Convergent cross mapping (CCM) is a statistical test for a cause-and-effect relationship between two variables that, like the Granger causality test, seeks to resolve the problem that correlation does not imply causation.
[1] 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.
As such, CCM is specifically aimed to identify linkage between variables that can appear uncorrelated with each other.
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,
, preserving instrinsic state space properties of
Convergent Cross Mapping (CCM) leverages a corollary to the Generalized Takens Theorem[2] 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
belong to the same dynamical system, their reconstructions via embeddings
leaves a signature on the affected variable
CCM leverages this property to infer causality by predicting
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
Cross mapping is generally asymmetric.
The basic steps of convergent cross mapping for a variable
are: CCM is used to detect if two variables belong to the same dynamical system, for example, can past ocean surface temperatures be estimated from the population data over time of sardines or if there is a causal relationship between cosmic rays and global temperatures.
As for the latter it was hypothesised that cosmic rays may impact cloud formation, therefore cloudiness, therefore global temperatures.
[3] Extensions to CCM include: Animations: