  To gain insight into complex systems it is a key challenge to infer nonlinear
causal directional relations from observational time-series data. Specifically,
estimating causal relationships between interacting components in large systems
with only short recordings over few temporal observations remains an important,
yet unresolved problem. Here, we introduce a large-scale Nonlinear Granger
Causality (lsNGC) approach for inferring directional, nonlinear, multivariate
causal interactions between system components from short high-dimensional
time-series recordings. By modeling interactions with nonlinear state-space
transformations from limited observational data, lsNGC identifies casual
relations with no explicit a priori assumptions on functional interdependence
between component time-series in a computationally efficient manner.
Additionally, our method provides a mathematical formulation revealing
statistical significance of inferred causal relations. We extensively study the
ability of lsNGC to recovering network structure from two-node to thirty-four
node chaotic time-series systems. Our results suggest that lsNGC captures
meaningful interactions from limited observational data, where it performs
favorably when compared to traditionally used methods. Finally, we demonstrate
the applicability of lsNGC to estimating causality in large, real-world systems
by inferring directional nonlinear, multivariate causal relationships among a
large number of relatively short time-series acquired from functional Magnetic
Resonance Imaging (fMRI) data of the human brain.
