# Visualization and Assessment of Spatio-temporal Covariance Properties

## TL;DR

This paper introduces a functional data analysis framework for visualizing and testing the properties of spatio-temporal covariance functions, specifically separability and symmetry, using real and simulated data.

## Contribution

It proposes a novel approach combining functional boxplots and nonparametric tests to assess covariance properties from spatio-temporal data.

## Key findings

- Effective visualization of covariance properties using functional boxplots.
- Nonparametric tests successfully detect non-separability and asymmetry.
- Method performs well on simulated and real-world datasets.

## Abstract

Spatio-temporal covariances are important for describing the spatio-temporal variability of underlying random processes in geostatistical data. For second-order stationary processes, there exist subclasses of covariance functions that assume a simpler spatio-temporal dependence structure with separability and full symmetry. However, it is challenging to visualize and assess separability and full symmetry from spatio-temporal observations. In this work, we propose a functional data analysis approach that constructs test functions using the cross-covariances from time series observed at each pair of spatial locations. These test functions of temporal lags summarize the properties of separability or symmetry for the given spatial pairs. We use functional boxplots to visualize the functional median and the variability of the test functions, where the extent of departure from zero at all temporal lags indicates the degree of non-separability or asymmetry. We also develop a rank-based nonparametric testing procedure for assessing the significance of the non-separability or asymmetry. The performances of the proposed methods are examined by simulations with various commonly used spatio-temporal covariance models. To illustrate our methods in practical applications, we apply it to real datasets, including weather station data and climate model outputs.