Emilio Porcu

Advances and challenges in space-time modelling of natural events

Introduction to the Book: Space-Time Random Fields by J.M. Montero, E. Porcu, M. Schlather and G. Fernandez-Aviles.- Random Fields Defined on Hilbert Spaces by Hao Zhang.- Some Problems Relating Isotropic Covariance Functions by Michael L. Stein and Emilio Porcu.- Multivariate Random Fields and Robustness in Spatial Statistics by Marc G. Genton.- Spatial and Space-Time Extreme Value Theory for Random Fields by Martin Schlather.- Space-Time Second Order Properties of Random Fields by Emilio Porcu, J.Maria Montero and Gema Fernandez-Aviles.- Gaussian Markov Random Fields by Finn Lindgren.- A Journey through non Gaussian Random Fields by Denis Allard.- Space-Time Design for Gaussian Random Fields by Werner Muller.- Simulation of Stochastic Processes and Inference by Maria Dolores Ruiz Medina.- Space Time Point Processes by Thordis Thorarinsdottir.

Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere

Abstract In this article, 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 to describe spatial dependence. Given the mathematical difficulties for the construction of covariance functions for processes defined over spheres cross time, approximations of the state of nature have been proposed in the literature by using the Euclidean (based on map projections) and the chordal distances. We present several methods of construction based on the great circle distance and provide closed-form expressions for both spatio-temporal and multivariate cases. A simulation study assesses the discrepancy between the great circle distance, chordal distance, and Euclidean distance based on a map projection both in terms of estimation and prediction in a space-time and a bivariate spatial setting, where the space is in this case the Earth. We revisit the analysis of Total Ozone Mapping Spectrometer (TOMS) data and investigate differences in terms of estimation and prediction between the aforementioned distance-based approaches. Both simulation and real data highlight sensible differences in terms of estimation of the spatial scale parameter. As far as prediction is concerned, the differences can be appreciated only when the interpoint distances are large, as demonstrated by an illustrative example. Supplementary materials for this article are available online.

Radial basis functions with compact support for multivariate geostatistics

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 compactly supported over the sphere of a d-dimensional space, of a given radius. In particular, we offer a representation based on scaled mixtures of Askey functions; we also suggest a method of construction based on B-splines. Finally, we show that the very appealing convolution arguments are indeed effective when working in one dimension, prohibitive in two and feasible, but substantially useless, when working in three dimensions. We exhibit the statistical performance of the proposed models through simulation study and then discuss the computational gains that come from our constructions when the parameters are estimated via maximum likelihood. We finally apply our constructions to a North American Pacific Northwest temperatures dataset.

Classes of compactly supported covariance functions for multivariate random fields

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 exploit a class of functions called here the Wendland–Gneiting class; their use is illustrated via both a simulation study and an application to a North American bivariate dataset of precipitation and temperature. Because for this dataset, as for others, the empirical covariances exhibit a hole effect, the turning bands operator is extended to matrix-valued covariance functions so as to obtain matrix-valued covariance models with negative covariances.

Dimension walks and Schoenberg spectral measures

Abstract. Schoenberg (1938) identified the class of positive definite radial (or isotropic) functions φ : Rd → R, φ(0) = 1, as having a representation φ(x) = ∫ R+ Ωd(tu)Gd(du), t = ‖x‖, for some uniquely identified probability measure Gd on R+ and Ωd(t) = E(e it〈e1,η〉), where η is a vector uniformly distributed on the unit spherical shell Sd−1 ⊂ Rd and e1 is a fixed unit vector. Call such Gd a d-Schoenberg measure, and let Φd denote the class of all functions f : R+ → R for which such a d-dimensional radial function φ exists with f(t) = φ(x) for t = ‖x‖. Mathéron (1965) introduced operators Ĩ and D̃, called Montée and Descente, that map suitable f ∈ Φd into Φd′ for some different dimension d′: Wendland described such mappings as dimension walks. This paper characterizes Mathéron’s operators in terms of Schoenberg measures and describes functions, even in the class Φ∞ of completely monotone functions, for which neither Ĩf nor D̃f is well defined. Because f ∈ Φd implies f ∈ Φd′ for d′ < d, any f ∈ Φd has a d′-Schoenberg measure Gd′ for 1 ≤ d′ < d and finite d ≥ 2. This paper identifies Gd′ in terms of Gd via another ‘dimension walk’ relating the Fourier transforms Ωd′ and Ωd that reflect projections on Rd ′ within Rd. A study of the Euclid hat function shows the indecomposability of Ωd.

An improved spectral turning-bands algorithm for simulating stationary vector Gaussian random fields

We propose a spectral turning-bands approach for the simulation of second-order stationary vector Gaussian random fields. The approach improves existing spectral methods through coupling with importance sampling techniques. A notable insight is that one can simulate any vector random field whose direct and cross-covariance functions are continuous and absolutely integrable, provided that one knows the analytical expression of their spectral densities, without the need for these spectral densities to have a bounded support. The simulation algorithm is computationally faster than circulant-embedding techniques, lends itself to parallel computing and has a low memory storage requirement. Numerical examples with varied spatial correlation structures are presented to demonstrate the accuracy and versatility of the proposal.

Study of spatial relationships between two sets of variables: a nonparametric approach

We propose a new method for estimating a codispersion coefficient to quantify the association between two spatial variables. Our proposal is based on a Nadaraya–Watson version of the codispersion coefficient through a suitable kernel. Under regularity conditions, we derive expressions for the bias and mean square error for a kernel version of the cross-variogram and establish the consistency of a Nadaraya–Watson estimator of the codispersion coefficient. In addition, we propose a bandwidth selection method for both the variogram and the cross-variogram. Monte Carlo simulations support the theoretical findings, and as a result, the new proposal performs better than the classic Matherons estimator. The proposed method is useful for quantifying spatial associations between two variables measured at the same location. Finally, we study forest data concerning the relationship among the tree height, basal area, elevation and slope of Pinus radiata plantations. A two-dimensional codispersion map is constructed to provide insight into the spatial association between these variables.

Nonstationary matrix covariances: compact support, long range dependence and quasi-arithmetic constructions

Flexible models for multivariate processes are increasingly important for datasets in the geophysical, environmental, economics and health sciences. Modern datasets involve numerous variables observed at large numbers of space–time locations, with millions of data points being common. We develop a suite of stochastic models for nonstationary multivariate processes. The constructions break into three basic categories—quasi-arithmetic, locally stationary covariances with compact support, and locally stationary covariances with possible long-range dependence. All derived models are nonstationary, and we illustrate the flexibility of select choices through simulation.

Determinantal point process models on the sphere

We consider determinantal point processes on the

Equivalence of Gaussian measures of multivariate random fields

d