Convergent cross mapping
Convergent cross mapping (CCM) is a
Theory
In the event one has access to system variables as
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:
- If needed, create the state space manifold from
- Define a sequence of library subset sizes ranging from a small fraction of to close to .
- Define a number of ensembles to evaluate at each library size.
- At each library subset size :
- For ensembles:
- Randomly select state space vectors from
- Estimate from the random subset of using the Simplex state space prediction
- Compute the correlation between and
- Compute the mean correlation over the ensembles at
- For ensembles:
- The spectrum of versus must exhibit convergence.
- 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:
See also
References
- S2CID 19749064.
- PMID 21483839.
- ISBN 978-3-319-58895-7, retrieved 2023-10-19
- PMID 26435402.
- S2CID 238859361.
Further reading
- Chang, CW., Ushio, M. & Hsieh, Ch. (2017). "Empirical dynamic modeling for beginners". Ecol Res. 32 (6): 785–796. hdl:2433/235326.)
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: CS1 maint: multiple names: authors list (link - Stephan B Munch, Antoine Brias, doi:10.1093/icesjms/fsz209.)
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: CS1 maint: multiple names: authors list (link
External links
Animations: