Parametric Modeling of Intrinsic Structure Covariance Functions for Non-Homogeneous and Non-Stationary Spatio-Temporal Random Processes on the Sphere

Abstract

Identifying appropriate models for random processes and their associated covariance functions is one of the primary goals in spatial and spatio-temporal statistics, as it enables researchers to analyze the dependence structure within the data. For this purpose, assumptions of spatial homogeneity and temporal stationarity are commonly used, and many models have been developed under these conditions. However, these assumptions are often overly strong and unrealistic in practical applications.

Moreover, when working on the sphere, standard approaches from Euclidean space may not be appropriate due to the unique geometric and topological properties of the spherical domain. Despite this, relatively fewer studies have addressed random process modeling and covariance function development specifically for the sphere.

In this research, we introduce a parametric modeling framework for intrinsic structure covariance functions (ISCFs), designed to address non-homogeneous and non-stationary spatio-temporal random processes and their covariance functions on the sphere. To alleviate the assumption of spatial homogeneity while accounting for the spherical domain, we apply the theory of intrinsic random functions (IRFs) on the sphere. Similarly, to address temporal non-stationarity, we use the concept of random processes with stationary increments, exploring their relationship with intrinsic random functions on the real line.

We also provide a methodology for estimating the parameters associated with the ISCF model. This is demonstrated through a simulation study and an application to a real-world dataset, highlighting the advantage of the model’s interpretable parameters.