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The ``hypercube''

For a $M$-dimensional hypercube, the cumulative distribution function of the distance $r$ between two randomly chosen points in that cube, $x$ and $y$, is given by [Smith, 1988]:
P(\parallel x-y \parallel \; \leq r) = {[r (2-r)]}^M
\quad (\approx 2^M r^M \mbox{ for small r})
\end{displaymath} (4.8)

The correlation integral $C(r,N)$ is an estimator of $P(r)$, hence
\nu(r) = \frac{d \ln (C(r))} {d \ln (r)} =
M \left( 1 - \frac{r}{(2-r)} \right) \quad (\approx M \mbox{ for small r})
\end{displaymath} (4.9)

From this equation, we observe that we will underestimate the dimension. This systematic error is due to edge effects. For instance, the dimension will be 75% of its ``true value'' at $r=0.4$. Note that this example resembles the estimation of correlation integrals from time series consisting of uncorrelated, uniformly distributed numbers. The effects caused by the finiteness of the length of the time series are addressed in section 6.4.