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The ultrametric correlation matrix for modelling hierarchical latent concepts

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Abstract

Many relevant multidimensional phenomena are defined by nested latent concepts, which can be represented by a tree-structure supposing a hierarchical relationship among manifest variables. The root of the tree is a general concept which includes more specific ones. The aim of the paper is to reconstruct an observed data correlation matrix of manifest variables through an ultrametric correlation matrix which is able to pinpoint the hierarchical nature of the phenomenon under study. With this scope, we introduce a novel model which detects consistent latent concepts and their relationships starting from the observed correlation matrix.

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Notes

  1. A matrix \(\mathbf {A} = [a_{ij}]\) is nonnegative if \(a_{ij} \ge 0 \; \forall i,j\) (see Horn and Johnson 2013, p. 519). It is important to stress that the definition of a nonnegative matrix is different from that of a nonnegative definite matrix.

  2. Negative values are not taken into account herein thanks to the assumption of nonnegative correlations which is introduced from this section on.

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Authors and Affiliations

  1. University of Rome La Sapienza, Piazzale Aldo Moro 5, 00185, Rome, Italy

    Carlo Cavicchia, Maurizio Vichi & Giorgia Zaccaria

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  1. Carlo Cavicchia
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  2. Maurizio Vichi
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  3. Giorgia Zaccaria
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Correspondence to Giorgia Zaccaria.

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Cavicchia, C., Vichi, M. & Zaccaria, G. The ultrametric correlation matrix for modelling hierarchical latent concepts. Adv Data Anal Classif 14, 837–853 (2020). https://doi.org/10.1007/s11634-020-00400-z

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  • DOI: https://doi.org/10.1007/s11634-020-00400-z

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