Spatial Analysis. Kanti V. Mardia

, as a function of the index

, both without a nugget effect (dashed line) and with a nugget effect (solid line). In addition, the log‐likelihood for each

, with the parameters estimated by MINQUE (described In Section 5.13), is shown (dotted line).

       Figure 6.1 Mercer–Hall data: bubble plot. See Example 6.1 for an interpretation.

       Figure 6.2 A plot of the sample and two fitted covariance functions (“biased‐mom‐ML”) for a CAR model fitted to the leftmost 13 columns of the Mercer–Hall data (Example 6.1). The data have been summarized by the biased sample covariance function. The four panels show the covariance function in the four principal directions with the sample covariances (open circles) together with the fitted covariances using moment estimation (solid lines) and maximum likelihood estimation (dashed lines).

       Figure 6.3 A plot of the sample and two fitted covariance functions (“fold‐mom‐ML”) for a CAR model fitted to the leftmost 13 columns of the Mercer–Hall data (Example 6.2). The data have been summarized by the folded sample covariance function. The four panels show the covariance function in the four principal directions with the sample covariances (open circles) together with the fitted covariances using moment estimation (solid lines) and maximum likelihood estimation (dashed lines).

       Figure 6.4 Relative efficiency of the composite likelihood estimator in AR(1) model relative to the ML estimator.

       Figure 7.1 Kriging predictor for

data points assumed to come from a stationary random field with a squared exponential covariance function 7.55, without a nugget effect, with mean 0. Panels (a)–(c) show the kriging predictor for three choices of the range parameter,

, respectively. Each panel shows the true unknown shifted sine function (solid), together with the fitted kriging curve (dashed), plus/minus twice the kriging standard errors (dotted).

Figure 7.2 Kriging predictor for

data points assumed to come from a stationary random field with a squared exponential covariance function 7.55, plus a nugget effect, with mean 0. The size of the relative nugget effect in Panels (a)–(c) is given by

, respectively. Each panel shows the true unknown shifted sine function (solid), together with the fitted kriging curve (dashed), plus/minus twice the kriging standard errors (dotted).

       Figure 7.3 Panel (a) shows the interpolated kriging surface for the elevation data, as a contour map. Panel (b) shows a contour map of the corresponding kriging standard errors. This figure is also included in Figure 1.3.

       Figure 7.4 Panel (a) shows a contour plot for the kriged surface fitted to the bauxite data assuming a constant mean and an exponential covariance function for the error terms. Panel (b) shows the same plot assuming a quadratic trend and independent errors. Panels (c) and (d) show the kriging standard errors for the models in (a) and (b), respectively.

       Figure 7.5 Kriging predictor and kriging standard errors for

data points assumed to come from an intrinsic random field,

, no nugget effect. The intrinsic drift is constant. Panel (a): no extrinsic drift; Panel (b): linear extrinsic drift. Each panel shows the fitted kriging curve (solid), plus/minus twice the kriging standard errors (dashed).

       Figure 7.6 Panel (a) shows the interpolated kriging surface for the gravimetric data, as a contour map. Panel (b) shows a contour map of the corresponding kriging standard errors.

       Figure 7.7 Kriging predictors for Example 7.6. For Panel (a), the kriging predictor is based on value constraints at sites 1,2,3. For Panel (b), the kriging predictor is additionally based on derivative constraints at the same sites.

       Figure 7.8 Deformation of a square (a) into a kite (b) using a thin‐plate spline. The effect of the deformation on

can also be visualized: it maps a grid of parallel lines to a bi‐orthogonal grid.

       Figure B.1 Creators of Kriging: Danie Krige and Georges Matheron.

       Figure B.2 Letter from Matheron to Mardia, dated 1990.

       Figure B.3 Translation of the letter from Matheron to Mardia, dated 1990.

List of Tables

       Table 1.1 Illustrative data

, on a

regular grid, represented in various ways,

.

       Table 1.2 Elevation data: elevation

in feet above the sea level, where

,

.

       Table 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