ABSTRACT
Two important features of a random field are its mean and covariance structure. The former represents the large-scale changes of Z(s), the latter the variability due to small- and micro-scale stochastic sources. In Chapter 2, we gave several different representations of the stochastic dependence (second-order structure) between spatial observations. Direct and indirect specifications based on model representations (§2.4.1), representations based on convolutions (§2.4.2) and spectral decompositions (§2.5). In the case of a spatial point process, the second-order structure is represented by the second-order intensity, and by the K-function in the isotropic case (Chapter 3). If a spatial random field has model representation Z(s) = μ (s)+e(s), where e(s) ∼ (0, Σ), the spatial dependence structure is expressed through the variance-covariance matrix Σ. The semivariogram and covariance function of a spatial process with fixed, continuous domain were introduced in §2.2, since these parameters require that certain stationarity conditions be met. The variance-covariance matrix of e(s) is not bound by any stationarity requirements, it simply captures the variances and covariances of the process. In addition, the model representation does not confine e(s) to geostatistical applications, the domain may be a lattice, for example. In practical applications, Σ is unknown and must be estimated from the data. Unstructured variance-covariance matrices that are common in multivariate statistical methods are uncommon in spatial statistics. There is typically structure to the spatial covariances, for example, they may decrease with increasing lag. And without true replications, there is no hope to estimate the entries in an unspecified variance-covariance matrix. Parametric forms are thus assumed so that Σ ≡ Σ( θ ) and θ is estimated from the data. The techniques employed to parameterize Σ vary with circumstances. In a lattice model Σ is defined indirectly by the choice of a neighborhood matrix and an autoregressive structure. For geostatistical data, Σ is constructed directly from a model for the continuous spatial autocorrelation among observations. The importance of choosing the correct model for Σ( θ ) also depends on the application. Consider a spatial model Z ( s ) = X β + e ( s ) , e ( s ) ~ ( 0 , Σ ( θ ) ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275086/f9447f91-7686-4ff8-bc9a-2190068cf9cc/content/unequ4_1.tif"/>
