Moreno Bevilacqua

• Ph.D.
• Professor (Associate) at Adolfo Ibáñez University

There exists a plethora of parametric models for positive definite kernels, and their use is ubiquitous in disciplines as diverse as statistics, machine learning, numerical analysis, and approximation theory. Usually, the kernel parameters index certain features of an associated process. Amongst those features, smoothness (in the sense of Sobolev s...

A huge literature in statistics and machine learning is devoted to parametric families of correlation functions, where the correlation parameters are used to understand the properties of an associated spatial random process in terms of smoothness and global or compact support. However, most of current parametric correlation functions attain only no...

The Gaussian copula is a powerful tool that has been widely used to model spatial and/or temporal correlated data with arbitrary marginal distributions. However, this kind of model can potentially be too restrictive since it expresses a reflection symmetric dependence. In this paper, we propose a new spatial copula model that makes it possible to o...

The Gaussian copula is a powerful tool that has been widely used to model spatial and/or temporal correlated data with arbitrary marginal distributions. However, this kind of model can potentially be too restrictive since it expresses a reflection symmetric dependence. In this paper, we propose a new spatial copula model that makes it possible to o...

The Mat\'ern model has been a cornerstone of spatial statistics for more than half a century. More recently, the Mat\'ern model has been central to disciplines as diverse as numerical analysis, approximation theory, computational statistics, machine learning, and probability theory. In this article we take a Mat\'ern-based journey across these disc...

The Matérn and the Generalized Cauchy families of covariance functions have a prominent role in spatial statistics as well as in a wealth of statistical applications. The Matérn family is crucial to index mean-square differentiability of the associated Gaussian random field; the Cauchy family is a decoupler of the fractal dimension and Hurst effect...

Cokriging is the common method of spatial interpolation (best linear unbiased prediction) in multivariate geostatistics. While best linear prediction has been well understood in univariate spatial statistics, the literature for the multivariate case has been elusive so far. The new challenges provided by modern spatial datasets, being typically mul...

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 wi...

In this paper we propose a highly scalable method for (non-)Gaussian random fields estimation.In particular, we propose a novel (a)symmetric weight function based on nearest neighboursfor the method of maximum weighted composite likelihood based on pairs (WCLP). The proposed weight function allows estimating massive (up to millions) spatial dataset...

The Mat\'ern and the Generalized Cauchy families of covariance functions have a prominent role in spatial statistics as well as in a wealth of statistical applications. The Mat\'ern family is crucial to index mean-square differentiability of the associated Gaussian random field; the Cauchy family is a decoupler of the fractal dimension and Hurst ef...

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 sch...

The Matérn 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 We...

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...

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 wi...

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 p...

This book offers essential, systematic information on the assessment of the spatial association between two processes from a statistical standpoint. Divided into eight chapters, the book begins with preliminary concepts, mainly concerning spatial statistics. The following seven chapters focus on the methodologies needed to assess the correlation be...

In this chapter, we study another measure to quantify the assessment of two spatial or temporal series. This coefficient, called the codispersion coefficient, was first introduced by Matheron (1965) and has been used in several applications (Goovaerts 1994, 1997, 1998; Chiles and Delfiner 1999; Blanco-Moreno et al. 2005; Vallejos 2008, 2012; Buckle...

Digital images are subject to a variety of contaminations (distortions) during the acquisition, processing, compression, storage transmission, and reproduction. This can significantly affect the posterior visualization of images. In image processing there are at least two ways to approach this issue, objective and subjective image quality assessmen...

Determining measures of association between two processes on the space is not a simple task. Clifford and Richardson (1985) and Clifford et al. (1989), (see also Dutilleul 1993) introduced a modified t statistic based on a correction of both, the sample covariance and the degrees of freedom of the distribution under the null hypothesis. This proced...

The goal of the current chapter is to introduce a nonparametric version of the codispersion coefficient. Extensions of this nature have previously been considered in the spatial statistics literature. For example, Hall et al. (1994) studied nonparametric estimations of the covariance function. Nonparametric estimations of the semivariogram were exa...

Addressing the spatial association between three or more processes is a challenging problem. Here, similarly as in the previous chapters we focus our attention in a continuous multivariate process with more than two components. Although motivation could be theoretical, there are several applications in the context of image processing. For instance,...

In the previous chapters, we studied the association between two georeferenced sequences from a hypothesis testing perspective. In the following three chapters, we focus on some coefficients of spatial association. These coefficients are not simple modifications of the correlation coefficient, but the underlying idea of its construction relies on t...

Assessing the significance of the correlation between the components of a bivariate random field is of great interest in the analysis of spatial-spatial data. In this chapter, testing the association between two georeferenced correlated variables is addressed for the components of a bivariate Gaussian random field using the asymptotic distribution...

The Mat{\'e}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 genera...

