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Conclusions

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 $r_{u}$ and for $e$. 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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Next: Bibliography Up: mlbmb Previous: An application
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