of two sets of multivariate random variables and (iv) the retained information and stability of a filtered correlation matrix. We apply the discussed concepts to a portfolio of stocks traded in a financial market. The paper is organized as follows. In Section \[corr\] we discuss how to obtain hierarchical trees and correlation based trees or networks from the correlation matrix of a complex system and we discuss about the role of bootstrap in the statistical validation of hierarchical trees and correlation based networks. In Section \[sectionH\] we discuss the definition and the properties of a factor model with independent factors which are hierarchically nested. In Section \[sectionEMP\] we present an empirical application of the hierarchically nested factor model. Section \[kullbackTHEOR\] discusses how to quantify the information and stability of a correlation matrix by using a Kullback-Leibler distance and Section \[sectionCOMP\] presents the quantitative comparison of different filtering procedures performed with the same distance. Section \[sectionCONC\] briefly presents some conclusions.

Correlation based hierarchical organization and networks {#corr}
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Hereafter we discuss a simple example illustrating two filtering procedures of a correlation matrix performed with methods of hierarchical clustering. In our approach, by using the correlation between elements as the similarity measure 