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The paper tackles the problem of simulating isotropic vector-valued Gaussian random fields defined over the unit two-dimensional sphere embedded in the three-dimensional Euclidean space. Such random fields are used in different disciplines of the natural sciences to model observations located on the Earth or in the sky, or direction-dependent subsoil properties measured along borehole core samples. The simulation is obtained through a weighted sum of finitely many spherical harmonics with random degrees and orders, which allows accurately reproducing the desired multivariate covariance structure, a construction that can actually be generalized to the simulation of isotropic vector random fields on the d-dimensional sphere. The proposed algorithm is illustrated with the simulation of bivariate random fields whose covariances belong to the $${{{\mathcal{F}}}}$$ , spectral Matern and negative binomial classes of covariance functions on the two-dimensional sphere.

The construction of valid and flexible cross-covariance functions is a fundamental task for modeling multivariate space– time data arising from, e.g., climatological and oceanographical phenomena. Indeed, a suitable specification of the covariance structure allows to capture both the space–time dependencies between the observations and the development of accurate predictions. For data observed over large portions of planet earth it is necessary to take into account the curvature of the planet. Hence the need for random field models defined over spheres across time. In particular, the associated covariance function should depend on the geodesic distance, which is the most natural metric over the spherical surface. In this work, we propose a flexible parametric family of matrix-valued covariance functions, with both marginal and cross structure being of the Gneiting type. We also introduce a different multivariate Gneiting model based on the adaptation of the latent dimension approach to the spherical context. Finally, we assess the performance of our models through the study of a bivariate space–time data set of surface air temperatures and precipitable water content.

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.

There is a growing interest in the analysis of geostatistical functional data. Such a fusion between geostatistical methods and functional data analysis has been shown to open promising area of research. The present paper is devoted to a kriging predictor for functional data where the functions are probability densities functions (PDFs for short), being also a special case of infinite dimensional compositional data. The predictor proposed in this paper is the analogue of the classic ordinary kriging predictor, defined in terms of scalar parameters, but considering PDFs (with support on a finite interval) instead of one-dimensional data. The methodology is applied to both simulated and real data. The statistical performance of our predictor is then evaluated through cross validation techniques using real and simulated data. From the mathematical point of view, in order to assess the properties of our predictor we need to work under the framework of Hilbert valued random fields as well as with Aitchison geometries. We present some original results being of interest for themselves, and that give a complete picture of the framework illustrated through the paper.

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.

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.

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.

Dynamic life tables arise as an alternative to the standard (static) life tables with the aim of incorporating the evolution of mortality over time. These tables can be considered as a two-way table on a grid equally spaced in either the vertical (age) or horizontal (year) directions, and the data can be decomposed into a deterministic large-scale variation (trend) plus a stochastic small-scale variation (residuals). In this context, space–time geostatistical methods can be used for fitting and predicting the dynamic mortality. We use four different space–time covariance functions for fitting and predicting mortality in Spain during the period 1980–2005. In particular, we aim at showing the behavior of separable versus nonseparable fitted structures on one hand, and the behavior of simple structures given by combinations of products and sums versus more complicated negative structures on the other hand.

Some special functional equations involving means and related to a problem of reducibility of some classes of correlation functions are considered. We show some characterizations of the reducibility problem under several choices of the mean operators and different weak regularity assumptions imposed on the involving functions. We find that mean-generated correlation functions are completely irreducible, in the sense that, for this broad class of correlation functions, there does not exist a non-trivial solution associated to the Perrin–Senoussi problem.

In the nonseparable spatio-temporal context, several efforts have been made in order to obtain general classes of spatio-temporal covariances. Our aim in this paper is to join several approaches coming from different authors and provide some ideas for the construction of new models of spatio-temporal covariance and spectral density functions. On one hand, we build new covariance families while removing some undesirable features of the previously proposed models, particularly following Stein’s (in J Am Stat Assoc 100:310–321, 2005) remark about Gneiting’s (in J Am Stat Assoc 97:590–600, 2002) approach and about some tensorial product covariance models. We show some of the theoretical results and examples obtained with the product or the sum of spatio-temporal covariance functions or even better with the mixed forms. On the other hand, we define new models for spectral densities through the product of two other spectral densities. We give some characterizations and properties as well as several examples. Finally, we present a practical modelling of Irish wind speed data based on some of the space-time covariance models presented in this paper.

There is a great demand for statistical modeling of phenomena that evolve in both space and time, and thus, there is a growing literature on correlation function models for spatio-temporal processes. In particular, various properties of these correlation functions have been studied only for the merely spatial or temporal case, fact that constitutes a strong motivation for our work. The goal of this paper is to inspect some properties, obtained with respect to partial differentiation and integration, of stationary spatio-temporal correlation functions for which anisotropy is obtained through isotropy between components as in Fernandez-Casal et al. (Stat Comput 13(2):127–136, 2003). We show that through partial differentiation and integration it is possible to obtain permissible spatio-temporal correlation functions in the space–time domain. Other new results regard specific classes of space–time correlations introduced in recent literature. A curious result arises by differentiating scale mixtures of Euclid’s hat.

Obtaining new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to introduce and develop new spatio-temporal covariance models taking into account the problem of spatial anisotropy. Recent literature has focused on the problem of full symmetry and the problem of anisotropy has been overcome. Here we propose a generalization of Gneiting’s (J Am Stat Assoc 97:590–600, 2002a) approach and obtain new classes of stationary nonseparable spatio-temporal covariance functions which are spatially anisotropic. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability.