Next: Methods for estimating the
Up: THE CORRELATION INTEGRAL
Previous: Introduction
Contents
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).
Next: Methods for estimating the
Up: THE CORRELATION INTEGRAL
Previous: Introduction
Contents
webmaster@rullf2.xs4all.nl