Fractal and Smoothness Properties of Space-Time Gaussian Models

TL;DR

This paper develops a broad class of space-time Gaussian models with stationary increments, analyzing their fractal and smoothness properties, and providing bounds on prediction errors for applications in statistics and related fields.

Contribution

It applies Yaglom's theory to construct and analyze new space-time Gaussian models, establishing their fractal and smoothness characteristics with practical bounds on prediction errors.

Findings

Established bounds on prediction errors for the models.

Determined fractal and smoothness properties of the models.

Applicable to existing stationary space-time models by Cressie, Huang, Gneiting, and Stein.

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 space-time models introduced by Cressie and Huang (1999), Gneiting (2002) and Stein (2005), respectively.