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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.

** Next:** The correlation integral
** Up:** mlbmb
** Previous:** mlbmb
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