithms. Details on theory, algorithms and advanced spatial modeling can be found in Cressie (1993), Stein (1999), Banerjee et al. (2004), and other standard texts. The reader is referred to the excellent monograph by Møller and Waagepetersen (2004) for details on modeling and computation for spatial point processes.