Next: Methods to estimate the
Up: NEAREST NEIGHBOUR METHODS
Previous: Introduction
Contents
For uncorrelated points in phase space, we expect the
probability to find exactly points in a given ball, centered
at a reference point ,with volume and radius to be binomial:

(3.1) 
where is the random variable denoting the number of points in
the ball; it is assumed that the density is constant within each ball.
The (possibly noninteger) dimensional volume equals
where the (fractal) hypersphere has radius 1.
The probability density function of the distance to the th
nearest neighbour can be obtained by [Pettis et al., 1979]:
For large and small (so that will also be small),
the binomial
distribution can be approximated by the Poisson distribution:

(3.3) 
Hence we obtain

(3.4) 
where .
For the expectation of
we write
since the gamma function is defined as [Abramowitz and Stegun, 1970]:

(3.6) 
Nearest neighbour distances will be averaged over a set of reference
points. The expectation of the averaged distance to the
th nearest neighbour is:

(3.7) 
where

(3.8) 
and is the number of reference points.
In general an attractor is inhomogeneous, i.e. the dimension depends
on the reference point, so that will depend on .
We will estimate with

(3.9) 
where

(3.10) 
since
and
for
large .
Here is called the ``correction function'' and it is close to unity
for large ; is independent of and .
is called the
``dimension function'' [Badii and Politi, 1985] and it is nondecreasing.
In this derivation we followed references
[Pettis et al., 1979,Somorjai, 1986,Grassberger, 1985]. A firm basis using the
formalism of local scaling indices is given in [van de Water and Schram, 1988].
Next: Methods to estimate the
Up: NEAREST NEIGHBOUR METHODS
Previous: Introduction
Contents
webmaster@rullf2.xs4all.nl