Estimating and Modeling Space-Time Variograms

Donald E. Myers
Department of Mathematics
University of Arizona
Tucson, AZ 85721 USA
Ph. 520 621 6859, Fax 520 621 6859
E-mail: myers@math.arizona.edu

Abstract
As with a spatial variogram or spatial covariance, a principal purpose of estimating and modeling a space-time variogram is to quantify the spatial temporal dependence reflected in a data set. The resulting model might then be used for spatial  interpolation and/or temporal prediction which might take several forms, e.g. kriging and Radial Basis functions. There are significant problems to overcome in both the estimation and the modeling stages for space-time problems unl ike the purely spatial appl ication where estimation is the more difficult step. The key point is that a spatial-temporal variogram as a function must be conditionally negative definite (not just semi-definite) which can be a difficult condition to verify in specific cases. In the purely spatial  context one rel ies on a known l ist of isotropic valid models, e.g., the Matern class as well as the exponential and gaussian models, as well as on positive linear combinations of known valid models. Bochner’s Theorem (or the extension given for generalized covariances by Matheron) characterizes non- negative definite functions but does not easily distinguish the strictly positive definite functions. Geometric anisotropies can be incorporated via an affine transformation and space-time might be viewed as simply a higher dimensional space but possibly with an anisotropy in the model. This approach impl ies that there is an appropriate and natural  choice of a norm (or metric) on space-time analogous to the usual Euclidean norm for space. The most obvious way to construct a model for space-time is to “separate” the dependence on space and on time. This is not new and in fact a similar problem can occur in spatial application, i.e., a zonal anisotropy. Early attempts used either the sum of two covariances or the sum of two variograms, in either case one component being defined on space and the other on time. It is easily shown that this leads to semi-definite models and hence if used in kriging equations, the result may be a non- invertible coefficient matrix. It is also easy to see that the product of two variograms (even on the same domain) can violate the growth condition. However it is well known that the product of two strictly positive definite functions is again strictly positive definite. In fact a gaussian covaiance model might be viewed as product (of several gaussian models each defined on a lower dimensional space). Likewise one form of the exponential covariance, often used in hydrology appl ications, is also a product. When converted to variogram form, there is not only a product (with a negative sign) but also a sum. It turns out that the variogram form is more convenient in the estimation stage. The simple product covariance is somewhat too limi ting however, each component effectively must have the same “si ll”. An obvious extension is the product-sum model which when converted to variogram form is the same as for the product (but with different coefficients), This can be further generalized to an integrated product sum model. In the estimation stage there are two separate problems, one is to determine the appropriate model type and the other is to estimate the model parameters. In a typical spatial  application the list of possible models is usually kept smal l and hence the primary emphasis is on estimating the model parameters. In the spatial temporal context the list of possible models is likely to be much greater and model type selection more difficult. De Ioca, Myers and Posa have shown that the use of marginal variograms is one way to attack this problem and have given an example of extending to the integrated product sum model as well as to the multivariate case using a Linear Coregional ization Model.

In: McRoberts, R. et al. (eds).  Proceedings of the joint meeting of The 6th International Symposium On Spatial Accuracy Assessment In Natural Resources and Environmental Sciences and The 15th Annual Conference of The International Environmetrics Society, June 28 – July 1 2004, Portland, Maine, USA.

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