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Our numerical results from the simulated data indicate that the expressions for
the asymptotic variances are reliable.
In combination with mathematical models for which the correlation
dimension and entropy are known theoretically these expressions can be used to
identify systematic errors and limitations of the dimension
and entropy estimators in general;
in applications they can be useful to identify chaos.
To obtain the most precise estimates for the correlation dimension and
the correlation entropy,
the maximum likelihood method suggests
to choose the largest possible values for and for .
We remark that our expressions are also valid
if the scaling regions are not the same at different embedding dimensions,
which occurs if one uses the Euclidean norm.
A potential disadvantage of this maximum likelihood approach
is that the distances used must be independent.
This is almost unavoidable in order to have a simple enough likelihood
function.
The results obtained by applying the derived estimators to
the electrogram recorded from the atrium of a conscious dog suggest that
some types of atrial fibrillation may be characterized
by low-dimensional chaotic dynamics.
As aptly pointed out by Ruelle (1990) there
is a real danger that
the present methods for detecting chaos are applied beyond their
domain of validity. However, in our application to the atrial
fibrillation data the time series and the scaling region seem long enough
for the estimation of a low value of the correlation dimension.
Of course to draw firm
conclusions about the dynamics of experimental time series in general
and atrial fibrillation in particular much more
work has to be done.

We are greatly indebted to Professors R.D. Gill and
W.R. van Zwet for suggestions and reading the manuscript
and to Prof. M.A. Allessie and his group for providing the electrograms.

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