Spatial Analysis. Kanti V. Mardia

1.1 Fingerprint of R A Fisher, taken from Mardia's personal collection. A blowup of the marked rectangular section is given in Figure 1.9.

       Figure 1.2 Elevation data: (a) raw plot giving the elevation at each site and (b) bubble plot where larger elevations are indicated by bigger circles.

       Figure 1.3 Panels (a), (b), and (c) show interpolated plots for the elevation data, as a contour map, a perspective plot (viewed from the top of the region), and an image plot, respectively. Panel (d) shows a contour map of the corresponding standard errors.

       Figure 1.4 Bauxite data: (a) raw plot giving the ore grade at each site and (b) bubble plot where larger ore grades are indicated by bigger circles.

       Figure 1.5 Landsat data (

pixels): image plot.

       Figure 1.6 Synthetic Landsat data: image plot.

       Figure 1.7 Typical semivariogram, showing the range, nugget variance, and sill.

       Figure 1.8 Angle convention for polar coordinates. Angles are measured clockwise from vertical.

       Figure 1.9 Fingerprint section data (218 pixels wide by 356 pixels high): (a) image plot and (b) directional semivariograms.

       Figure 1.10 Elevation data: (a) directional semivariograms and (b) omnidirectional semivariogram.

       Figure 1.11 Bauxite data: (a) directional semivariograms and (b) omnidirectional semivariogram.

Figure 1.12 Directional semivariograms for (a) the Landsat data and (b) the synthetic Landsat data.

       Figure 1.13 Gravimetric data: (a) bubble plot and (b) directional semivariograms.

       Figure 1.14 Soil data: (a) bubble plot and (b) directional semivariograms.

       Figure 1.15 Mercer–Hall wheat data: log–log plot of variance vs. block size.

       Figure 2.1 Matérn covariance functions for varying index parameters. The range and scale parameters have been chosen so that the covariance functions match at lags

and

.

       Figure 3.1 Examples of radial semivariograms: the power schemes

for

and the exponential scheme

. All the semivariograms have been scaled to take the same value for

.

       Figure 3.2 A linear semivariogram with a nugget effect:

.

       Figure 4.1 Panels (a) and (b) illustrate the first‐order basic and full neighborhoods of the origin in the plane. Panel (c) illustrates the second‐order basic neighborhood.

       Figure 4.2 Three notions of “past” of the origin in

: (a) quadrant past (

), (b) lexicographic past (

), and (c) weak past (

). In each plot, ○ denotes the origin,

denotes a site in the past, and

denotes a site in the future.

       Figure 4.3 (a) First‐order basic neighborhood (nbhd) of the origin ○ in

dimensions. Neighbors of the origin are indicated by

. (b) Two types of clique in addition to singleton cliques: horizontal and vertical edges.

       Figure 4.4 (a) First‐order full neighborhood (nbhd) of the origin ○ in

dimensions. Neighbors of the origin are indicated by

. (b) Seven types of clique in addition to singleton cliques: horizontal and vertical edges, four shapes of triangle and a square.

       Figure 5.1 Bauxite data: Bubble plot and directional semivariograms.

       Figure 5.2 Elevation data: Bubble plot and directional semivariograms.

       Figure 5.3 Bauxite data: Profile log‐likelihoods together with 95% confidence intervals. Exponential model, no nugget effect.

       Figure 5.4 Bauxite data: sample isotropic semivariogram values and fitted Matérn semivariograms with a nugget effect, for

(solid),

(dashed), and

(dotted).

Figure 5.5 Elevation data: Profile log‐likelihoods together with 95% confidence intervals. Exponential model, no nugget effect.

       Figure 5.6 Unilateral lexicographic neighborhood of full size

for lattice data; current site marked by

; neighborhood sites in the lexicographic past marked by

. Other sites are marked by a dot.

       Figure 5.7 Profile log‐likelihoods for self‐similar models of intrinsic order