A novel adaptive multi-scale Rényi transfer entropy based on kernel density estimation

The utilization of Rényi Transfer Entropy (RTE) as a powerful model for analyzing causal relationships among variables has become pervasive in the field of complex systems time series causality detection. However, there are two major challenges in using RTE: the inherent multilevel structure of causal associations in the systems, and the precision in RTE estimation. To address these challenges, this paper proposes an adaptive multi-scale Rényi transfer entropy based on kernel density estimation. The framework of causal detection based on the novel RTE consists of two parts: adaptive discrete wavelet transform (ADWT)-based time series decomposition and multivariate kernel density estimation (MKDE)-based causality network generation. In the ADWT-based time series decomposition, the original series are decomposed into different frequency bands by optimal wavelet coefficients, which is generated adaptively by an auto-encoder. In the MKDE-based causality network generation, the causal network between the variables is represented by an adjacency matrix composed of their decomposition components in each layer, and the values of the matrix are the RTE values between the variables. In order to accurate estimation of RTE values an evaluation criterion for KDE under a uniform measure in both univariate and multivariate cases and the optimal bandwidth selection is provided in this part. To validate the effectiveness of the novel causal measure in this paper, the proposed method is tested on the synthetic and real data, and the results show that it can effectively detect causal relationships among variables at different levels in non-stationary time series of both bidirectional and undirectional complex systems. Compared to the other RTE estimators, the proposed method can detect the causality accurately and avoid the spurious causality.