Lumped permutation entropy: A robust complexity measure on noisy time series under state transitions

Abstract

Measuring complexity from non-stationary time series provides an important clue to the understanding of dynamic patterns of a given dynamical system. One of the widely used complexity measures is the permutation entropy, which quantifies the entropy of the symbolic patterns of the observed time series. In this study, we propose the lumped permutation entropy (LPE) which provides a robust complexity measure that helps to overcome some limitations of previous algorithms under noisy environments. LPE allows a joint rank on the pattern formation, enhancing the robustness over strong background noise. The performance of LPE is demonstrated for chaotic time series from some dynamical models and empirical electroencephalogram (EEG) data from anesthetic studies. In particular, our results show that LPE of the EEG complexity is anti-correlated with the concentration of anesthetics throughout the anesthesia phase, allowing the quantitative monitoring of the state transitions of the brain through the entropic complexity of EEG patterns. In summary, the lumped permutation entropy (LPE) can be useful for measuring and monitoring the complexity of dynamical systems under strong noise such as EEGs with continuous state transitions.