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Information dimension

Another way of attaching weight to tex2html_wrap_inline6495-balls, which is more natural, is the probability tex2html_wrap_inline6569 itself. The resulting scaling exponent is called the information dimension tex2html_wrap_inline7809. Since the Kaplan-Yorke dimension of Sec.gif is an approximation of tex2html_wrap_inline7809, the computation of tex2html_wrap_inline7809 through scaling properties is a relevant cross-check for highly deterministic data. tex2html_wrap_inline7809 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 tex2html_wrap_inline6495 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
 equation5952
The routine c1 computes the (geometric) mean length scale tex2html_wrap_inline7827 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 tex2html_wrap_inline7837 of k has to be replaced by the digamma function tex2html_wrap_inline7841. The resulting expression is implemented in c1. Given m and tex2html_wrap_inline6553, 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 tex2html_wrap_inline7853 (default: K=100), all N available points are searched for neighbors and k is varied. For tex2html_wrap_inline7861, k=K is kept fixed and N is decreased. The result for the NMR laser data is shown in Fig. gif (d), where a nice scaling with tex2html_wrap_inline7867 can be discerned. For comparability, the logarithmic derivative of k/N is plotted versus tex2html_wrap_inline7871 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, tex2html_wrap_inline7809 is close to tex2html_wrap_inline7567, and multifractality cannot be established positively.




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

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