Convergent cross mapping

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 fundamental idea of this test was first published by Cenys et al. in 1991 and used in a series of statistical approaches (see for example,). It was then further elaborated in 2012 by the lab of George Sugihara of the Scripps Institution of Oceanography. 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 fundamental idea of this test was first published by Cenys et al. in 1991 and used in a series of statistical approaches (see for example,). It was then further elaborated in 2012 by the lab of George Sugihara of the Scripps Institution of Oceanography. 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, X {displaystyle X} . This reconstructed or shadow attractor M X {displaystyle M_{X}} is diffeomorphic (has a one-to-one mapping) to the true manifold, M {displaystyle M} . Consequently, if two variables X and Y belong to the same dynamics system, the shadow manifolds M X {displaystyle M_{X}} and M Y {displaystyle M_{Y}} will also be diffeomorphic. Time points that are nearby on the manifold M X {displaystyle M_{X}} will also be nearby on M Y {displaystyle M_{Y}} . Therefore, the current state of variable Y {displaystyle Y} can be predicted based on M X {displaystyle M_{X}} . Cross mapping need not be symmetric. If X {displaystyle X} forces Y {displaystyle Y} unidirectionally, variable Y {displaystyle Y} will contain information about X {displaystyle X} , but not vice versa. Consequently, the state of X {displaystyle X} can be predicted from M Y {displaystyle M_{Y}} , but Y {displaystyle Y} will not be predictable from M X {displaystyle M_{X}} . The basic steps of the convergent cross mapping test according to