S. De Iaco

Space–time analysis using a general product–sum model

A generalization of the product-sum covariance model introduced by De Cesare et al. (Statist. Probab. Lett. 51 (2001) 9) is given in this paper. This generalized model is non-separable and in general is non-integrable, hence, it cannot be obtained from the Cressie-Huang representation. Moreover, the product-sum model does not correspond to the use of a metric in space-time. It is shown that there are simple methods for estimating and modeling the covariance or variogram components of the product-sum model using data from realizations of spatial-temporal random fields.

Nonseparable Space-Time Covariance Models: Some Parametric Families

By extending the product and product–sum space-time covariance models, new families are generated as integrated products and product–sums. These include nonintegrable space-time covariance models not obtainable by the Cressie–Huang representation. It is shown how to fit the spatial and temporal components of the models as well as the probability density function. The methods are illustrated by a case study.

Space-time variograms and a functional form for total air pollution measurements

A space-time functional form for some contaminants is obtained and used for estimating total air pollution (TAP) in the district of Milan, Italy, during selected high-risk days of 1999. This functional form is determined through a space-time product-sum variogram model for TAP measurements and the dual form of kriging, i.e., radial basis functions. Data for nitric oxide (NO), nitrogen dioxide (NO2) and carbon monoxide (CO) collected in Milan district, Italy are used to generate a combined indicator of traffic pollution, called TAP. In a previous study the weightings were obtained by multiple principal component analyses of the daily concentration levels. It was found that the first component explains approximately 70% of the total variance for each day and this component is treated as samples defined over space and time. A systematic pattern, which follows the corridor along which survey stations, characterized by heavy traffic are located, has been observed for TAP throughout Milan district, for all days considered. Note that the pollution data set is just an illustration for the new statistical method proposed.

The Linear Coregionalization Model and the Product–Sum Space–Time Variogram

The product covariance model, the product–sum covariance model, and the integrated product and integrated product–sum models have the advantage of being easily fitted by the use of “marginal” variograms. These models and the use of the marginals are described in a series of papers by De Iaco, Myers, and Posa. Such models allow not only estimating values at nondata locations but also prediction in future times, hence, they are useful for analyzing air pollution data, meteorological data, or ground water data. These three kinds of data are nearly always multivariate and because the processes determining the deposition or dynamics will affect all variates, a multivariate approach is desirable. It is shown that the use of marginal variograms for space–time modeling can be extended to the multivariate case and in particular to the use of the Linear Coregionalization Model (LCM) for cokriging in space–time. An application to an environmental data set is given.

Modeling and prediction of multivariate space-time random fields

Abstract In various environmental studies multivariate spatial–temporal correlated data are involved, hence appropriate techniques to enhance space–time prediction are in great demand. An extension of multivariate spatial geostatistics to a spatio-temporal domain might be a straightforward task; nevertheless, up to now, little has been done in a multivariate spatial–temporal context. Modeling and prediction techniques are described for a multivariate space–time random field, moreover some theoretical and practical aspects are investigated for a bivariate space–time random field through a case study.

Space-time radial basis functions

Abstract Radial basis functions are “isotropic”; i.e., under a rotation, the basis function is left unchanged and is obtained as a function of a distance on the space. For Euclidean space this is not a problem since there is a natural metric. To extend radial basis functions to space-time, i.e., Rm × T, either a zonal anisotropy has to be incorporated or a metric must be defined on space-time. While the sum of two valid radial basis functions defined on different dimensional spaces is generally only semidefinite on the product space, the product of two positive definite functions on lower dimensional spaces is positive definite on the product space. This construction can be extended in several ways including a product-sum, integrated product, and the integrated product-sum. Examples are given for each construction and an application is given. The constructions are equally applicable to extending from space to space-time or for splitting higher-dimensional Euclidean spaces into the product of several lower-dimensional spaces.

Strict Positive Definiteness of a Product of Covariance Functions

Positive definiteness represents an admissibility condition for a function to be a covariance. Nevertheless, the more restricted condition of strict positive definiteness has received attention in literature, especially in spatial statistics, since it ensures that the kriging system has a unique solution. Most known covariance functions are isotropic but there are applications where isotropy is not appropriate, e.g., space-time covariance functions. One way to construct non-isotropic covariance functions is to use a product or a product-sum. In this article, it is given a necessary as well as a sufficient condition for a product of two covariance functions to be strictly positive definite. This result is extended to the well-known product-sum covariance model.

Total Air Pollution And Space-Time Modelling

Two general problems occur in the analysis of air pollution data; multiple contaminants and a dependence on both spatial location and time of observation. Principal Component Analysis (PCA) provides a tool for removing the interdependence of the contaminant concentrations, in addition an analysis of the principal components, eigenvectors and eigenvalues provides additional insight into the dispersion and occurrence of the pollution plume. New models for space-time variograms and techniques for modelling them have been introduced by De laco, Myers and Posa.

Using Simultaneous Diagonalization to Identify a Space–Time Linear Coregionalization Model

Although there are multiple methods for modeling matrix covariance functions and matrix variograms in the geostatistical literature, the linear coregionalization model is still widely used. In particular it is easy to check to ensure whether the matrix covariance function is positive definite or that the matrix variogram is conditionally negative definite. One of the difficulties in using a linear coregionalization model is in determining the number of basic structures and the corresponding covariance functions or variograms. In this paper, a new procedure is given for identifying the basic structures of the space–time linear coregionalization model and modeling the matrix variogram. This procedure is based on the near simultaneous diagonalization of the sample matrix variograms computed for a set of spatiotemporal lags. A case study using a multivariate spatiotemporal data set provided by the Environmental Protection Agency of Lombardy, Italy, illustrates how nearly simultaneous diagonalization of the empirical matrix variograms simplifies modeling of the matrix variograms. The new methodology is compared with a previous one by analyzing various indices and statistics.

Strict positive definiteness in geostatistics

Geostatistical modeling is often based on the use of covariance functions, i.e., positive definite functions. However, when interpolation problems have to be solved, it is advisable to consider the subset of strictly positive definite functions. Indeed, it will be argued that ensuring strict positive definiteness for a covariance function is convenient from a theoretical and practical point of view. In this paper, an extensive analysis on strictly positive definite covariance functions has been given. The closure of the set of strictly positive definite functions with respect to the sum and the product of covariance functions defined on the same Euclidean dimensional space or on factor spaces, as well as on partially overlapped lower dimensional spaces, has been analyzed. These results are particularly useful (a) to extend strict positive definiteness in higher dimensional spaces starting from covariance functions which are only defined on lower dimensional spaces and/or are only strictly positive definite in lower dimensional spaces, (b) to construct strictly positive definite covariance functions in space–time as well as (c) to obtain new asymmetric and strictly positive definite covariance functions.