Modeling Non-Stationary Processes Through Dimension Expansion

In this paper, we propose a novel approach to modeling nonstationary spatial fields. The proposed method works by expanding the geographic plane over which these processes evolve into higher dimensional spaces, transforming and clarifying complex patterns in the physical plane. By combining aspects of multi-dimensional scaling, group lasso, and latent variables models, a dimensionally sparse projection is found in which the originally nonstationary field exhibits stationarity. Following a comparison with existing methods in a simulated environment, dimension expansion is studied on a classic test-bed data set historically used to study nonstationary models. Following this, we explore the use of dimension expansion in modeling air pollution in the United Kingdom, a process known to be strongly influenced by rural/urban effects, amongst others, which gives rise to a nonstationary field

On potentially negative space time covariances obtained as sum of products of marginal ones

Most of the literature on spatio-temporal covariance models proposes structures that are positive in the whole domain. However, problems of physical, biological or medical nature need covariance models allowing for negative values or oscillations from positive to negative values. In this paper we propose an easy-to-implement and interpretable class of models that admits this type of covariances. We show particular analytical examples that may be of interest in the biometrical contex

Some covariance models based on normal scale mixtures

Modelling spatio-temporal processes has become an important issue in current research. Since Gaussian processes are essentially determined by their second order structure, broad classes of covariance functions are of interest. Here, a new class is described that merges and generalizes various models presented in the literature, in particular models in Gneiting (J. Amer. Statist. Assoc. 97 (2002) 590--600) and Stein (Nonstationary spatial covariance functions (2005) Univ. Chicago). Furthermore, new models and a multivariate extension are introduced.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ226 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Extrapolation of Stationary Random Fields

We introduce basic statistical methods for the extrapolation of stationary random fields. For square integrable fields, we set out basics of the kriging extrapolation techniques. For (non--Gaussian) stable fields, which are known to be heavy tailed, we describe further extrapolation methods and discuss their properties. Two of them can be seen as direct generalizations of kriging.Comment: 52 pages, 25 figures. This is a review article, though Section 4 of the article contains new results on the weak consistency of the extrapolation methods as well as new extrapolation methods for $\alpha$-stable fields with $0<\alpha\leq 1

Application-driven Sequential Designs for Simulation Experiments: Kriging Metamodeling

This paper proposes a novel method to select an experimental design for interpolation in simulation.Though the paper focuses on Kriging in deterministic simulation, the method also applies to other types of metamodels (besides Kriging), and to stochastic simulation.The paper focuses on simulations that require much computer time, so it is important to select a design with a small number of observations.The proposed method is therefore sequential.The novelty of the method is that it accounts for the specific input/output function of the particular simulation model at hand; i.e., the method is application-driven or customized.This customization is achieved through cross-validation and jackknifing.The new method is tested through two academic applications, which demonstrate that the method indeed gives better results than a design with a prefixed sample size.experimental design;simulation;interpolation;sampling;sensitivity analysis;metamodels

Model-based geostatistics: some issues in modelling and model diagnostics

Spatial modelling is examined in a model-based geostatistical context using the Gaussian linear mixed model in a likelihood framework. Complex spatial models developed provide practitioners with a practical and best-practice guide for spatial analysis. Adequate modelling theory and matrix algebra are provided to ground the methods demonstrated. A multivariate model over two time points and three-dimensional space is developed which is novel to the field of soil science. Soil organic carbon measurements at three soil depths and two time points from a cropping field with four soil classes are used. The spatial process is assessed for second-order stationarity and anisotropic correlation. Univariate spatial modelling is used to inform bivariate spatial modelling of pre- and post-harvest soil organic carbon at each soil depth. Bivariate modelling is extended to the multivariate level, where both time points and the three soil depths are incorporated in a single model to pool maximum information. A common correlation structure is tested and is supported for the response variable at each of the six time-depth combinations. Separable correlation structures are used for computational efficiency. The difficulty of estimating nugget effects suggests a sub-optimal sampling design. Preferred fitted models are all isotropic. Equations for predictions and the variance of prediction errors are extended from well-known results and maps of predicted values and variance of prediction errors are produced and show close correspondence with observed values. Finally, univariate models for spatially referenced seed counts from small sampling plots are examined within a Gaussian framework using Box-Cox transformations. The discrete nature of the data, small sample size and computational problems hamper model fitting. Anisotropy is examined using a variogram envelope diagnostic technique. ASReml-R software is shown to be a powerful analytical tool for spatial processes