Brown‒Resnick processes are max-stable processes that are associated to Gaussian processes. Their simulation is often based on the corresponding spectral representation which is not unique. We study to what extent simulation accuracy and efficiency can be improved by minimizing the maximal variance of the underlying Gaussian process. Such a minimization is a difficult mathematical problem that also depends on the geometry of the simulation domain. We extend Matheron's (1974) seminal contribution in two directions: (i) making his description of a minimal maximal variance explicit for convex variograms on symmetric domains, and (ii) proving that the same strategy also reduces the maximal variance for a huge class of nonconvex variograms representable through a Bernstein function. A simulation study confirms that our noncostly modification can lead to substantial improvements among Gaussian representations. We also compare it with three other established algorithms.

The nonuniform spatial distribution of weeds complicates sampling, modeling, and management of weed populations. Principles of a rational approach to analysis of weed spatial distribution, combining classical and spatial statistics, are presented using data for cumulative emergence of common lambsquarters in no-tillage soybean fields in 1990 and 1993. Classical statistics, dispersion indices, mean/variance relationships, and frequency histograms confirmed that raw and loge-transformed data were not normally distributed, that populations were aggregated, and that large-scale trends in population means violated assumptions of spatial statistics. Detrending was accomplished by median polishing loge-transformed data and confirmed by evaluation of standardized residuals and frequency histograms. Detrended residuals were used to construct omni-directional and uni-directional semivariograms to describe the spatial structure of the populations. A spherical model fit to the data was verified by cross validation. Semivariograms showed that common lambsquarters density was spatially autocorrelated at distances to 16 m, with more than 30% of the variance in density due to distance between field locations. Comparisons of kriged estimates and their standard deviations with and without detrending indicated that estimates using detrended data were more appropriate and more precise. Kriged estimates of common lambsquarters density were used to draw contour maps of the populations.

The aim of this paper is to investigate local spatial dependency with regard to Salmonella seropositivity in data from the Danish swine salmonellosis control programme and its application in informing surveillance strategies. We applied inhomogeneous and observed-difference K-function estimation, and geo-statistical modelling to data from the Danish swine salmonellosis control programme. Slaughter-pig farm density showed large variation at both the country-wide and local level in Denmark (median 0·23, range 0·02–0·47 farms/km2). The spatial distribution of pig farms followed a random inhomogeneous Poisson process but was not aggregated. We found evidence for aggregation of Salmonella case farms over that of all farms at distances of up to 6 km and semivariogram analyses of Salmonella seropositivity revealed spatial dependency between pairs of farms up to 4 km apart. The strength of the spatial dependency was positively associated with slaughter-pig farm density. We proposed sampling more intensively those farms within a 4 km radius of farms that were identified with a high Salmonella status, and reduced sampling of farms that are within this radius of ‘Salmonella-free’ farms. Our approach has the potential to optimize sampling strategies while maintaining consumer confidence in food safety and also has potential to be used for other zoonotic disease surveillance systems.

Variograms and covariance functions are the fundamental tools for modeling dependent data observed over time, space, or space-time. This paper aims at constructing nonseparable spatio-temporal variograms and covariance models. Special attention is paid to an intrinsically stationary spatio-temporal random field whose covariance function is of Schoenberg-Lévy type. The correlation structure is studied for its increment process and for its partial derivative with respect to the time lag, as well as for the superposition over time of a stationary spatio-temporal random field. As another approach, we investigate the permissibility of the linear combination of certain separable spatio-temporal covariance functions to be a valid covariance, and obtain a subclass of stationary spatio-temporal models isotropic in space.

This paper studies a class of stationary covariance models, in both the discrete- and the continuous-time domains, which possess a simple functional form γ(τ + τ0)+γ(τ − τ0)− 2γ(τ), where τ 0 is a fixed lag andγ(τ) is an intrinsically stationary variogram, and include the fractional Gaussian noise of Kolmogorov (1940) and a stochastic volatility model of Barndorff-Nielsen and Shephard (2001), (2002) as special cases. Properties of the class, and interesting special cases with long memory, are studied. We also characterize the covariance function via the variogram.

Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.

For modelling non-stationary spatial random fields Z = {Z(x) : x∊ℝn, n≥2} a recent method has been proposed to deform bijectively the index space so that the spatial dispersion D(x,y) = var[Z(x)-Z(y)], (x,y)∊ℝnxℝn, depends only on the Euclidean distance in the deformed space through an isotropic variogram γ. We prove uniqueness of this model in two different cases: (i) γ is strictly increasing; (ii) γ(u) is differentiable for u > 0.

In time series analysis, it is well-known that the differencing operator ∇d may transform a non-stationary series, {Z(t)} say, to a stationary one, {W(t)} = ∇dZ(t)}; and there are many procedures for analysing and modelling {Z(t)} which exploit this transformation. Rather differently, Matheron (1973) introduced a set of measures on Rn that transform an appropriate non-stationary spatial process to stationarity, and Cressie (1988) then suggested that specialized low-order analogues of these measures, called increment-vectors, be used in time series analysis. This paper develops a general theory of increment-vectors which provides a more powerful transformation tool than mere simple differencing. The methodology gives a handle on the second-moment structure and divergence behaviour of homogeneously non-stationary series which leads to many important applications such as determining the correct degree of differencing, forecasting and interpolation.