Causality detection with matrix-based transfer entropy

Wanqi Zhou, Shujian Yu, Badong Chen

Abstract: Transfer entropy (TE<math><mrow is="true"><mi mathvariant="normal" is="true">TE</mi></mrow></math>) is a powerful tool for analyzing causality between time series and complex systems. However, it faces two key challenges. First, TE<math><mrow is="true"><mi mathvariant="normal" is="true">TE</mi></mrow></math> is often used to quantify the pairwise causal direction; yet, in real-world applications, one is always interested in identifying more complex causal relationships, such as indirect causation, common causation, and synergistic effect. Second, the estimation of TE usually relies on probability estimation, which is particularly complicated, or even infeasible for high-dimensional data. In this work, we take TE<math><mrow is="true"><mi mathvariant="normal" is="true">TE</mi></mrow></math> one step further and develop a pair of measures, the matrix-based conditional transfer entropy (CTEM<math><mrow is="true"><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">CTE</mi></mrow><mrow is="true"><mi mathvariant="normal" is="true">M</mi></mrow></msub></mrow></math>) and the matrix-based high-order transfer entropy (HTEM<math><mrow is="true"><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">HTE</mi></mrow><mrow is="true"><mi mathvariant="normal" is="true">M</mi></mrow></msub></mrow></math>). The former can detect both indirect and common causation, while the latter can detect synergistic effect. Making use of the recently proposed matrix-based Rényi’s α<math><mrow is="true"><mi is="true">α</mi></mrow></math>-order entropy functional, CTEM<math><mrow is="true"><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">CTE</mi></mrow><mrow is="true"><mi mathvariant="normal" is="true">M</mi></mrow></msub></mrow></math> and HTEM<math><mrow is="true"><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">HTE</mi></mrow><mrow is="true"><mi mathvariant="normal" is="true">M</mi></mrow></msub></mrow></math> are defined on the eigenspectrum of a normalized Hermitian matrix of the projected data in kernel space, which avoids the necessity of density estimation and the curse of dimensionality. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of our measures in high-dimensional space, and their superiority in recovering complex causal structures for more than two time series.