Spatial Analysis. Kanti V. Mardia

functions.

       Table 2.2 Special cases of the Matérn covariance function in (2.34) for half‐integer

with scale parameter

.

       Table 2.3 Some examples of stationary covariance functions

on the circle, together with the terms

in their Fourier series.

       Table 3.1 Self‐similar random fields with spectral density

: some particular cases

Table 5.1 Parameter estimates (and standard errors) for the bauxite data using the Matérn model with a constant mean and with various choices for the index

.

       Table 5.2 Parameter estimates (and standard errors) for the elevation data using the Matérn model with a constant mean and with various choices for the index

.

       Table 5.3 Parameter estimates (with standard errors in parentheses) for Vecchia's composite likelihood in Example 5.7 for the synthetic Landsat data, using different sizes of neighborhood

       Table 7.1 Notation used for kriging at the data sites

, and at the prediction and data sites

.

       Table 7.2 Various methods of determining the kriging predictor

, where

can be defined in terms of transfer matrices by

or in terms of

by

.

       Table 7.3 Comparison between the terminology and notation of this book for simple kriging and Rasmussen and Williams (2006, pp. 13–17) for simple Bayesian kriging. The posterior mean and covariance function take the same form in both formulations, given by (7.61) and (7.62).

       Table A.1 Types of domain.

       Table A.2 Domains for Fourier transforms and inverse Fourier transforms in various settings.

       Table A.3 Three types of boundary condition for a one‐dimensional set of data,

.

       Table A.4 Some examples of Toeplitz, circulant, and folded circulant matrices

, respectively, for

and

.

       Table A.5 First and second derivatives of

and

with respect to

and

.

Preface

Spatial statistics is concerned with data collected at various spatial locations or sites, typically in a Euclidean space

. The important cases in practice are

, corresponding to the data on the line, in the plane, or in 3‐space, respectively. A common property of spatial data is “spatial continuity,” which means that measurements at nearby locations will tend to be more similar than measurements at distant locations. Spatial continuity can be modeled statistically using a covariance function of a stochastic process for which observations at nearby sites are more highly correlated than at distant sites. A stochastic process in space is also known as a random field.

      One distinctive feature of spatial statistics, and related areas such as time series, is that there is typically just one realization of the stochastic process to analyze. Other branches of statistics often involve the analysis of independent replications of data.

The purpose of this book is to develop the statistical tools to analyze spatial data. The main emphasis in the book is on Gaussian processes. Here is a brief summary of the contents. A list of Notation and Terminology is given at the start for ease of reference. An introduction to the overall objectives of spatial analysis, together with some exploratory methods, is given in Chapter 1. Next is the specification of possible covariance functions (Chapter 2 for the stationary case and Chapter 3 for the intrinsic case). It is helpful to distinguish discretely indexed, or lattice, processes from continuously indexed processes. In particular, for lattice processes, it is possible to specify a covariance function through an autoregressive model (the SAR and CAR models of Chapter 4), with specialized estimation procedures (Chapter 6). Model fitting through maximum likelihood and related ideas for continuously indexed processes is covered in Chapter 5. An important use of spatial models is kriging, i.e. the prediction of the process at a collection of new sites, given the values of the process at a collection of training sites (Chapter 7), and in particular the links to machine learning are explained. Some additional topics, for which there was not space for in the book, are summarized in Chapter 8. The technical mathematical tools have been collected in Appendix A for ease of reference. Appendix B contains a short historical review of the spatial linear model.

      The development of statistical methodology for spatial data arose somewhat separately in several academic disciplines over the past century.

      1 Agricultural field trials. An area of land is divided into long, thin plots, and different crop is grown on each plot. Spatial correlation in the soil fertility can cause spatial correlation in the crop yields (Webster and Oliver, 2001).

      2 Geostatistics. In mining applications, the concentration of a mineral of interest will often show spatial continuity in a body of ore. Two giants in the field of spatial analysis came out of this field. Krige (1951) set out the methodology for spatial prediction (now known as kriging) and Matheron (1963) developed a comprehensive theory for stationary and intrinsic random fields; see Appendix