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The estimation of the dimension

The correlation dimension has been defined by (see Chapter 4):
\nu = \lim_{r \rightarrow 0} \lim_{N \rightarrow \infty}
\frac{\ln C(r,N)} {\ln (r)}
\end{displaymath} (5.3)

for a sufficiently large embedding dimension. It is assumed that $C(r) \approx P(r)$ (eq. (5.1)) for small $r$ [Grassberger and Procaccia, 1984]. In the following subsections we derive maximum likelihood estimators of the correlation dimension.