SDSS DR7 superclusters. Principal component analysis
  We apply the principal component analysis and Spearman's correlation test to
study the properties of superclusters drawn from the SDSS DR7. We analyse
possible selection effects in the supercluster catalogue, study the physical
and morphological properties of superclusters, find their possible subsets, and
determine scaling relations for superclusters. We show that the parameters of
superclusters do not correlate with their distance. The correlations between
the physical and morphological properties of superclusters are strong.
Superclusters can be divided into two populations according to their total
luminosity. High-luminosity superclusters form two sets, more elongated systems
with the shape parameter K_1/K_2 < 0.5 and less elongated ones with K_1/K_2 >
0.5. The first two principal components account for more than 90% of the
variance in the supercluster parameters and define the fundamental plane, which
characterises the physical and morphological properties of superclusters. We
use principal component analysis to derive scaling relations for superclusters,
in which we combine the physical and morphological parameters of superclusters.
Structure formation simulations for different cosmologies, and more data about
the local and high redshift superclusters are needed to understand better the
evolution and the properties of superclusters.
