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Introduction

At present there is a considerable interest in analysing experimental time series using methods from nonlinear dynamical systems theory. For a quantitative characterization of dynamical systems from a measured signal, algorithms have been developed to estimate the dimension spectrum and the generalized Kolmogorov entropies [Broggi, 1988,Schuster, 1988]. These quantities can be used to differentiate between (quasi-) periodic, chaotic and stochastic processes. Time series from living organisms usually show irregular behaviour. It is interesting to determine whether the underlying dynamics is low-dimensional chaotic, because then it can in principle be modelled by a small set of deterministic nonlinear differential equations, implying that the irregularities are an intrinsic part of the process. Examples could be the brain activity and the functioning of the heart in both health and disease. The correlation integral method [Grassberger and Procaccia, 1983b,Takens, 1983] is widely used to estimate the correlation dimension and entropy. A sufficient condition for chaos is that the correlation entropy is positive. Thus to identify chaos, we need a measure of the uncertainty with which the correlation entropy can be estimated. Although attention has been given to the statistical error of estimators of the correlation dimension [Denker and Keller, 1986,Ramsey and Yuan, 1989,Abraham et al., 1990,Theiler, 1990a,Theiler, 1990b], this is not so for estimators of the correlation entropy.

This paper is organized as follows. In section 2, we describe the correlation integral method. In section 3 we will derive estimators of the correlation dimension and entropy together with expressions of their uncertainties using the maximum likelihood approach. We demonstrate their validity by Monte Carlo simulations in section 4. An example of the usage of the expressions is given in section 5, for a recording of atrial fibrillation. Finally, some conclusions are given in section 6.


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Next: The correlation integral Up: mlbmb Previous: mlbmb
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