Improved convergent cross mapping method for causal inference based on decomposition of the Lorenz trajectory

When applying the convergent cross mapping (CCM) algorithm to test the causal relationships among the three variables in the Lorenz equations, no causal association of variables X and Y with variable Z can be detected. The reason for this is that the reconstructed manifold M_Z for variable Z cannot reproduce the complete dynamics of the original Lorenz system, that is, the dynamic behavior of the points on the manifold M_Z and its optimal neighbor points is inconsistent. Accordingly, this paper proposes an improved CCM algorithm known as local dynamic behavior-consistent CCM (LdCCM). The core concept of LdCCM lies in selecting optimal nearest neighbors so as to ensure that any point and its neighbors show consistent local dynamic behavior. Compared with detection using traditional CCM, the LdCCM algorithm demonstrates significantly enhanced performance in identifying causal strength. Notably, there is considerable improvement in detection of the causal influence of variables X and Y on variable Z. Finally, LdCCM can detect causal relationships between atmospheric observation data, which confirms the effectiveness and reliability of the method.