convergent cross mapping について

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convergent cross mapping ：ウィキペディア英語版

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 test was developed in 2012 by the lab of George Sugihara of the Scripps Institution of Oceanography, La Jolla, California, USA.〔(Michael Marshall in New Scientist magazine 2884: Causality test could help preserve the natural world, 28 September 2012 )〕
==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,

X

. This reconstructed or shadow attractor

M_X

is diffeomorphic (has a one-to-one mapping) to the true manifold,

M

. Consequently, if two variables X and Y belong to the same dynamics system, the shadow manifolds

M_X

and

M_Y

will also be diffeomorphic (have a one-to-one mapping). Time points that are nearby on the manifold

M_X

will also be nearby on

M_Y

. Therefore, the current state of variable

Y

can be predicted based on

M_X

.
Cross mapping need not be symmetric. If

X

forces

Y

unidirectionally, variable

Y

will contain information about

X

, but not vice versa. Consequently, the state of

X

can be predicted from

M_Y

, but

Y

will not be predictable from

M_X

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