Cokriging is the common method of spatial interpolation (best linear unbiased prediction) in multivariate geostatistics. While best linear prediction has been well understood in univariate spatial statistics, the literature for the multivariate case has been elusive so far. The new challenges provided by modern spatial datasets, being typically mul...

In this paper, we concentrate on an alternative modeling strategy for positive data that exhibit spatial or spatiotemporal dependence. Specifically, we propose to consider stochastic processes obtained through a monotone transformation of scaled version of χ ² random processes. The latter is well known in the specialized literature and originates b...

This paper proposes a new class of covariance functions for bivariate random fields on spheres, having the same properties as the bivariate Matérn model proposed in Euclidean spaces. The new class depends on the geodesic distance on a sphere; it allows for indexing differentiability (in the mean square sense) and fractal dimensions of the component...

In this paper we propose a spatio-temporal blockwise Euclidean likelihood method for the estimation of covariance models when dealing with large spatio-temporal Gaussian data. The method uses moment conditions coming from the score of the pairwise composite likelihood. The blockwise approach guarantees considerable computational improvements over t...

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 con...

we consider a new operator acting on rescaled weighted differences between two members of the class Φd of positive definite radial functions. In particular, we study the positive definiteness of the operator for the Matérn, Generalized Cauchy and Wendland families.

We study the problem of estimating the covariance parameters of a one-dimensional Gaussian process with exponential covariance function under fixed-domain asymptotics. We show that the weighted pairwise maximum likelihood estimator of the microergodic parameter can be consistent or inconsistent. This depends on the range of admissible parameter val...

We propose new covariance functions for bivariate Gaussian random fields that are very general and include as special cases other popular models proposed in earlier literature, namely, the bivariate Matérn and bivariate Cauchy models. The proposed model allows the covariance margins to belong to different parametric families with. To our knowledge,...

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 t...

In this paper we aim to propose two models for regression and dependence analysis when dealing with positive spatial or spatio-temporal continuous data. Specifically we propose two (possibly non stationary) random processes with Gamma and Weibull marginals. Both processes stem from the same idea, namely from the transformation of a sum of independe...

We study composite likelihood estimation of the covariance parameters with data from a one-dimensional Gaussian process with exponential covariance function under fixed domain asymptotics. We show that the weighted pairwise maximum likelihood estimator of the microergodic parameter can be consistent or inconsistent , depending on the range of admis...

The equivalence of Gaussian measures is a fundamental tool to establish the asymptotic properties of both prediction and estimation of Gaussian fields under fixed domain asymptotics. The paper solves Problem 18 in the list of open problems proposed by Gneiting (2013). Specifically, necessary and sufficient conditions are given for the equivalence o...

We study estimation and prediction of Gaussian processes with covariance model belonging to the generalized Cauchy (GC) family, under fixed domain asymptotics. Gaussian processes with this kind of covariance function provide separate characterization of fractal dimension and long range dependence, an appealing feature in many physical, biological o...

We study estimation and prediction of Gaussian processes with covariance model belonging to the generalized Cauchy (GC) family, under fixed domain asymptotics. Gaussian processes with this kind of covariance function provide separate characterization of fractal dimension and long range dependence, an appealing feature in many physical, biological o...

We offer a dual view of the dimple problem related to space-time correlation functions in terms of their contours.We find that the dimple property (Kent et al., 2011) in the Gneiting class of correlations is in oneto-one correspondence with nonmonotonicity of the parametric curve describing the associated contour lines. Further, we show that given...

This paper considers a multivariate spatial random field, with each component having univariate marginal distributions of the skew-Gaussian type. We assume that the field is defined spatially on the unit sphere embedded in $\mathbb{R}^3$, allowing for modeling data available over large portions of planet Earth. This model admits explicit expression...

This paper considers a multivariate spatial random field, with each component having univariate marginal distributions of the skew-Gaussian type. We assume that the field is defined spatially on the unit sphere embedded in $\mathbb{R}^3$, allowing for modeling data available over large portions of planet Earth. This model admits explicit expression...

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 Mat\'ern case, this class allows a continuous parameterization of smoothness of the underlying Gaussian random field, being additionally compactly supported. The paper is divide...

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 Mat\'ern case, this class allows a continuous parameterization of smoothness of the underlying Gaussian random field, being additionally compactly supported. The paper is divide...

In the recent years, there has been a growing interest in proposing covariance models for multivariate Gaussian random fields. Some of these covariance models are very flexible and can capture both the marginal and the cross-spatial dependence of the components of the associated multivariate Gaussian random field. However, effective estimation meth...

We consider the Buhmann class of compactly supported covariance functions, that includes the most prominent classes used in the spatial statistics literature. 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 consider maximum likelihood estimation with data from a bivariate Gaussian process with a separable exponential covariance model under fixed domain asymptotics. We first characterize the equivalence of Gaussian measures under this model. Then consistency and asymptotic normality for the maximum likelihood estimator of the microergodic parameters...

