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For a -dimensional hypercube, the cumulative
distribution function of the distance between two randomly chosen points
in that cube, and ,
is given by [Smith, 1988]:
The correlation integral is an estimator of , hence
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
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.