Spatial Analysis. Kanti V. Mardia

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1.3 Bauxite data: percentage ore grade for bauxite at locations.

       Table 1.4 Gravimetric data: local gravity measurements in Quebec, Canada.

       Table 1.5 Semivariograms in each direction for the gravimetric data.

       Table 1.6 Soil data: surface pH in on an

grid.

       Table 1.7 Mercer–Hall wheat yield (in lbs.) for 20 (rows) 25 (columns) agricultural plots (Mercer and Hall, 1911), where top–bottom corresponds to West–East, and left–right corresponds to North–South.

       Table 1.8 Aggregated Mercer–Hall wheat data for plots aggregated into blocks, giving the array layout for the blocks, the block dimensions, the block size

(number of plots in each block), the number of blocks

, and sample variance

.

       Table 2.1 Some radial covariance 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

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