The intrinsic random functions and their applications

Abstract

The intrinsic random functions (IRF) are a particular case of the Guelfand generalized processes with stationary increments. They constitute a much wider class than the stationary RF, and are used in practical applications for representing non-stationary phenomena. The most important topics are: existence of a generalized covariance (GC) for which statistical inference is possible from a unique realization; theory of the best linear intrinsic estimator (BLIE) used for contouring and estimating problems; the turning bands method for simulating IRF; and the models with polynomial GC, for which statistical inference may be performed by automatic procedures.

Spatial autocorrelation and sampling design in plant ecology

“…In the present study we produced interpolated maps by kriging, which is a geostatistical method developed by mining engineers (Matheron 1973;David 1977;Journel & Huijbregts 1978). Kriging is a method of interpolating that makes use of the spatial autocorrelation structure of the variable.…”

Spatial pattern and ecological analysis

“…Stratified (or zonal) anisotropy (different sills, same range) refers to the fact that the sills of the variograms may not be the same in different directions. In the presence of one or the other type of anisotropy, or both, there are three solutions to obtain acceptable interpolated maps by kriging: one can compute compromise variogram parameters, using the formulas in David (1977) or in Journel & Huijbregts (1978); secondly, one can use a kriging program that makes use of the parameters of variograms computed separately in different directions of the physical space (2 or 3, depending on the problem); or finally, one can use 'generalized intrinsic random functions of order k' (Matheron 1973) that allow for linear or quadratic trends in the data.…”

“…Kriging, developed by mining engineers and named after Krige (1966) to estimate mineral resources, usually produces a more detailed map than ordinary interpolation. Contrary to trend surface analysis, kriging uses a local estimator that takes into account only data points located in the vicinity of the point to be estimated, as well as the autocorrelation structure of the phenomenon; this information can be provided either by the variogram (see above), or by generalized intrinsic random functions of order k (Matheron 1973) that allow to make valid interpolation in the case of non-stationary variables (Journel & Huijbregts 1978). The variogram is used as follows during kriging: the kriging interpolation method estimates a point by considering all the other data points located in the observation cone of the variogram (given by the direction and window aperture angles), and weighs them using the values read on the adjusted theoretic variogram at the appropriate distances; furthermore, kriging splits this weight among neighbouring points, so that the result does not depend upon the local density of points.…”

A kernel approximation to the kriging predictor of a spatial process

“…We will only discuss here the specific intrinsic random function that yields the order 2 thin plate smoothing spline as the optimal predictor; more complete expositions of intrinsic random function theory are given by Matheron (1973) and Delfiner (1976). For the model in which we are interested, the mean of z(x) is taken to be linear in x --(s, t); that is,…”

“…Universal kriging is the geostatistical term for best linear unbiased prediction under a class of nonstationary spatial processes known as intrinsic random functions (Matheron (1973)). This paper develops a kernel approximation for the universal kriging predictor under a particular intrinsic random function model in two dimensions that is appropriate as the number of observations near the point being predicted increases.…”