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Emilio Porcu

Publications and source records attributed to Emilio Porcu.

49 records · Page 3Linked to original sources
arXiv2017

Modeling Temporally Evolving and Spatially Globally Dependent Data

The last decades have seen an unprecedented increase in the availability of data sets that are inherently global and temporally evolving, from remotely sensed networks to climate model ensembles. This paper provides a view of statistical modeling techniques for space-time processes, where space is the sphere representing our planet. In particular, we make a distintion between (a) second order-based, and (b) practical approaches to model temporally evolving global processes. The former are based on the specification of a class of space-time covariance functions, with space being the two-dimensional sphere. The latter are based on explicit description of the dynamics of the space-time process, i.e., by specifying its evolution as a function of its past history with added spatially dependent noise. We especially focus on approach (a), where the literature has been sparse. We provide new models of space-time covariance functions for random fields defined on spheres cross time. Practical approaches, (b), are also discussed, with special emphasis on models built directly on the sphere, without projecting the spherical coordinate on the plane. We present a case study focused on the analysis of air pollution from the 2015 wildfires in Equatorial Asia, an event which was classified as the year's worst environmental disaster. The paper finishes with a list of the main theoretical and applied research problems in the area, where we expect the statistical community to engage over the next decade.

math.ST↗
arXiv2017

Asymmetric Matrix-Valued Covariances for Multivariate Random Fields on Spheres

Matrix-valued covariance functions are crucial to geostatistical modeling of multivariate spatial data. The classical assumption of symmetry of a multivariate covariance function is overlay restrictive and has been considered as unrealistic for most of real data applications. Despite of that, the literature on asymmetric covariance functions has been very sparse. In particular, there is some work related to asymmetric covariances on Euclidean spaces, depending on the Euclidean distance. However, for data collected over large portions of planet Earth, the most natural spatial domain is a sphere, with the corresponding geodesic distance being the natural metric. In this work, we propose a strategy based on spatial rotations to generate asymmetric covariances for multivariate random fields on the $d$-dimensional unit sphere. We illustrate through simulations as well as real data analysis that our proposal allows to achieve improvements in the predictive performance in comparison to the symmetric counterpart.

math.ST↗
arXiv2017

Schoenberg's theorem for real and complex Hilbert spheres revisited

Schoenberg's theorem for the complex Hilbert sphere proved by Christensen and Ressel in 1982 by Choquet theory is extended to the following result: Let L denote a locally compact group and let \overline{\D} denote the closed unit disc in the complex plane. Continuous functions f:\overline{\D}\times L\to \C such that f(ξ\cdot η,u^{-1}v) is a positive definite kernel on the product of the unit sphere in \ell_2(\C) and L are characterized as the functions with a uniformly convergent expansion f(z,u)=\sum_{m,n=0}^\infty φ_{m,n}(u)z^m\overline{z}^n, where φ_{m,n} is a double sequence of continuous positive definite functions on L such that \sumφ_{m,n}(e_L)<\infty (e_L is the neutral element of L). It is shown how the coefficient functions φ_{m,n} are obtained as limits from expansions for positive definite functions on finite dimensional complex spheres via a Rodrigues formula for disc polynomials. Similar results are obtained for the real Hilbert sphere.

math.CA↗
arXiv2016

Orthogonal expansions related to compact Gelfand pairs

Given a compact Gelfand pair (G,K) and a locally compact group L, we characterize the class P_K^\sharp(G,L) of continuous positive definite functions f:G\times L\to \C which are bi-invariant in the G-variable with respect to K. The functions of this class are the functions having a uniformly convergent expansion \sum_{φ\in Z} B(φ)(u)φ(x) for x\in G,u\in L, where the sum is over the space Z of positive definite spherical functions φ:G\to\C for the Gelfand pair, and (B(φ))_{φ\in Z} is a family of continuous positive definite functions on L such that \sum_{φ\in Z}B(φ)(e_L)<\infty. Here e_L is the neutral element of the group L. For a compact abelian group G considered as a Gelfand pair (G,K) with trivial K=\{e_G\}, we obtain a characterization of P(G\times L) in terms of Fourier expansions on the dual group \widehat{G}. The result is described in detail for the case of the Gelfand pairs (O(d+1),O(d)) and (U(q),U(q-1)) as well as for the product of these Gelfand pairs. The result generalizes recent theorems of Berg-Porcu (2016) and Guella-Menegatto (2016)

math.CA↗
arXiv2016

Determinantal point process models on the sphere

We consider determinantal point processes on the $d$-dimensional unit sphere $\mathbb S^d$. These are finite point processes exhibiting repulsiveness and with moment properties determined by a certain determinant whose entries are specified by a so-called kernel which we assume is a complex covariance function defined on $\mathbb S^d\times\mathbb S^d$. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. Particularly, we characterize and construct isotropic DPPs models on $\mathbb{S}^d$, where it becomes essential to specify the eigenvalues and eigenfunctions in a spectral representation for the kernel, and we figure out how repulsive isotropic DPPs can be. Moreover, we discuss the shortcomings of adapting existing models for isotropic covariance functions and consider strategies for developing new models, including a useful spectral approach.

