A Spectral Domain Test for Stationarity of Spatio‐Temporal Data

Abstract

Many random phenomena in the environmental and geophysical sciences are functions of both space and time; these are usually called spatio‐temporal processes. Typically, the spatio‐temporal process is observed over discrete equidistant time and at irregularly spaced locations in space. One important aim is to develop statistical models based on what is observed. While doing so a commonly used assumption is that the underlying spatio‐temporal process is stationary. If this assumption does not hold, then either the mean or the covariance function is misspecified. This can, for example, lead to inaccurate predictions. In this article we propose a test for spatio‐temporal stationarity. The test is based on the dichotomy that Fourier transforms of stochastic processes are near uncorrelated if the process is second‐order stationary but correlated if the process is second‐order nonstationary. Using this as motivation, a discrete Fourier transform for spatio‐temporal data over discrete equidistant times but on irregularly spaced spatial locations is defined. Two statistics which measure the degree of correlation in the discrete Fourier transforms are proposed. These statistics are used to test for spatio‐temporal stationarity. It is shown that the same statistics can also be adapted to test for the one‐way stationarity (either spatial or temporal stationarity). The proposed methodology is illustrated with a small simulation study.

A Test for Weak Stationarity in the Spectral Domain

“…As we pointed out in the introduction, our statistic (2.4) draws some similarities to those given in Dwivedi and Subba Rao (2011) see also Jentsch and Subba Rao (2015) and Bandyopadhy et al (2017). More specifically, denote ω k = 2πk/T and J…”

“…We consider three block sizes for every sample size. 4 In this sense, the optimal block size for T = 256 is n = 32; for T = 512, n = 32; and, finally, for T = 1024, n = 128. For the sake of clarity, in the tables that follow we mark those pairs with the " " sign.…”

Change-Point Analysis of Time Series with Evolutionary Spectra

“…More recently there has been a growing interest about functional and time-varying parameter models where the latter are characterized by infinite-dimensional parameters which change continuously over time [see, e.g., Dahlhaus (1997), Neumann and von Sachs (1997), Hörmann and Kokoszka (2010), Dette, Preuß, and Vetter (2011), Zhang and Wu (2012), Panaretos and Tavakoli (2013), Aue, Dubart Nourinho, and Hormann (2015) and van Delft and Eichler (2018)]. Several authors have extended the scope of the stationarity tests originally introduced by Priestley and Subba Rao (1969), and further developed by, e.g., Dwivedi and Subba Rao (2010), Jentsch and Subba Rao (2015) and Bandyopadhyay, Carsten, and Subba Rao (2017), to these settings. In the context of locally stationary times series, Paparoditis (2009) proposed a test based on comparing a local estimate of the spectral density to a global estimate and Preuß, Vetter, and Dette (2013) proposed a test for stationarity using empirical process theory.…”