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Bevilacqua, M., Camano-Carrillo, C., & Porcu, E. (2022). Unifying compactly supported and Matern covariance functions in spatial statistics. J. Multivar. Anal., 189, 104949.
Abstract: The Matern family of covariance functions has played a central role in spatial statistics for decades, being a flexible parametric class with one parameter determining the smoothness of the paths of the underlying spatial field. This paper proposes a family of spatial covariance functions, which stems from a reparameterization of the generalized Wendland family. As for the Matern case, the proposed family allows for a continuous parameterization of the smoothness of the underlying Gaussian random field, being additionally compactly supported.
More importantly, we show that the proposed covariance family generalizes the Matern model which is attained as a special limit case. This implies that the (reparametrized) Generalized Wendland model is more flexible than the Matern model with an extra-parameter that allows for switching from compactly to globally supported covariance functions.
Our numerical experiments elucidate the speed of convergence of the proposed model to the Matern model. We also inspect the asymptotic distribution of the maximum likelihood method when estimating the parameters of the proposed covariance models under both increasing and fixed domain asymptotics. The effectiveness of our proposal is illustrated by analyzing a georeferenced dataset of mean temperatures over a region of French, and performing a re-analysis of a large spatial point referenced dataset of yearly total precipitation anomalies.
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Morales-Navarrete, D., Bevilacqua, M., Caamano-Carrillo, C., & Castro, L. M. (2023). Modeling Point Referenced Spatial Count Data: A Poisson Process Approach. J. Am. Stat. Assoc., Early Access.
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.
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Morales-Onate, V., Crudu, F., & Bevilacqua, M. (2021). Blockwise Euclidean likelihood for spatio-temporal covariance models. Econ. Stat., 20, 176–201.
Abstract: A spatio-temporal blockwise Euclidean likelihood method for the estimation of covariance models when dealing with large spatio-temporal Gaussian data is proposed. The method uses moment conditions coming from the score of the pairwise composite likelihood. The blockwise approach guarantees considerable computational improvements over the standard pairwise composite likelihood method. In order to further speed up computation, a general purpose graphics processing unit implementation using OpenCL is implemented. The asymptotic properties of the proposed estimator are derived and the finite sample properties of this methodology by means of a simulation study highlighting the computational gains of the OpenCL graphics processing unit implementation. Finally, there is an application of the estimation method to a wind component data set. (C) 2021 EcoSta Econometrics and Statistics. Published by Elsevier B.V. All rights reserved.
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