Convergent Cross-Mapping and Pairwise Asymmetric Inference
  Convergent Cross-Mapping (CCM) is a technique for computing specific kinds of correlations between sets of times series. It was introduced by Sugihara et al. and is reported to be "a necessary condition for causation" capable of distinguishing causality from standard correlation. We show that the relationships between CCM correlations proposed in \cite{Sugihara2012} do not, in general, agree with intuitive concepts of "driving", and as such, should not be considered indicative of causality. It is shown that CCM causality analysis implies causality is a function of system parameters for simple linear and nonlinear systems. For example, in a RL circuit, both voltage and current can be identified as the driver depending on the frequency of the source voltage. It is shown that CCM causality analysis can, however, be modified to identify asymmetric relationships between pairs of time series that are consistent with intuition for the considered example systems for which CCM causality analysis provided non-intuitive driver identifications. This modification of the CCM causality analysis is introduced as "pairwise asymmetric inference" (PAI) and examples of its use are presented. 
James M. McCracken, Robert S. Weigel
physics.data-an