arXiv subjects

E. Porcu

Publications and source records attributed to E. Porcu.

5 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↗

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.

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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.

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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↗

arXiv2006

Quasi-arithmetic means of covariance functions with potential applications to space-time data

The theory of quasi-arithmetic means is a powerful tool in the study of covariance functions across space-time. In the present study we use quasi-arithmetic functionals to make inferences about the permissibility of averages of functions that are not, in general, permissible covariance functions. This is the case, e.g., of the geometric and harmonic averages, for which we obtain permissibility criteria. Also, some important inequalities involving covariance functions and preference relations as well as algebraic properties can be derived by means of the proposed approach. In particular, we show that quasi-arithmetic covariances allow for ordering and preference relations, for a Jensen-type inequality and for a minimal and maximal element of their class. The general results shown in this paper are then applied to study of spatial and spatiotemporal random fields. In particular, we discuss the representation and smoothness properties of a weakly stationary random field with a quasi-arithmetic covariance function. Also, we show that the generator of the quasi-arithmetic means can be used as a link function in order to build a space-time nonseparable structure starting from the spatial and temporal margins, a procedure that is technically sound for those working with copulas. Several examples of new families of stationary covariances obtainable with this procedure are shown. Finally, we use quasi-arithmetic functionals to generalise existing results concerning the construction of nonstationary spatial covariances and discuss the applicability and limits of this generalisation.

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