*Astron. Astrophys. 351, 405-412 (1999)*
## 4. Method of analysis
We first extract various subsamples of LCRS using the criteria given
in Table 1 for each of the subsample. For each subsample we next
calculate the weight function
(Eq. 11) for all the galaxies in the subsample. In addition the
3-dimensional distribution of galaxies in the sub-sample is converted
into a corresponding 2-dimensional distribution using the steps
outlined in the previous section and we finally have a collection of
*N* galaxies distributed over a region of a plane.
We next choose *M* of these galaxies at random and count the
number of galaxies inside a circle of radius *r* drawn around
each of these *M* randomly chosen galaxies. In determining this
we use a modified version of Eq. (3) where each galaxy in the circle
has an extra weight factor as
calculated in the previous section, i.e.
The different moments of this quantity are averaged over the
*M* galaxies to obtain defined
in Eq. (6) for a range of *q*. The exercise is repeated with
circles of different radii (different values of *r*) to finally
obtain for a large range of
*r*.
It should be noted that the region from which the *M* points
can be chosen at random depends on the size of the circle which we are
considering. For very large values of *r* a large region around
the boundaries of the survey has to be excluded because a circle of
radius *r* drawn around a galaxy in that region will extend
beyond the boundaries of the survey. As a consequence for large values
of *r* we do not have many galaxies which can serve as centers,
while for small values of *r* there are many galaxies which can
serve as centers for circles of radius *r*. For *r* between
to
we use which is of the same order
as the the total number of galaxies available for use as centers. To
estimate the statistical significance of our results at this range of
length-scales we have randomly divided the 60 centers into independent
groups of 20 centers and repeated the analysis for each of these. We
have used the variation in the results from the different subsamples
to estimate the statistical errors for our results on large scales. In
the range we have used
which is only a small fraction of
the total number of galaxies which can possibly serve as centers which
is around 1500. At this range of length-scales it is possible to
choose many independent sets of 100 centers. We have performed the
analysis for a large number of such sets of 100 centers and these have
been used to estimate the mean generalized dimension
and the statistical errors in the
estimated at small scales. For both
the range of length-scales considered we have tried the analysis
making changes in the number of centers and we find that the results
are insensitive to variation in the number of centers used in the
analysis.
The value of the generalized dimension
is determined for a fixed value of
*q* by looking at the scaling behaviour of
as a function of *r* (e.g.
Figs. 4 and 5) We have considered *q* in the range
. In principle we could have
considered arbitrarily large (or small) values of *q* also, but
the fact that there are only a finite number of galaxies in the survey
implies that only a finite number of the moments can have independent
information. This point has been discussed in more detail by Bouchet
et al. (Bouchet et al. 1991).
In addition to the subsamples of galaxies listed in Table 1,
we have also carried out our analysis for mock versions of these
subsamples of galaxies. The mock versions of each subsample contains
the same number of galaxies as the actual subsample. The galaxies in
the mock versions are selected from a homogeneous random distribution
using the same selection function and geometry as the actual
subsample. We have carried out the whole analysis for many different
random realizations of each of the subsamples listed in Table 1.
The main aim of this exercise was to test the reliability of the
method of analysis adopted here.
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999
helpdesk.link@springer.de |