Abstract: In this paper, we propose a new class of non-Gaussian random fields named two-piece random fields. The proposed class allows to generate random fields that have flexible marginal distributions, possibly skewed and/or heavy-tailed and, as a consequence, has a wide range of applications. We study the second-order properties of this class and provide analytical expressions for the bivariate distribution and the associated correlation functions. We exemplify our general construction by studying two examples: two-piece Gaussian and two-piece Tukey-h random fields. An interesting feature of the proposed class is that it offers a specific type of dependence that can be useful when modeling data displaying spatial outliers, a property that has been somewhat ignored from modeling viewpoint in the literature for spatial point referenced data. Since the likelihood function involves analytically intractable integrals, we adopt the weighted pairwise likelihood as a method of estimation. The effectiveness of our methodology is illustrated with simulation experiments as well as with the analysis of a georeferenced dataset of mean temperatures in Middle East.

Abstract: This article gives a narrative overview of what constitutes climatological data and their typical features, with a focus on aspects relevant to statistical modeling. We restrict the discussion to univariate spatial fields and focus on maximum likelihood estimation. To address the problem of enormous datasets, we study three common approximation schemes: tapering, direct misspecification, and composite likelihood for Gaussian and nonGaussian distributions. We focus particularly on the so-called 'sinh-arcsinh distribution', obtained through a specific transformation of the Gaussian distribution. Because it has flexible marginal distributions – possibly skewed and/or heavy-tailed – it has a wide range of applications. One appealing property of the transformation involved is the existence of an explicit inverse transformation that makes likelihood-based methods straightforward. We describe a simulation study illustrating the effects of the different approximation schemes. To the best of our knowledge, a direct comparison of tapering, direct misspecification, and composite likelihood has never been made previously, and we show that direct misspecification is inferior. In some metrics, composite likelihood has a minor advantage over tapering. We use the estimation approaches to model a high-resolution global climate change field. All simulation code is available as a Docker container and is thus fully reproducible. Additionally, the present article describes where and how to get various climate datasets. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

Abstract: Random fields are useful mathematical tools for representing natural phenomena with complex dependence structures in space and/or time. In particular, the Gaussian random field is commonly used due to its attractive properties and mathematical tractability. However, this assumption seems to be restrictive when dealing with counting data. To deal with this situation, we propose a random field with a Poisson marginal distribution considering a sequence of independent copies of a random field with an exponential marginal distribution as “inter-arrival times ” in the counting renewal processes framework. Our proposal can be viewed as a spatial generalization of the Poisson counting process. Unlike the classical hierarchical Poisson Log-Gaussian model, our proposal generates a (non)-stationary random field that is mean square continuous and with Poisson marginal distributions. For the proposed Poisson spatial random field, analytic expressions for the covariance function and the bivariate distribution are provided. In an extensive simulation study, we investigate the weighted pairwise likelihood as a method for estimating the Poisson random field parameters. Finally, the effectiveness of our methodology is illustrated by an analysis of reindeer pellet-group survey data, where a zero-inflated version of the proposed model is compared with zero-inflated Poisson Log-Gaussian and Poisson Gaussian copula models. for this article, including technical proofs and R code for reproducing the work, are available as an online supplement.