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, X. This reconstructed or shadow manifold M_X is diffeomorphic to the true manifold, M, preserving instrinsic state space properties of M in 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 X and Y, X causes Y. Since X and Y belong to the same dynamical system, their reconstructions via embeddings M_{X} and M_{Y}, also map to the same system.

The causal variable X leaves a signature on the affected variable Y, and consequently, the reconstructed states based on Y can be used to cross predict values of X. CCM leverages this property to infer causality by predicting X using the 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 M_{Y} are used. If the prediction skill of X increases and saturates as the entire M_{Y} is used, this provides evidence that X is causally influencing Y.

Cross mapping is generally asymmetric. 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.

Algorithm

The basic steps of convergent cross mapping for a variable X of length N against variable Y are:

1. If needed, create the state space manifold M_Y from Y
2. Define a sequence of library subset sizes L ranging from a small fraction of N to close to N.
3. Define a number of ensembles N_E to evaluate at each library size.
4. At each library subset size L_i:
5. For N_E ensembles: 1. Randomly select L_i state space vectors from M_Y 1. Estimate \hat{X} from the random subset of M_Y using the Simplex state space prediction 1. Compute the correlation \rho between \hat{X} and X
6. Compute the mean correlation \bar{\rho} over the N_E ensembles at L_i
7. The spectrum of \bar{\rho} versus L must exhibit convergence.
8. Assess significance. One technique is to compare \bar{\rho} to \bar{\rho_S} computed from S random realizations (surrogates) of X.

Applications

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.

Extensions

Extensions to CCM include:

• Extended Convergent Cross Mapping
• Convergent Cross Sorting

References

References

1. (2012). "Detecting Causality in Complex Ecosystems". Science.
2. (2011). "Generalized Theorems for Nonlinear State Space Reconstruction". PLOS ONE.
3. Tsonis, Anastasios A.. (2018). "Convergent Cross Mapping: Theory and an Example". Springer International Publishing.
4. (2015). "Distinguishing time-delayed causal interactions using convergent cross mapping". Scientific Reports.
5. (2021). "Convergent cross sorting for estimating dynamic coupling". Scientific Reports.