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# The expected value of nearest neighbour distances

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 non-integer) -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]:
 (3.2)

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
 (3.5)

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 non-decreasing. 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].

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