where is the sample time, is the embedding dimension and is an appropriate lag. The correlation integral is defined as

where denotes a norm. It is assumed that distances between two randomly chosen points in phase space obey the cumulative probability distribution function [Grassberger and Procaccia, 1983b]

and that for sufficiently small and large . We also assume that and do not depend on (see, however, Theiler, 1988) and that does not depend on the embedding dimension. The correlation dimension is now given by [Eckmann and Ruelle, 1985]

(4) |

With experimental data,
a ``scaling region'' must be identified,
for which the distribution (3) is valid. The
dimension can be estimated by the slope of a straight
line through a number of points
in the scaling region.
The entropy can be estimated by observing the behaviour of
as a function of but using equation (5)
directly may result in slow convergence. Therefore one usually studies
the quotient of two correlation integrals, at embedding dimensions
and [Grassberger and Procaccia, 1983a] :