stat.ME↗
arXiv2016

Fast and exact simulation of complex-valued stationary Gaussian processes through embedding circulant matrix

This paper is concerned with the study of the embedding circulant matrix method to simulate stationary complex-valued Gaussian sequences. The method is, in particular, shown to be well-suited to generate circularly-symmetric stationary Gaussian processes. We provide simple conditions on the complex co-variance function ensuring the theoretical validity of the minimal embedding circulant matrix method. We show that these conditions are satisfied by many examples and illustrate the algorithm. In particular, we present a simulation study involving the circularly-symmetric fractional Brownian motion, a model introduced in this paper.

math.ST↗
arXiv2016

A Flexible Class of Non-separable Cross-Covariance Functions for Multivariate Space-Time Data

Multivariate space-time data are increasingly available in various scientific disciplines. When analyzing these data, one of the key issues is to describe the multivariate space-time dependencies. Under the Gaussian framework, one needs to propose relevant models for multivariate space-time covariance functions, i.e. matrix-valued mappings with the additional requirement of non-negative definiteness. We propose a flexible parametric class of cross-covariance functions for multivariate space-time Gaussian random fields. Space-time components belong to the (univariate) Gneiting class of space-time covariance functions, with Matérn or Cauchy covariance functions in the spatial margins. The smoothness and scale parameters can be different for each variable. We provide sufficient conditions for positive definiteness. A simulation study shows that the parameters of this model can be efficiently estimated using weighted pairwise likelihood, which belongs to the class of composite likelihood methods. We then illustrate the model on a French dataset of weather variables.

stat.ME↗
arXiv2015

From Schoenberg coefficients to Schoenberg functions

In his seminal paper, Schoenberg (1942) characterized the class P(S^d) of continuous functions f:[-1,1] \to \R such that f(\cos θ) is positive definite over the product space S^d \times S^d, with S^d being the unit sphere of \R^{d+1} and θbeing the great circle distance. In this paper, we consider the product space S^d \times G, for G a locally compact group, and define the class P(S^d, G) of continuous functions f:[-1,1]\times G \to \C such that f(\cos θ, u^{-1}\cdot v) is positive definite on S^d \times S^d \times G \times G. This offers a natural extension of Schoenberg's Theorem. Schoenberg's second theorem corresponding to the Hilbert sphere S^\infty is also extended to this context. The case G=\R is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of Planet Earth.

math.CA↗
arXiv2013

From Schoenberg to Pick-Nevanlinna: Toward a complete picture of the variogram class

We show that a large subclass of variograms is closed under products and that some desirable stability properties, such as the product of special compositions, can be obtained within the proposed setting. We introduce new classes of kernels of Schoenberg-Lévy type and demonstrate some important properties of rotationally invariant variograms.

math.ST↗
arXiv2013

Validity of covariance models for the analysis of geographical variation

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 are valid mathematical models have been overlooked so far. We provide rigorous results for the construction of valid covariance models in this family. We also outline how to construct alternative covariance models for the analysis of geographical variation that are both mathematically well behaved and easily implementable.

stat.AP↗
arXiv2009

Space-time covariance functions with compact support

We characterize completely the Gneiting class of space-time covariance functions and give more relaxed conditions on the involved functions. We then show necessary conditions for the construction of compactly supported functions of the Gneiting type. These conditions are very general since they do not depend on the Euclidean norm. Finally, we discuss a general class of positive definite functions, used for multivariate Gaussian random fields. For this class, we show necessary criteria for its generator to be compactly supported.

stat.ME↗
arXiv2008

The Dagum family of isotropic correlation functions

A function $ρ:[0,\infty)\to(0,1]$ is a completely monotonic function if and only if $ρ(\Vert\mathbf{x}\Vert^2)$ is positive definite on $\mathbb{R}^d$ for all $d$ and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function \[ρ(β,γ)(x)=1-\biggl(\frac{x^β}{1+x^β}\biggr )^γ,\qquad x\ge 0, β,γ>0,\] called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any $d$-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.

math.ST↗