Fractal and smoothness properties of space-time Gaussian models

Abstract

Spatio-temporal models are widely used for inference in statistics and many applied areas. In such contexts, interests are often in the fractal nature of the sample surfaces and in the rate of change of the spatial surface at a given location in a given direction. In this paper, we apply the theory of Yaglom (1957) to construct a large class of space-time Gaussian models with stationary increments, establish bounds on the prediction errors, and determine the smoothness properties and fractal properties of this class of Gaussian models. Our results can be applied directly to analyze the stationary spacetime models introduced by Cressie and Huang (1999), Gneiting (2002), and Stein (2005), respectively.

Keywords

Space-time model / anisotropic Gaussian field / prediction error / mean square differentiability / sample path differentiability / Hausdorff dimension

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