Convergent cross mapping

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, ${\displaystyle X}$. This reconstructed or shadow attractor ${\displaystyle M_{X}}$ is diffeomorphic (has a one-to-one mapping) to the true manifold, ${\displaystyle M}$. Consequently, if two variables X and Y belong to the same dynamics system, the shadow manifolds ${\displaystyle M_{X}}$ and ${\displaystyle M_{Y}}$ will also be diffeomorphic. Time points that are nearby on the manifold ${\displaystyle M_{X}}$ will also be nearby on ${\displaystyle M_{Y}}$. Therefore, the current state of variable ${\displaystyle Y}$ can be predicted based on ${\displaystyle M_{X}}$.

Cross mapping need not be symmetric. If ${\displaystyle X}$ forces ${\displaystyle Y}$ unidirectionally, variable ${\displaystyle Y}$ will contain information about ${\displaystyle X}$, but not vice versa. Consequently, the state of ${\displaystyle X}$ can be predicted from ${\displaystyle M_{Y}}$, but ${\displaystyle Y}$ will not be predictable from ${\displaystyle M_{X}}$.

The basic steps of the convergent cross mapping test according to[8]

1. Create the shadow manifold for ${\displaystyle X}$, called ${\displaystyle M_{X}}$
2. Find the nearest neighbors to a point in the shadow manifold at time t
3. Create weights using the nearest neighbors
4. Estimate Y using the weights; (this estimate is called ${\displaystyle Y}$ | ${\displaystyle M_{X}}$ )
5. Compute the correlation between ${\displaystyle Y}$ and ${\displaystyle Y}$ | ${\displaystyle M_{X}}$

• Demonstrating that the apparent correlation between sardine and anchovy in the California Current is due to shared climate forcing and not direct interaction.[1]

Animations:

• State Space Reconstruction: Time Series and Dynamic Systems on YouTube
• State Space Reconstruction: Takens' Theorem and Shadow Manifolds on YouTube
• State Space Reconstruction: Convergent Cross Mapping on YouTube