Convergent cross mapping (CCM) is a statistical test for a cause-and-effect relationship between two time series variables that, like the Granger causality test, seeks to resolve the problem that correlation does not imply causation. While Granger causality is best suited for purely stochastic systems where the influences of the causal variables are separable (independent of each other), CCM is based on the theory of dynamical systems and can be applied to systems where causal variables have synergistic effects. The test was developed in 2012 by the lab of George Sugihara of the Scripps Institution of Oceanography, La Jolla, California, USA.
Contents
Theory
Convergent cross mapping is based on Takens' embedding theorem, which states that generically the attractor manifold of a dynamical system can be reconstructed from a single observation variable of the system,
Cross mapping need not be symmetric. If
Algorithm
The basic steps of the convergent cross mapping test according to
- Create the shadow manifold for
X , calledM X - Find the nearest neighbors to a point in the shadow manifold at time t
- Create weights using the nearest neighbors (e.g. using K-nearest neighbors algorithm)
- Estimate Y using the weights; (this estimate is called
Y |M X - Compute the correlation between
Y andY |M X