Читать книгу Spatial Analysis - Kanti V. Mardia - Страница 12

Figure 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 , 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.