D. Posa

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.

Product‐sum covariance for space‐time modeling: an environmental application

In this paper a product-sum covariance for space-time modeling of nitrogen dioxide in the Milan district is introduced. Residuals have been generated for all stations after the removal of daily and seasonal trends, which are readily interpretable, in order to estimate and model the spatial-temporal variogram. The trend component and the residual variogram model have been used to predict the hourly averages of nitrogen dioxide for the first two days of January 1997. GSLIB programs were modified for sample variogram computations, cross-validation and kriging. Copyright

FORTRAN programs for space-time modeling

Modified GSLIB FORTRAN 77 routines are given in this paper for estimating and modeling space-time variograms. Two general families of models are incorporated in the programs: these are the product model and the product-sum model, both based on the decomposition of the space-time covariance in terms of a space covariance and a time covariance. The GSLIB kriging program has also been modified to incorporate these space-time models. One of the programs detects and removes temporal periodicities in the data. The program removes them and generates residuals for all monitoring stations, in order to estimate and model the spatial-temporal variogram using residuals. The modified kriging program also allows the use of cross-validation in conjunction with fitting of space-time variogram models. The trend component and the residual variogram model can be used for prediction. To illustrate the use of the programs, hourly averages of NO2 for the first ten months of 1998 in Lombardy were used.

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.

Characteristic behavior and order relations for indicator variograms

Heuristic models for indicator variograms and their parameters (practical nugget effect and range) are proposed for a bivariate normal distribution with spherical correlogram. These models can be used conveniently as a preliminary check for bivariate normality. In the general non-Gaussian case, indicator variogram models for multiple threshold values must verify a certain number of order relations (inequalities) established directly from the properties of a general bivariate cumulative distribution function. An interesting, little-known maximum hole effect for indicator correlation is pointed out.

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.

Conditioning of the stationary kriging matrices for some well-known covariance models

In this paper, the condition number of the stationary kriging matrix is studied for some well-known covariance models. Indeed, the robustness of the kriging weights is strongly affected by this measure. Such an analysis can justify the choice of a covariance function among other admissible models which could fit a given experimental covariance equally well.

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.