Causal discovery in symmetric dynamic systems with convergent cross mapping

This paper systematically discusses how the inherent properties of chaotic attractors influence the results of discovering causal relationships from time series using convergent cross mapping, particularly how convergent cross mapping misleads the bidirectional causal relationship as unidirectional when the chaotic attractor exhibits symmetry. We propose a novel method based on the k-means clustering method to address the challenges when the measurement function defines even parity, which may cause this issue. This method is demonstrated to recover the symmetry of the latent chaotic attractor and discover the correct causal links between time series without introducing information from other variables. We validate the accuracy of this method using time series derived from numerical simulations and a real-world system for which convergent cross mapping may conclude erroneous results.