A Test for Non-Stationarity of Time-Series

Abstract

We consider the problem of testing a given time-series for stationarity. The approach is based on evolutionary spectral analysis, and the proposed method consists essentially in testing the "homogeneity" of a set of evolutionary spectra evaluated at different instants of time.Using a logarithmic transformation, we show that the mechanics of the test are formally equivalent to a two-factor analysis of variance procedure when the residual variance is known, a priori.In addition to testing stationarity, the analysis provides also a method for testing whether the observed series fits a "uniformly modulated" model, and a test for "randomness" (constancy of spectra).

A study on the Gaussianity and stationarity of the random noise in the seismic exploration

“…A nonstationarity test makes use of the concept of the evolutionary spectrum of a non-stationary process, and the basis of the method consists essentially in testing the "uniformity" of a set of evolutionary spectra evaluated at different instants of time. (Priestley and Subba Rao, 1968). …”

A formal test for nonstationarity of spatial stochastic processes

“…This test is a generalization of the test for stationarity of time series presented by Priestley and Rao [23] to spatial processes. We first estimate the spatial spectral density f s i ( ) at m nodes s 1 , .…”

“…Then, the expected value of the smoothed spatial periodogram,f x ( ), for ∈ [− / , / ] 2 , is approximately f x ( ). The bandwidth of | ( )| 2 is related to the magnitude of 1/B g and it can be approximated by the distance between the "half-power" points on the main lobe of | ( 1 )|| 2 [23]. If the bandwidth of |W (s)| 2 is small compared with the "space-domain bandwidth" of f x ( ), Then,f 2 x ( ), is approximately f 2 x ( ), and the asymptotic variance off x ( ) is…”

“…Several authors have proposed methods for testing whether or not a given time-series may be regarded as stationary, some of these methods are designed to detect nonstationary "trends". Priestley and Rao [23] used the concept of the "evolutionary (time-dependent) spectrum" to test the lack of stationarity of a time series, and this is the key idea for the work developed in this article. The literature does not offer many approaches to test for nonstationarity of spatial processes.…”

Fitting dynamic factor models to non-stationary time series

“…The second test that we considered has been proposed by Priestley and Subba Rao (1969) and takes into account the full dynamic structure of the process. The test is based on a factorization of the time-varying spectrum into two components:…”