Testing nonlinear Markovian hypotheses in dynamical systems

We present a statistical approach for detecting the Markovian character of dynamical systems by analyzing their flow of information. Especially in the presence of noise which is mostly the case for real-world time series, the calculation of the information flow of the underlying system via the concept of symbolic dynamics is rather problematic since one has to use infinitesimal partitions. We circumvent this difficulty by measuring the information flow indirectly. More precisely, we calculate a measure based on higher order cumulants which quantifies the statistical dependencies between the past values of the time series and the point r steps ahead. As an extension of Theiler's method of surrogate data (Theiler et al., 1992) this cumulant based information flow (a function of the look-ahead r) is used as the discriminating statistic in testing the observed dynamics against a hierarchy of null hypotheses corresponding to nonlinear Markov processes of increasing order. This procedure is iterative in the sense that whenever a null hypothesis is rejected new data sets can be generated corresponding to better approximations of the original process in terms of information flow. Since we use higher order cumulants for calculating the discriminating statistic our method is also applicable to small data sets. Numerical results on artificial and real-world examples including non-chaotic, nonlinear processes, autoregressive models and noisy chaos show the effectiveness of our approach.