Important information on the structure of complex systems, consisting of more
than one component, can be obtained by measuring to which extent the individual
components exchange information among each other. Such knowledge is needed to
reach a deeper comprehension of phenomena ranging from turbulent fluids to
neural networks, as well as complex physiological signals. The linear Granger
approach, to detect cause-effect relationships between time series, has emerged
in recent years as a leading statistical technique to accomplish this task.
Here we generalize Granger causality to the nonlinear case using the theory of
reproducing kernel Hilbert spaces. Our method performs linear Granger causality
in the feature space of suitable kernel functions, assuming arbitrary degree of
nonlinearity. We develop a new strategy to cope with the problem of
overfitting, based on the geometry of reproducing kernel Hilbert spaces.
Applications to coupled chaotic maps and physiological data sets are presented.Comment: Revised version, accepted for publication on Physical Review Letter
