Determining the Stationarity Distance via a Reversible Stochastic Process

The problem of controlling stationarity involves an important aspect of forecasting, in which a time series is analyzed in terms of levels or differences. In the literature, non-parametric stationary tests, such as the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests, have been shown to be very important; however, they are affected by problems with the reliability of lag and sample size selection. To date, no theoretical criterion has been proposed for the lag-length selection for tests of the null hypothesis of stationarity. Their use should be avoided, even for the purpose of so-called 'confirmation'. The aim of this study is to introduce a new method that measures the distance by obtaining each numerical series from its own time-reversed series. This distance is based on a novel stationary ergodic process, in which the stationary series has reversible symmetric features, and is calculated using the Dynamic Time-warping (DTW) algorithm in a self-correlation procedure. Furthermore, to establish a stronger statistical foundation for this method, the F-test is used as a statistical control and is a suggestion for future statistical research on resolving the problem of a sample of limited size being introduced. Finally, as described in the theoretical and experimental documentation, this distance indicates the degree of non-stationarity of the times series.