arXiv subjects

M. Bevilacqua

Publications and source records attributed to M. Bevilacqua.

6 recordsLinked to original sources

arXiv2022

Unifying Compactly Supported and Matern Covariance Functions in Spatial Statistics

The Mat{é}rn 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 new family of spatial covariance functions, which stems from a reparameterization of the generalized Wendland family. As for the Mat{é}rn case, the new class 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 Mat{é}rn model which is attained as a special limit case. The practical implication of our theoretical results questions the effective flexibility of the Mat{é}rn covariance from modeling and computational viewpoints. Our numerical experiments elucidate the speed of convergence of the proposed model to the Mat{é}rn model. We also inspect the level of sparseness of the associated (inverse) covariance matrix and the asymptotic distribution of the maximum likelihood estimator under increasing and fixed domain asymptotics. The effectiveness of our proposal is illustrated by analyzing a georeferenced dataset on maximum temperatures over the southeastern United States, and performing a re-analysis of a large spatial point referenced dataset of yearly total precipitation anomalies

math.ST↗

arXiv2020

On modelling positive continuous data with spatio-temporal dependence

In this paper we concentrate on an alternative modeling strategy for positive data that exhibit spatial or spatio-temporal dependence. Specifically we propose to consider stochastic processes obtained trough a monotone transformation of scaled version of $χ^2$ random processes. The latter are well known in the specialized literature and originates by summing independent copies of a squared Gaussian process. However their use as stochastic models and related inference have not been much considered. Motivated by a spatio-temporal analysis of wind speed data from a network of meteorological stations in the Netherlands, we exemplify our modeling strategy by means of a non-stationary process with Weibull marginal distributions. For the proposed Weibull process we study the second-order and geometrical properties and we provide analytic expressions for the bivariate distribution. Since the likelihood is intractable, even for relatively small data-set, we suggest to adopt the pairwise likelihood as a tool for the inference. Moreover we tackle the prediction problem and we propose a linear prediction. The effectiveness of our modeling strategy is illustrated through the analysis of the aforementioned Netherland wind speed data that we supplement with a simulation study.

stat.ME↗

arXiv2019

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 Australia

math.ST↗

arXiv2019

Zastavnyi Operators and Positive Definite Radial Functions

Positive definite functions are fundamental to many areas of applied mathematics, probability theory, spatial statistics and machine learning, amogst others. Motivated by a problem coming from the maximum likelihood estimation under fixed domain asymptotics, we consider a new operator acting on rescaled weighted differences between two members of the class $Φ_d$ of positive definite radial functions,. In particular, we study the positive definiteness of the operator for the Matérn, Generalized Cauchy and Generalized Wendland families. It turns out that proposed operator allows to govern differentiability at the origin, and to attain negative correlations.

math.SP↗

arXiv2017

Estimation and Prediction using generalized Wendland Covariance Functions under fixed domain asymptotics

We study estimation and prediction of Gaussian random fields with covariance models belonging to the generalized Wendland (GW) class, under fixed domain asymptotics. As the Matérn case, this class allows a continuous parameterization of smoothness of the underlying Gaussian random field, being additionally compactly supported. The paper is divided into two parts: First, we characterize the equivalence of two Gaussian measures with GW covariance function, and we provide sufficient conditions for the equivalence of two Gaussian measures with Matérn and GW covariance functions. We elucidate the consequences of these facts in terms of (misspecified) best linear unbiased predictors. In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GW covariance model, under fixed domain asymptotics. Our findings are illustrated through a simulation study: The first compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter with the given asymptotic distribution. We then compare the finite-sample behavior of the prediction and its associated mean square error when using two equivalent Gaussian measures with Matérn and GW covariance model, using covariance tapering as benchmark.

math.ST↗

arXiv2016

Buhmann covariance functions, their compact supports, and their smoothness

We consider the Buhmann class of compactly supported radial basis functions, whih includes a wealth of special cases that have been studied in both numerical analysis and spatial statistics literatures. In particular, the celebrated Wu, Wendland and Missing Wendland functions are notable special cases of this class. We propose a very simple difference operator and show the conditions for which the application of it to Buhmann functions preserves positive definiteness on $m$-dimensional Euclidean spaces. We also show that the application of the difference operator increases smoothness at the origin, whilst keeping positive definiteness in the same $m$-dimensional Euclidean space, as well as compact support. Thus, our operator is a competitor of the celebrated Mont{é}e operator, which allows to increase the smoothness at the origin, at the expense of losing positive definiteness in the space where the radial basis function is originally defined. The proofs of our results highlight surprising connections with past literatures on celebrated class of functions. Amongst them, absolute and completely monotone functions.

math.ST↗