Assessing Spatial Stationarity and Segmenting Spatial Processes into Stationary Components

Abstract

In this research, we propose a novel technique for visualizing nonstationarity in geostatistics, particularly when confronted with a single realization of data at irregularly spaced locations. Our method hinges on formulating a statistic that tracks a stable microergodic parameter of the exponential covariance function, allowing us to address the intricate challenges of nonstationary processes that lack repeated measurements. We implement the fused lasso technique to elucidate nonstationary patterns at various resolutions. For prediction purposes, we segment the spatial domain into stationary sub-regions via Voronoi tessellations. Additionally, we devise a robust test for stationarity based on contrasting the sample means of our proposed statistics between two selected Voronoi subregions. The effectiveness of our method is demonstrated through simulation studies and its application to a precipitation dataset in Colorado. Supplementary materials accompanying this paper appear online.

Appendix A

This section contains Figs. 7, 8, 9 and 10 showing the distributions of p-values for diverse scenarios under the null hypothesis $H_0$, as discussed in Sect. 4.1.

Fig. 7

The distributions of p-values under $H_0$ for various scenarios with $\tau ^2=0$ under a uniform sampling design

Fig. 8

The distributions of p-values under $H_0$ for various scenarios with $\tau ^2=0.01$ under a uniform sampling design

Fig. 9

The distributions of p-values under $H_0$ for various scenarios with $\tau ^2=0$ under a clustered sampling design

Fig. 10

The distributions of p-values under $H_0$ for various scenarios with $\tau ^2=0.01$ under a clustered sampling design