Multivariate type G Mat\'ern stochastic partial differential equation random fields

For many applications with multivariate data, random field models capturing departures from Gaussianity within realisations are appropriate. For this reason, we formulate a new class of multivariate non-Gaussian models based on systems of stochastic partial differential equations with additive type G noise whose marginal covariance functions are of Mat\'ern type. We consider four increasingly flexible constructions of the noise, where the first two are similar to existing copula-based models. In contrast to these, the latter two constructions can model non-Gaussian spatial data without replicates. Computationally efficient methods for likelihood-based parameter estimation and probabilistic prediction are proposed, and the flexibility of the suggested models is illustrated by numerical examples and two statistical applications

Asymptotic Multivariate Kriging Using Estimated Parameters with Bayesian Prediction Methods for Non-linear Predictands

The motivation for this dissertation is the need that often arises in spatial settings to perform a data transformation to achieve a stationary process and/or variance stabilization. The transformation may be a non-linear transformation, and the desired predictand may be multivariate in that it is necessary to interpolate predictions at multiple sites. We assume the underlying spatial model is a Gaussian random field with a parametrically specified covariance structure, but that the predictions of interest are for multivariate nonlinear functions of the Gaussian field. This induces new complications in the spatial interpolation known as kriging. For instance, it is no longer possible to derive the predictive distribution function in closed form. The underlying process of a spatial model is a stochastic process, and spatial prediction techniques are founded on the assumption that the realizations come from a Gaussian process with a known covariance structure, and known, specified parameters. A difficulty that arises with traditional kriging methods is the fact that the standard formula for the mean squared prediction error does not take into account the estimation of the covariance parameters. This generally leads to underestimated prediction errors, even if the model is correct. Smith and Zhu (2004) establish a second-order expansion for predictive distributions in Gaussian processes with estimated covariances. Here, we establish a similar expansion for multivariate kriging with non-linear predictands. Bayesian methods provide a possible resolution to errors encountered through employing frequentist estimation techniques for obtaining spatial parameters. Bayesian analysis seeks to utilize prior and nonexperimental sources of information. Bayesian methods evaluate procedures for a repeated sampling experiment of parameters drawn from the posterior distribution given a set of observed data. An important property of Bayesian methods is the ability to deal with the uncertainty in a particular model. A Bayesian paradigm enables a more realistic assessment of the variability inherent in estimating parameters of interest. Here we explore a Laplace approximation to Bayesian techniques that provides an alternative to common iterative Bayesian methods, such as Markov Chain Monte Carlo. The theoretical Bayesian coverage probability bias for prediction intervals is computed and compared with the plug-in method using the restricted maximum likelihood estimates of the covariance parameters. The form of the Bayesian predictive distribution can be expressed as partial derivatives of the restricted log-likelihood. This leads to possible analytical evaluation, and a bootstrap method is explored to obtain predictions for general, non-linear predictands. The main results are asymptotic formulae for a general, non-linear predictand for the expected length of a Bayesian prediction interval, which has possible applications in network design, and for the coverage probability bias, which can lead to the development of a matching prior. The matching prior may be difficult to compute in practice. As an alternative, an asymptotic estimator is developed

A multivariate semiparametric Bayesian spatial modeling framework for hurricane surface wind fields

Storm surge, the onshore rush of sea water caused by the high winds and low pressure associated with a hurricane, can compound the effects of inland flooding caused by rainfall, leading to loss of property and loss of life for residents of coastal areas. Numerical ocean models are essential for creating storm surge forecasts for coastal areas. These models are driven primarily by the surface wind forcings. Currently, the gridded wind fields used by ocean models are specified by deterministic formulas that are based on the central pressure and location of the storm center. While these equations incorporate important physical knowledge about the structure of hurricane surface wind fields, they cannot always capture the asymmetric and dynamic nature of a hurricane. A new Bayesian multivariate spatial statistical modeling framework is introduced combining data with physical knowledge about the wind fields to improve the estimation of the wind vectors. Many spatial models assume the data follow a Gaussian distribution. However, this may be overly-restrictive for wind fields data which often display erratic behavior, such as sudden changes in time or space. In this paper we develop a semiparametric multivariate spatial model for these data. Our model builds on the stick-breaking prior, which is frequently used in Bayesian modeling to capture uncertainty in the parametric form of an outcome. The stick-breaking prior is extended to the spatial setting by assigning each location a different, unknown distribution, and smoothing the distributions in space with a series of kernel functions. This semiparametric spatial model is shown to improve prediction compared to usual Bayesian Kriging methods for the wind field of Hurricane Ivan.Comment: Published at http://dx.doi.org/10.1214/07-AOAS108 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Non-Gaussian Geostatistical Modeling using (skew) t Processes

We propose a new model for regression and dependence analysis when addressing spatial data with possibly heavy tails and an asymmetric marginal distribution. We first propose a stationary process with $t$ marginals obtained through scale mixing of a Gaussian process with an inverse square root process with Gamma marginals. We then generalize this construction by considering a skew-Gaussian process, thus obtaining a process with skew-t marginal distributions. For the proposed (skew) $t$ process we study the second-order and geometrical properties and in the $t$ case, we provide analytic expressions for the bivariate distribution. In an extensive simulation study, we investigate the use of the weighted pairwise likelihood as a method of estimation for the $t$ process. Moreover we compare the performance of the optimal linear predictor of the $t$ process versus the optimal Gaussian predictor. Finally, the effectiveness of our methodology is illustrated by analyzing a georeferenced dataset on maximum temperatures in Australi