Assessing Serial Dependence in Ordinal Patterns Processes Using Chi-squared Tests with Application to EEG Data Analysis

Abstract

We extend Elsinger’s work on chi-squared tests for independence using ordinal patterns and investigate the general class of m-dependent ordinal patterns processes, to which belong ordinal patterns processes derived from random walk, white noise, and moving average processes. We describe chi-squared asymptotically distributed statistics for such processes that take into account necessary constraints on ordinal patterns probabilities and propose a test for m-dependence, with which we are able to quantify the range of serial dependence in a process. We apply the test to epilepsy electroencephalography time series data and observe shorter m-dependence associated with seizures, suggesting that the range of serial dependence decreases during those events. The ordinal patterns symbolization transforms a real-valued time series into a sequence of symbols called ordinal patterns, which simplifies statistical analysis while keeping information about up and down movements. Despite the increasing interest in its application due to the need to understand complex nonlinear dynamics based on observed time series, analytical properties of the distributions of ordinal patterns frequencies are not yet fully known. By modeling a sequence of ordinal patterns as the output of a symbolic process, we study m-dependent ordinal patterns processes, i.e., symbolic processes in the space of ordinal patterns whose maximum dependence range is m. We derive chi-squared asymptotically distributed statistics for this class of processes and use them to evaluate the range of serial dependence in general ordinal patterns processes. Applying the results to analyze epilepsy electroencephalography (EEG) time series, we find that seizure events are characterized by a decrease in the range of serial dependence of the ordinal dynamics.