Second-order spatial models and geostatistics

Abstract

Suppose S ⊆ R d is a spatial set. A random field X on S taking values in a state space E means a collection X = {X s , s ∈ S} of random variables (r.v.) indexed by S taking values in E. This chapter is devoted to the study of second-order random fields, i.e., real-valued random fields where each X s has finite variance. We also study the broader class of intrinsic random fields, that is, random fields with increments of finite variance. We consider two approaches. In the geostatistics approach, S is a continuous subset of R d and we model X in a " second-order " way with its covariance function or its variogram. For example, for d = 2, s = (x, y) ∈ S is characterized by fixed geographic coordinates and if d = 3, we add altitude (or depth) z. Spatio-temporal evolution in space can also be modeled at space-time " sites " (s,t) ∈ R 3 × R + , where s represents space and t time. Initially developed for predicting mineral reserves in an exploration zone S ⊆ R 3 , geostatis-tics is today used in a variety of domains (cf. Chilès and Delfiner (43); Diggle and Ribeiro (63)). These include, among others, earth science and mining exploration (134; 152), epidemiology, agronomy and design of numerical experiments (193). A central goal of geostatistics is to predict X by kriging over all of S using only a finite number of observations. The second approach involves autoregressive (AR) models, used when S is a discrete network of sites (we will also use the word " lattice "). S may have a regu-lar form, for example S ⊂ Z d (images, satellite data, radiography; (42), (224)) or it may not (econometrics, epidemiology; (45), (7), (105)). Here, the spatial correlation structure is induced by the AR model chosen. Such models are well adapted to sit-uations where measurements have been aggregated over spatial zones: for example, in econometrics this might be the percentages of categories of a certain variable in an administrative unit, in epidemiology, the number of cases of an illness per district s and in agronomy, the total production in each parcel of land s. C. Gaetan, X. Guyon, Spatial Statistics and Modeling, Springer Series in Statistics, 1 DOI 10.1007/978-0-387-92257-7_1, c Springer Science+Business Media, LLC 2010 2 1 Second-order spatial models and geostatistics 1.1 Some background in stochastic processes Let (Ω , F , P) be a probability space, S a set of sites and (E, E) a measurable state space. Definition 1.1. Stochastic process A stochastic process (or process or random field) taking values in E is a family X = {X s , s ∈ S} of random variables defined on (Ω , F , P) and taking values in (E, E). (E, E) is called the state space of the process and S the (spatial) set of sites at which the process is defined. For any integer n ≥ 1 and n-tuple (s 1 , s 2 ,...,s n) ∈ S n , the distribu-tion of (X s 1 , X s 2 ,...,X s n) is the image of P under the mapping ω −→ (X s 1 (ω), X s 2 (ω),...,X s n (ω)): that is, for A i ∈ E , i = 1,...,n,