On the frequency variogram and on frequency domain methods for the analysis of spatio-temporal data

TL;DR

This paper explores frequency domain methods, especially Discrete Fourier Transforms, for modeling and analyzing intrinsically stationary spatio-temporal data, addressing estimation challenges due to the inclusion of the time dimension.

Contribution

It introduces an alternative frequency domain approach for spatio-temporal data analysis, focusing on estimation and testing under intrinsic stationarity assumptions.

Findings

DFT of stationary time series at different frequencies are asymptotically independent

Derived properties of estimators for spatio-temporal covariance functions

Proposed tests of independence based on frequency domain methods

Abstract

The covariance function and the variogram play very important roles in modelling and in prediction of spatial and spatio-temporal data. The assumption of second order stationarity, in space and time, is often made in the analysis of spatial data and the spatio-temporal data. Several times the assumption of stationarity is considered to be very restrictive, and therefore, a weaker assumption that the data is Intrinsically stationary both in space and time is often made and used, mainly by the geo-statisticians and other environmental scientists. In this paper we consider the data to be intrinsically stationary. Because of the inclusion of time dimension,the estimation and derivation of the sampling properties of various estimators related to spatio-temporal data become complicated. In this paper our object is to present an alternative way, based on Frequency Domain methods for modelling the data. Here we consider Discrete Fourier Transforms (DFT) defined for the (Intrinsic) time series data observed at several locations as our data, and then consider the estimation of the parameters of spatio-temporal covariance function, estimation of Frequency Variogram, tests of independence etc. We use the well known property that the Discrete Fourier Transforms of stationary time series evaluated at distinct Fourier Frequencies are asymptotically independent and distributed as complex normal in deriving many results considered in this paper.