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.

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, . This reconstructed or shadow manifold is diffeomorphic to the true manifold, , preserving instrinsic state space properties of in .

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 and , causes . Since and belong to the same dynamical system, their reconstructions via embeddings and , also map to the same system.

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

Cross mapping is generally asymmetric. If forces unidirectionally, variable will contain information about , but not vice versa. Consequently, the state of can be predicted from , but will not be predictable from .

Algorithm

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

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

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. [3]

Extensions

Extensions to CCM include:

  • Extended Convergent Cross Mapping[4]
  • Convergent Cross Sorting[5]

See also

References

  1. Sugihara, George; May, Robert; Ye, Hao; Hsieh, Chih-hao; Deyle, Ethan; Fogarty, Michael; Munch, Stephan (2012). "Detecting Causality in Complex Ecosystems". Science. 338 (6106): 496–500. Bibcode:2012Sci...338..496S. doi:10.1126/science.1227079. PMID 22997134. S2CID 19749064.
  2. Deyle, Ethan R.; Sugihara, George (2011). "Generalized Theorems for Nonlinear State Space Reconstruction". PLOS ONE. 6 (3): e18295. Bibcode:2011PLoSO...618295D. doi:10.1371/journal.pone.0018295. PMC 3069082. PMID 21483839.
  3. Tsonis, Anastasios A.; Deyle, Ethan R.; Ye, Hao; Sugihara, George (2018), Tsonis, Anastasios A. (ed.), "Convergent Cross Mapping: Theory and an Example", Advances in Nonlinear Geosciences, Cham: Springer International Publishing, pp. 587–600, doi:10.1007/978-3-319-58895-7_27, ISBN 978-3-319-58895-7, retrieved 2023-10-19
  4. Ye, Hao; Deyle, Ethan R.; Gilarranz, Luis J.; Sugihara, George (2015). "Distinguishing time-delayed causal interactions using convergent cross mapping". Scientific Reports. 5: 14750. Bibcode:2015NatSR...514750Y. doi:10.1038/srep14750. PMC 4592974. PMID 26435402.
  5. Breston, Leo; Leonardis, Eric J.; Quinn, Laleh K.; Tolston, Michael; Wiles, Janet; Chiba, Andrea A. (2021). "Convergent cross sorting for estimating dynamic coupling". Scientific Reports. 11 (1): 20374. Bibcode:2021NatSR..1120374B. doi:10.1038/s41598-021-98864-2. PMC 8514556. PMID 34645847. S2CID 238859361.

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