Testing for Structural Breaks via Ordinal Pattern Dependence

Abstract

We propose new concepts in order to analyze and model the dependence structure between two time series. Our methods rely exclusively on the order structure of the data points. Hence, the methods are stable under monotone transformations of the time series and robust against small perturbations or measurement errors. Ordinal pattern dependence can be characterized by four parameters. We propose estimators for these parameters, and we calculate their asymptotic distributions. Furthermore, we derive a test for structural breaks within the dependence structure. All results are supplemented by simulation studies and empirical examples.For three consecutive data points attaining different values, there are six possibilities how their values can be ordered. These possibilities are called ordinal patterns. Our first idea is simply to count the number of coincidences of patterns in both time series, and to compare this with the expected number in the case of independence. If we detect a lot of coincident patterns, this means that the up-and-down behavior is similar. Hence, our concept can be seen as a way to measure non-linear 'correlation'. We show in the last section, how to generalize the concept in order to capture various other kinds of dependence. The authors gratefully acknowledge financial support of the DFG (German science Foundation) SFB 823: Statistical modeling of nonlinear dynamic processes (projects C3 and C5).

Change-Point Detection Using the Conditional Entropy of Ordinal Patterns

“…The time points where the signals are glued correspond to abrupt changes in the properties of the process and are called change-points . The first ideas of using ordinal patterns for detecting change-points were formulated in [ 23 , 24 , 34 , 36 , 37 , 38 ]. The advantage of the ordinal-patterns-based methods is that they require less information than most of the existing methods for change-point detection: it is assumed that the stochastic process is not from a specific family and that the change does not affect specific characteristics of the process.…”

Ordinal pattern dependence between hydrological time series

“…Here, we suggest a simple approach to describe correlation between time series by ordinal patterns ( [15], [16]), where the probability measures do not need to have second moments. The time series we consider do not have to be stationary.…”

“…Let us recall some of the advantages of the method which have been emphasized in [16]: the whole analysis is stable under monotone transformations of the state space. The ordinal structure is not destroyed by measurement errors or small perturbations of the data.…”

Ordinal patterns in clusters of subsequent extremes of regularly varying time series

“…Unakafov and Keller 2018). Dealing with 'correlated' time series, one could analyze the dependence between extreme events in a non-linear fashion as it has been developed in Schnurr (2014) and Schnurr and Dehling (2017). This might be advantageous in particular if the time series are on totally different scales.…”