Another way of attaching weight to -balls, which is more natural, is
the probability itself. The resulting scaling exponent is called the
information dimension . Since the Kaplan-Yorke dimension of
Sec. is an approximation of , the computation of
through scaling properties is a relevant cross-check for highly deterministic
data. can be computed from a modified correlation sum, where, however,
unpleasant systematic errors occur. The *fixed mass*
approach [81] circumvents these problems, so that, including
finite sample corrections [77], a rather robust estimator
exists. Instead of counting the number of points in a ball one asks here for
the diameter which a ball must have to contain a certain number *k*
of points when a time series of length *N* is given. Its scaling with *k* and
*N* yields the dimension in the limit of small length scales by

The routine c1 computes the (geometric) mean length scale
for which *k* neighbors are found in *N* data
points, as a function of *k*/*N*. Unlike the correlation sum, finite sample
corrections are necessary if *k* is small [77]. Essentially, the
of *k* has to be replaced by the digamma function . The
resulting expression is implemented in c1. Given *m* and , the
routine varies *k* and *N* such that the largest reasonable range of *k*/*N* is
covered with moderate computational effort. This means that for (default: *K*=100), all *N* available points are searched for neighbors
and *k* is varied. For , *k*=*K* is kept fixed and *N* is
decreased. The result for the NMR laser data is shown in
Fig. **(d)**, where a nice scaling with can
be discerned. For comparability, the logarithmic derivative of *k*/*N* is plotted
versus and not vice versa, although *k*/*N*
is the independent variable. One easily detects again the violations of
scaling discussed before: Cut-off on the large scales, noise on small scales,
fluctuations on even smaller scales, and a scaling range in between. In this
example, is close to , and multifractality cannot be established
positively.

Wed Jan 6 15:38:27 CET 1999