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. (
