Convergent cross mapping

Theory

In the event one has access to system variables as time series observations, Takens' embedding theorem can be applied. Takens' theorem generically proves that the state space of a dynamical system can be reconstructed from a single observed time series of the system, ${\displaystyle X}$. This reconstructed or shadow manifold ${\displaystyle M_X}$ is diffeomorphic to the true manifold, ${\displaystyle M}$, preserving instrinsic state space properties of ${\displaystyle M}$ in ${\displaystyle M_X}$.

Convergent Cross Mapping (CCM) leverages a corollary to the Generalized Takens Theorem that it should be possible to cross predict or cross map between variables observed from the same system. Suppose that in some dynamical system involving variables ${\displaystyle X}$ and ${\displaystyle Y}$, ${\displaystyle X}$ causes ${\displaystyle Y}$. Since ${\displaystyle X}$ and ${\displaystyle Y}$ belong to the same dynamical system, their reconstructions via embeddings ${\displaystyle M_X}$ and ${\displaystyle M_Y}$, also map to the same system.

The causal variable ${\displaystyle X}$ leaves a signature on the affected variable ${\displaystyle Y}$, and consequently, the reconstructed states based on ${\displaystyle Y}$ can be used to cross predict values of ${\displaystyle X}$. CCM leverages this property to infer causality by predicting ${\displaystyle X}$ using the ${\displaystyle M_Y}$ library of points (or vice-versa for the other direction of causality), while assessing improvements in cross map predictability as larger and larger random samplings of ${\displaystyle M_Y}$ are used. If the prediction skill of ${\displaystyle X}$ increases and saturates as the entire ${\displaystyle M_Y}$ is used, this provides evidence that ${\displaystyle X}$ is causally influencing ${\displaystyle Y}$.

Cross mapping is generally asymmetric. 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 ${\displaystyle M_X}$.