Kullback-Leibler distance as a measure of the information filtered from multivariate data

We show that the Kullback-Leibler distance is a good measure of the statistical uncertainty of correlation matrices estimated by using a finite set of data. For correlation matrices of multivariate Gaussian variables we analytically determine the expected values of the Kullback-Leibler distance of a sample correlation matrix from a reference model and we show that the expected values are known also when the specific model is unknown. We propose to make use of the Kullback-Leibler distance to estimate the information extracted from a correlation matrix by correlation filtering procedures. We also show how to use this distance to measure the stability of filtering procedures with respect to statistical uncertainty. We explain the effectiveness of our method by comparing four filtering procedures, two of them being based on spectral analysis and the other two on hierarchical clustering. We compare these techniques as applied both to simulations of factor models and empirical data. We investigate the ability of these filtering procedures in recovering the correlation matrix of models from simulations. We discuss such ability in terms of both the heterogeneity of model parameters and the length of data series. We also show that the two spectral techniques are typically more informative about the sample correlation matrix than techniques based on hierarchical clustering, whereas the latter are more stable with respect to statistical uncertainty.