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Gaussian kernel correlation integral

The correlation sum Eq.(gif) can be regarded as an average density of points where the local density is obtained by a kernel estimator with a step kernel tex2html_wrap_inline7655. A natural modification for small point sets is to replace the sharp step kernel by a smooth kernel function of bandwidth tex2html_wrap_inline6495. A particularly attractive case that has been studied in the literature [80] is given by the Gaussian kernel, that is, tex2html_wrap_inline7655 is replaced by tex2html_wrap_inline7661. The resulting Gaussian kernel correlation sum tex2html_wrap_inline7663 has the same scaling properties as the usual tex2html_wrap_inline7623. It has been observed in [3] that tex2html_wrap_inline7663 can be obtained from tex2html_wrap_inline7623 via
 equation5759
without having to repeat the whole computation. If tex2html_wrap_inline7623 is given at discrete values of tex2html_wrap_inline6495, the integrals in Eq.(gif) can be carried out numerically by interpolating tex2html_wrap_inline7623 with pure power laws . This is done in c2g which uses a 15 point Gauss-Kronrod rule for the numerical integration.

 figure1650
Figure:   Dimension estimation for the (noise filtered) NMR laser data. Embedding dimensions 2 to 7 are shown. From above: (a) slopes are determined by straight line fits to the log-log plot of the correlation sum, Eq. (gif). (b) Takes-Theiler estimator of the same slope. (c) Slopes are obtained by straight line fits to the Gaussian kernel correlation sum, Eq.(gif). (d) Instead of the correlation dimension, it has been attempted to estimate the information dimension.


next up previous
Next: Information dimension Up: Correlation dimension Previous: Takens-Theiler estimator

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