Performance Analysis of Permutation Entropy and Entropy of Difference applied to EEG Data

Abstract

Objective. Permutation Entropy (PE) is a measure of the complexity of a time series that has been applied to biomedical data, among others to EEG signals to analyse the brain activity at different states, distinguishing for example sleep stages or states of consciousness during anaesthesia. The concept of PE is based on the idea of looking at the order of values in sequences of neighbouring data points and measuring the relative frequencies of the occurring ordinal patterns. Entropy of Difference (EoD) is a similar concept but reduces the number of patterns taken into account by only considering differences between neighbouring data points. More importantly, when the length of these sequences, that is the order of the Entropy, is increased, the number of difference patterns grows much slower than the number of ordinal patterns. The question to be answered is whether EoD can serve as a substitute with shorter calculation time for PE when analysing EEG data. Methods. To quantify the advantage of EoD over PE regarding computational effort, first, the theoretical computational complexity was determined for each of the two parameters. Then, a MATLAB implementation was run on a dataset of EEG recordings of sleeping patients, and the runtimes were measured. To compare the quality in distinguishing different sleep phases, the correlation was measured and boxplots of values of PE and EoD during different sleep stages were produced. The same was done on an EEG data set from patients receiving anaesthesia. Results. For the EoD two pattern encoding algorithms were implemented. Using the MATLAB-optimized version, faster for low Entropy orders, the EoD runtime outperformed the PE runtime from order 3 on and was about twice as fast at order 7. At order 9 memory problems occurred for PE, for EoD at order 17. The second algorithm for encoding difference patterns achieved a runtime that increases linearly with the length of the EEG signal but does not grow with higher Entropy orders. The boxplots did not show a worse ability of the EoD to distinguish sleep phases. However, the analysis is yet to be refined and a conclusion to be drawn.