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Astron. Astrophys. 351, 405-412 (1999)

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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 [FORMULA] (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 [FORMULA] as calculated in the previous section, i.e.

[EQUATION]

The different moments of this quantity are averaged over the M galaxies to obtain [FORMULA] 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 [FORMULA] 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 [FORMULA] to [FORMULA] we use [FORMULA] 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 [FORMULA] we have used [FORMULA] 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 [FORMULA] and the statistical errors in the estimated [FORMULA] 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 [FORMULA] is determined for a fixed value of q by looking at the scaling behaviour of [FORMULA] as a function of r (e.g. Figs. 4 and 5) We have considered q in the range [FORMULA]. 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.

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© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999
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