Source-linked AI summary
Kernel method for nonlinear Granger causality
Daniele Marinazzo, Mario Pellicoro, Sebastiano Stramaglia
TL;DR
The paper addresses the need to detect cause-effect relationships in time series beyond linear models. It generalizes Granger causality with reproducing kernel Hilbert spaces, controls overfitting geometrically, and demonstrates the approach on coupled maps and physiological data.
Problem
Existing Granger causality analyzes cause-effect relationships through linear regression, motivating a method that can handle arbitrary nonlinearities while avoiding false causalities from overfitting.
Method
The method performs linear Granger causality in kernel-induced feature spaces and uses RKHS geometry, statistical filtering, and Bonferroni correction to control overfitting.
Results
The filtered index identifies directional and nonlinear causal structure in coupled maps and physiological signals, including robust L→R causality in rat EEG after lesion and H→B causality in sleep-apnea data.
Takeaways & Limitations
Kernel-based Granger causality extends causal analysis to nonlinear interactions while providing sharper, statistically filtered statements about directional relationships.
Takeaways & Limitations
The transfer-entropy connection relies on the generalized Markov property; under that condition, knowledge of Y does not improve prediction of x.
Abstract
from arXiv · showhide
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.