Date: Feb. 27, 2020
Causality detection in deterministic systems: Cross convergent mapping
Granger uses the nature of predictability to distinguish causal relations between time-series variables from correlations. That is, variable X is said to cause Y in Granger’s definition if given all the information in the universe, the predictability of Y decreases when X is removed from the universe. However, Granger’s causality test is expected to fail when the time-series variables are from deterministic systems due to its assumption of separability. In stochastic systems, new information is injected by the noise in the cause variable X so the cause variable X and the effect variable Y are separable. In deterministic systems, information about the cause X will be redundantly present in the history of the effect Y itself and cannot formally be removed from the universe.
Alternatively, Sugihara proposes a new definition of causality in deterministic settings. That is, time series variables are causally related if they are from the same dynamic systems (they share a common attractor manifold and each variable can identify the state of the other). To detect such causality in deterministic dynamic systems, Sugihara introduces the convergent cross mapping (CCM) method. Given two time series X and Y, the idea of the method is to see whether the time indices of near-by points on the X manifold can be used to identify the near-by points on the Y manifold. The method generates a cross-mapped estimate of Y from the nearest neighbors in the constructed X shadow manifold. The convergence of the cross-mapped estimates is a necessary condition for causation. In this project, we use the CCM test to detect causality in various dynamic systems and perform sensitivity analysis to the delay time and the dimension of the manifold embedding. Also, we study examples of some CCM test failures.
Discussions
In the Lorenz system, the CCM test is successful in detecting causality among all three times-series variables X, Y and Z The convergence of the cross-mapped estimates for X and Y is expected due to the diffeomorphism between the X and Y shadow manifolds and the Lorenz attractor M. The test works surprisingly well even when there the Z shadow manifold is topologically different from the M_X and M_Y shadow manifolds and not diffeomorphic with M. In the sensitivity analysis, we conclude that the CCM test is fairly robust when it comes to time lag change, despite that the structures of the shadow manifolds are affected. However, the test exhibits a high sensitivity to the embedding dimension. The cross-mapped estimation skills decrease as we increase the dimension. Note that the CCM test can falsely detect causality between two non-coupled time-series variables in a system with external force. Also, the test fails to converge when the time-series data is very noisy.
Code: https://github.com/aprilzhizhou/data_driven_methods/tree/main/cross_convergent_mapping
Full report: