where is and (see section 2.5). For example, is an estimate of in eqs (2.3) and (2.5). From these definitions we see that the correlation integral should scale as both and . For more details, see [Schuster, 1988].

If we calculate correlation integrals for different values of and , we are able to estimate both the dimension spectrum and the entropy spectrum . We assume that the scaling law eq. (4.5) is also valid for finite , and .

In practice,
a log-log plot of the correlation integral vs. the distances for which
it is calculated usually shows three regions:
a linear region where eq. (4.5) holds, the ``scaling region'';
the lower region is distorted
by fluctuations due to noise or the small number of points
and the upper region
is distorted due to the finite size of the attractor.
The slope of the scaling region will equal
if the embedding dimension is large enough (see section 2.5).
The dimension spectrum is usually estimated by fitting straight lines
using a linear squares method. However, the standard expression for
the variance of the slope is invalid
since and
are correlated so that the
variance will be underestimated [Theiler, 1990,Holzfuss and Mayer-Kress, 1986].
The entropy spectrum is given by [Eckmann and Ruelle, 1985]

for some within the scaling region. By increasing , fluctuations in the displacement factor can be reduced. For example, Grassberger and Procaccia [Grassberger and Procaccia, 1983a,Grassberger and Procaccia, 1984] use . It would also be possible to fit straight lines through a number of correlation integrals under the condition that these lines are displaced by the factor containing the entropy.