Analysing large-scale structure -: I.: Weighted scaling indices and constrained randomization

The method of constrained randomization, which was originally developed in the field of time-series analysis for testing for non-linearities, is extended to the case of three-dimensional point distributions as they are typical in the analysis of the large-scale structure of galaxy distributions in the Universe. With this technique it is possible to generate for a given data set so-called surrogate data sets that have the same linear properties as the original data whereas higher order or non-linear correlations are not preserved. The analysis of the original and surrogate data sets with measures, which are sensitive to non-linearities, yields valuable information about the existence of non-linear correlations in the data. On the other hand one can test whether given statistical measures are able to account for higher order or non-linear correlations by applying them to original and surrogate data sets. We demonstrate how to generate surrogate data sets from a given point distribution, which have the same linear properties (power spectrum) as well as the same density amplitude distribution but different morphological features. We propose weighted scaling indices, which measure the local scaling properties of a point set, as a non-linear statistical measure to quantify local morphological elements in large-scale structure. Using surrogates it is shown that the data sets with the same two-point correlation functions have slightly different void probability functions and especially a different set of weighted scaling indices. Thus a refined analysis of the large-scale structure becomes possible by calculating local scaling properties whereby the method of constrained randomization yields a vital tool for testing the performance of statistical measures in terms of sensitivity to different topological features and discriminative power.