Kanti V. Mardia - Spatial Analysis

Library of Congress Cataloging‐in‐Publication Data applied for

ISBN 9780471632054 (hardback); ISBN 9781118763568 (adobe pdf);

ISBN 9781118763575 (epub); 9781118763551 (obook)

Cover Design: Wiley

Cover Images: Courtesy of John Kent and Kanti Mardia

To my wife Sue for all her patience during the many years it has taken to complete the book

(John Kent)

To my son Hemant and daughter‐in‐law Preeti

— with Jainness

(Kanti Mardia)

“Whatever there is in all the three worlds, which are possessed of moving and non‐moving beings, cannot exist as apart from the ‘Ganita’ (mathematics/statistics).”

Acharya Mahavira (Jain monk, AD 850)

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

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

Figure 1.3Panels (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 1.4Bauxite 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 1.5Landsat data ( pixels): image plot.

Figure 1.6Synthetic Landsat data: image plot.

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

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

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

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

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

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

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

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

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

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

Figure 3.1Examples of radial semivariograms: the power schemes for and the exponential scheme . All the semivariograms have been scaled to take the same value for .

Figure 3.2A linear semivariogram with a nugget effect: .

Figure 4.1Panels (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 4.2Three 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 5.1Bauxite data: Bubble plot and directional semivariograms.

Figure 5.2Elevation data: Bubble plot and directional semivariograms.

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

Figure 5.4Bauxite data: sample isotropic semivariogram values and fitted Matérn semivariograms with a nugget effect, for (solid), (dashed), and (dotted).

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

Figure 5.6Unilateral 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.