The correlation sum Eq.() can be regarded as an average density
of points where the local density is obtained by a kernel estimator with a step
kernel . A natural modification for small point sets is to
replace the sharp step kernel by a smooth kernel function of *bandwidth*
. A particularly attractive case that has been studied in the
literature [80] is given by the Gaussian kernel, that is,
is replaced by . The
resulting Gaussian kernel correlation sum
has the same scaling properties as the usual . It has been
observed in [3] that can be
obtained from via

without having to repeat the whole computation. If is given
at discrete values of , the integrals in Eq.() can be
carried out numerically by interpolating with pure power laws
. This is done in
c2g which uses a 15 point Gauss-Kronrod rule for the numerical integration.

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. ().(b)Takes-Theiler estimator of the same slope.(c)Slopes are obtained by straight line fits to the Gaussian kernel correlation sum, Eq.().(d)Instead of the correlation dimension, it has been attempted to estimate the information dimension.

Wed Jan 6 15:38:27 CET 1999