Inferring causal influences from expansive distortions between state space reconstructions

State space reconstructions of nonlinear dynamical system contain within their metric and topological properties information about the causal influences between different observables. The expansive distortions among different observables not only reflect the directed coupling strengths, but also the dependency of effective influences on the systems temporally varying state. Estimation of expansions from pairs of time series is straightforward, either directly from intra neighborhood relations or the mapping between reconstructions. Two approaches to compute expansive distortions are demonstrated using analytical and numerical analysis in a range of complex dynamical systems. The biggest challenge for the inference of causal influences is reached in synchronising systems or system perturbed by large amounts of noise. Remarkably, expansive distortions no only give in sight into just the interaction scheme, but provide a time-dependent measure for these interaction. These new methods offer a potential tool to gain insight into interactions of (nonlinear) dynamical system for a wide range of disciplines.