The correlation function can be generalized to higher order
[1,
52,
53]:
The *N*-point correlation
functions. This allows to statistically characterize the galaxy
distribution with a hierarchy of quantities which progressively
provide us with more and more information about the clustering of
the point process. These measures, however, had been difficult to
derive with reliability from the galaxy catalogs. The new
generation of surveys will surely overcome this problem.

There are, nevertheless, other clustering measures which provide
complementary information to the second-order quantities describe
above. For example, the topology of the large-scale structure
measured by the genus statistic
[54] provides
information about the phase correlations of the density fluctuations in
*k*-space. To obtain this quantity, first the point process has to
be smoothed by means of a kernel function with a given bandwidth.
The topological genus of a surface is the number of holes minus
the number of isolated regions plus 1. This quantity is calculated
for the isodensity surfaces of the smoothed data corresponding to
a given density threshold. The genus has an analytically
calculable expression for a Gaussian field
[55].

Minkowski functionals are very effective clustering measures commonly
used in stochastic geometry
[20].
These quantities are adequate to study the shape and connectivity
of a union of convex bodies. They can easily be adapted to
point processes
[56]
by considering the covering of the point field formed by sets
*A*_{r} =
_{i=1}^{N}
*B*_{r}(**x**_{i}) where
*r* is the diagnostic parameter,
{**x**_{i}}_{i = 1}^{N}
represents the galaxy positions, and
*B*_{r}(**x**_{i}) is a ball
of radius *r* centered at point **x**_{i}. Minkowski
functionals
[57]
are applied to sets *A*_{r} as *r*
increases. In IR^{3} there are four functionals: the volume
*V*, the surface area *A*, the integral mean curvature
*H*, and the Euler-Poincaré characteristic
, related with the genus
of the boundary of *A*_{r}.

Several quantities based on distances to nearest neighbors have
been used in the cosmological literature. The empty space function
*F*(*r*) is the distribution function of the distance between a
given random test particle in IR^{3} and its nearest galaxy.
It is related with the void probability function
[58]
*P*_{0}(*r*)
- the probability that a ball of radius *r* randomly placed
contains no galaxies - by
*F*(*r*) = 1 -
*P*_{0}(*r*). *G*(*r*) is the
distribution function of the distance *r* of a given galaxy to its
nearest neighbor. The quotient
*J*(*r*) = [1 - *G*(*r*) ]/ [1 - *F*(*r*)]
has been successfully applied to describe the spatial pattern interaction
in the galaxy distribution
[59].
Related with the
nearest neighbor distances, the minimal spanning tree is a
structure descriptor that has shown powerful capabilities to
reveal the clustering properties of different point patterns
[60,
46,
61]. The
minimal spanning tree
(MST) is the unique network connecting the *N* points of the
process with a route formed by *N* - 1 edges, without closed loops
and having minimal total length (the total length is
the sum of the lengths of the edges). The frequency histograms of the
MST edge lengths can be used to analyze the galaxy distribution and to
compare it with the simulated models
[61].

The use of wavelets and related integral transforms is an extremely promising tool in the clustering analysis of 3-D catalogs. Some of these techniques are introduced in other contributions published in this volume [62, 63].

**Acknowledgments**

We are grateful to our collaborators Martin Snethlage and Dietrich Stoyan for common results on the correlation function of shifted Cox processes. We thank Valerie de Lapparent, Luigi Guzzo, Jon Loveday, and John Peacock for kindly providing us with some illustrations. This work was supported by the Spanish MCyT project AYA2000-2045 and by the Estonian Science Foundation under grant 2882.