Quantifying interdependence using the missing joint ordinal patterns.

Abstract

In this paper, we develop the concept of forbidden/missing ordinal patterns into the forbidden/missing joint ordinal patterns and propose the ratio of the number of missing joint ordinal patterns (RMJPs) as a sign of interdependence. RMJP in a surrogate analysis can be used to differentiate the forbidden joint ordinal patterns from the missing joint ordinal patterns due to small sample effects. We first apply RMJP to the simulated time series: a two-component autoregressive fractionally integrated moving average process, the Hénon map, and the Rössler system using active control and discuss the effect of the length of the time series, embedding dimension, and noise contamination. RMJP has been proven to be capable of measuring the interdependence in the numerical simulation. Then, RMJP is further used on the electroencephalogram (EEG) time series for empirical analysis to explore the interdependence of brain waves. With results by RMJP obtained from a widely used open dataset of the sleep EEG time series from healthy subjects, we find that RMJP can be used to quantify the brain wave interdependence under different sleep/wake stages, reveal the overall sleep architecture, and indicate a higher level of interdependence as sleep gets deeper. The findings are consistent with existing knowledge in sleep medicine. The proposed RMJP method has shown its validity and applicability and may assist automatic sleep quantification or bring insight into the understanding of the brain activity during sleep. Furthermore, RMJP can be used on sleep EEG under various pathological conditions and in large-scale sleep studies, helping to investigate the mechanisms of the sleep process and neuron synchronization.