Belief Tsallis-Deng Structure Entropy and its uniform framework for analyzing multivariate time-series complexity based on evidence theory

In the framework of evidence theory, this paper proposes a new measure of uncertainty and introduces an entirely new unified framework for analyzing the complexity of multivariate time series. The theoretical foundation of this paper is random sets, viewing multivariate sequences as a new type of set variable, and using the belief theory framework to describe their behavior. Set variables are a natural extension of point variables, thus providing stronger universality in the analysis of multivariate time series. We tested our proposed new analytical framework on the logistic map and used the Tsallis-Deng Structure Entropy introduced in this paper to analyze the generated multivariate time series. Compared to the traditional Deng entropy, Tsallis-Deng Structure Entropy can adjust the sensitivity to time series complexity by tuning its structural parameter p. As the parameters of the logistic map change, the changes in Tsallis-Deng Structure Entropy show a high correlation with the Lyapunov exponent. In addition, we applied the proposed framework to analyze a multivariate epileptic EEG dataset. The results showed that using the novel evidence theory-based analysis framework for the 23-channel multivariate time series, the average classification accuracies of Deng entropy, Tsallis-Deng entropy, and Tsallis-Deng structural entropy for 24 epilepsy patients were 79 %, 74 %, and 82 %, respectively. In contrast, the average recognition accuracy using the traditional probabilistic framework based on multivariate embedding theory was 64 %. The evidence theory-based framework generally outperformed the multivariate embedding theory, with Tsallis-Deng structural entropy achieving the best performance. These results indicate that the proposed evidence theory-based framework not only avoids the dimensionality curse of the multivariate embedding theory but also better quantifies the complexity of multivariate time series from the same system and distinguishes different types of multivariate time series.