   Next: Entropy estimates Up: Dimensions and entropies Previous: Gaussian kernel correlation integral

## Information dimension

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  circumvents these problems, so that, including finite sample corrections , 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 . 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.   Next: Entropy estimates Up: Dimensions and entropies Previous: Gaussian kernel correlation integral

Thomas Schreiber
Wed Jan 6 15:38:27 CET 1999