Detecting non-linear dynamics in financial time series

Characterizes a given dynamical system in the sense that the order of the Markov process which approximates its statistical dependencies appropriately, is determined. Our method measures the information flow of the dynamics indirectly via higher-order cumulants, considering linear and nonlinear correlations. The main point of the iterative procedure is that the raw data are tested against a hierarchy of nonlinear Markov processes, which correspond to the hypotheses in the surrogate mechanism. We extend the method of surrogate data in two ways to study the information flow in dynamical systems. First, we test the observable dynamics against a hierarchy of null hypotheses corresponding to nonlinear Markov processes of increasing order, the probability density function of which is estimated by neural networks. Second, the discriminating statistic is not a single number but a function of the look-ahead r. More precisely, we calculate a measure based on higher-order cumulants which quantifies the independence between the past values of the time series and the point r steps ahead. 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 memory. We define cumulant-based measures of statistical independence which characterize the loss of information with the look-ahead. The iterative procedure for testing against nonlinear Markov processes is explained and our experiments with a DAX-time series are described