Driver-Response Relationships cannot be Learned from Synchronization Patterns

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Abstract
We test recent claims that causal (driver/response) relationships can be deduced from interdependencies between simultaneously measured time series. We apply two recently proposed interdependence measures which should give similar results as cross predictabilities used by previous authors. The systems which we study are asymmetrically coupled simple models (Lorenz, Roessler, and Henon models), the couplings being such as to lead to generalized synchronization. If the data were perfect (noisefree, infinitely long), we should be able to detect, at least in some cases which of the coupled systems is the driver and which the response. This is no longer true if the time series has finite length. Instead, estimated interdependencies and mutual cross predictabilities depend strongly on which of the systems has a higher effective dimension at the typical neighborhood sizes used to estimate them. Thus, while interdependencies and cross predictabilities are useful to detect which of the systems is more `active' at a typical level of resolution, they cannot be used to detect causal relationships. We also show that slightly different variants of the interdependence measure can have quite different sensitivities.

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