We consider maximum likelihood estimation with data from a bivariate Gaussian process with a separable exponential covariance model under fixed domain asymptotic. We first characterize the equivalence of Gaussian measures under this model. Then consistency and asymptotic distribution for the microergodic parameters are established. A simulation stu...

Directional spatial data, typically represented through angles, are of central importance in many scientific disciplines, such as environmental sciences, oceanography and meteorology, among others. We propose a wrapped-Gaussian field to model directions in a multivariate spatial or spatio-temporal context. The n-dimensional distributions of a wrapp...

In recent literature there has been a growing interest in the construction of covariance models for multivariate Gaussian random fields. However, effective estimation methods for these models are somehow unexplored. The maximum likelihood method has attractive features, but when we deal with large data sets this solution becomes impractical, so com...

In this paper, we propose stationary covariance functions for processes that evolve temporally over a sphere, as well as cross-covariance functions for multivariate random fields defined over a sphere. For such processes, the great circle distance is the natural metric that should be used in order to describe spatial dependence. Given the mathemati...

Discussion of "Cross-Covariance Functions for Multivariate Geostatistics" by Genton and Kleiber [arXiv:1507.08017].

In this paper we propose a blockwise Euclidean likelihood method for the estimation of a spatial binary field obtained by thresholding a latent Gaussian random field. The moment conditions used in the Euclidean likelihood estimator derive from the score of the composite likelihood based on marginal pairs. A feature of this approach is that it is po...

Statistical analysis based on random fields has become a widely used approach in order to better understand real processes in many fields such as engineering, environmental sciences, etc. Data analysis based on random fields can be sometimes problematic to carry out from the inferential prospective. Examples are when dealing with: large dataset, co...

The paper combines simple general methodologies to obtain new classes of matrix-valued covariance functions that have two important properties: (i) the domains of the compact support of the several components of the matrix-valued functions can vary between components; and (ii) the overall differentiability at the origin can also vary. These models...

Assessing the significance of the correlation between the components of a bivariate random field is of great interest in the analysis of spatial data. This problem has been addressed in the literature using suitable hypothesis testing procedures or using coefficients of spatial association between two sequences. In this paper, testing the associati...

Due to the availability of large molecular data-sets, covariance models are increasingly used to describe the structure of genetic variation as an alternative to more heavily parametrised biological models. We focus here on a class of parametric covariance models that received sustained attention lately and show that the conditions under which they...

In the last years there has been a growing interest in proposing methods for estimating covariance functions for geostatistical data. Among these, maximum likelihood estimators have nice features when we deal with a Gaussian model. However maximum likelihood becomes impractical when the number of observations is very large. In this work we review s...

Matrix-valued radially symmetric covariance functions (also called radial basis functions in the numerical analysis literature) are crucial for the analysis, inference and prediction of Gaussian vector-valued random fields. This paper provides different methodologies for the construction of matrix-valued mappings that are positive definite and comp...

In this article, we propose two methods for estimating space and space-time covariance functions from a Gaussian random field, based on the composite likelihood idea. The first method relies on the maximization of a weighted version of the composite likelihood function, while the second one is based on the solution of a weighted composite score equ...

Testing for separability of space-time covariance functions is of great interest in the analysis of space-time data. In this paper we work in a parametric framework and consider the case when the parameter identifying the case of separability of the associated space-time covariance lies on the boundary of the parametric space. This situation is fre...

The problem of modelling dynamic mortality tables is considered. In this context, the influence of age on data graduation needs to be properly assessed through a dynamic model, as mortality progresses over the years. After detrending the raw data, the residuals dependence structure is analysed, by considering them as a realisation of a homogeneous...

In this paper, we investigate the problem of negative empirical covariances for temperature data in a three-dimensional geothermal field. We make use of permissibility criteria that allow a linear combination of covariance functions to be positive definite even if some of the weights of the linear combination are set to be negative. We thus propose...

There is a great demand for statistical modelling of phenomena that evolve in both space and time, and thus, there is a growing literature on covariance function models for spatio-temporal processes. Although several nonseparable space–time covariance models are available in the literature, very few of them can be used for spatially anisotropic dat...

We propose two methods based on composite likelihood that can be used to fit space-time data. We perform an intesive simulation study in order to evaluate the performances of the proposed methods, by keeping a constant comparison with well known methods proposed by the Geostatistical literature. The methods we propose are shown to have considerable...

Modelisation and prediction of environmental phenomena, which typically show dependence in space and time, has been one of the most important challenges over the last years. The classical steps of the spatial modeling approach can be resumed as follows: (1) a model-oriented step, in which a random fields assumption is considered; (2) estimation of...