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# The original definition and some modifications

Grassberger and Procaccia defined the correlation integral as [Grassberger and Procaccia, 1983b]:
 (4.1)

where denote -dimensional vectors (points in phase space) constructed from a time series using the method of delays (see eq. (2.6)), denotes the number of points and denotes a norm.

The correlation dimension (usually denoted by instead of ) is then defined by the limit [Eckmann and Ruelle, 1985]:

 (4.2)

for a sufficiently large embedding dimension (see section 2.5). The Renyi dimension spectrum can be assessed for integer by using -tuples of points [Grassberger and Procaccia, 1984]. For any it can be assessed by averaging, using reference points [Pawelzik and Schuster, 1987,Schuster, 1988]. This generalization can be written as:
 (4.3)

where the counting of the distances is symbolized by the Heaviside step function , is the number of reference points and is the number of distances calculated for every reference point. The Heaviside step function is defined by:
 (4.4)

To avoid spurious correlations, two points should not be too close in time [Theiler, 1986]. Therefore, let , where the number is called the Theiler correction (see also section 6.5).

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