An algorithm based on the convergent cross mapping method for the detection of causality in uni-directionally connected chaotic systems

In this paper, we present some improvements to the convergent cross mapping (CCM) algorithm for detecting causality in uni-directionally connected chaotic systems. The basic concept of the CCM algorithm is that the causal influence of system X on system Y appears as mapping of the neighbouring states in the reconstructed d-dimensional manifold, My, to the neighbouring states in the reconstructed d-dimensional manifold, Mx, and this effect is evaluated using the correlation coefficient between the estimated and observed values of Mx. We proposed a composite indicator of causality as the ratio between the correlation coefficient and the Shannon entropy of the distribution of the residuals between the estimated and observed values of Mx. Application of the proposed approach to four master-slave Rössler and Lorenz systems and real-world data showed that the new algorithm allowed a slight increase in capability to reveal the presence and direction of couplings.