Exploring Hierarchical Concepts: Theoretical and Application Comparisons

Cavicchia, Carlo; Vichi, Maurizio; Zaccaria, Giorgia

doi:10.1007/978-981-15-2700-5_19

Cited by 2 publications

(3 citation statements)

References 18 publications

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“…Further developments of this work can be explored. Indeed, it would be interesting to compare the performances of our proposal with the ones of traditional hierarchical clustering algorithms, i.e., to extended the proof in Cavicchia et al (2020a) to the more general case of mixed-type data.…”

Section: Discussionmentioning

confidence: 99%

“…It is worthy to highlight that the proposal differs from hierarchical clustering methods applied on variables (see Cavicchia et al, 2020a, for the quantitative case). Indeed, even if the latter can be implemented on variables after a proper transformation of similarity measures into dissimilarity ones, they are sequential and greedy procedures affected by misclassification at the bottom of the hierarchy that can have an effect on higher levels.…”

Section: Methodsmentioning

confidence: 99%

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Complex Dimensionality Reduction: Ultrametric Models for Mixed-Type Data

Mingione

Vichi

Zaccaria

2022

Building Bridges Between Soft and Statistical Methodologies for Data Science

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The factorial latent structure of variables, if present, can be complex and generally identified by nested latent concepts ordered in a hierarchy, from the most specific to the most general one. This corresponds to a tree structure, where the leaves represent the observed variables and the internal nodes coincide with latent concepts defining the general one (i.e., the root of the tree). Although several methodologies have been proposed in the literature to study hierarchical relationships among quantitative variables, very little has been done for more general mixed-type data sets. Hence, it is of the utmost importance to extend these methods and make them suitable to the even more frequent availability of mixed-type data matrices, as complex real phenomena are often described by both qualitative and quantitative variables. In this work, we propose a new exploratory model to study the hierarchical statistical relationships among variables of mixed-type nature by fitting an ultrametric matrix to the general dependence matrix, where the former is one-to-one associated with a hierarchical structure.

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Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Complex Dimensionality Reduction: Ultrametric Models for Mixed-Type Data

Mingione

Vichi

Zaccaria

2022

Building Bridges Between Soft and Statistical Methodologies for Data Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…The correlation coefficient is thereby transformed into distance in order to apply a hierarchical clustering algorithm (Cliff et al 1995;Gordon 1999;Strauss et al 1973). Cavicchia et al (2020a) showed that this approach has several limitations.…”

Section: Remarkmentioning

confidence: 99%

Gaussian mixture model with an extended ultrametric covariance structure

Cavicchia

Vichi

Zaccaria

2022

Adv Data Anal Classif

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Gaussian Mixture Models (GMMs) are one of the most widespread methodologies for model-based clustering. They assume a multivariate Gaussian distribution for each component of the mixture, centered at the mean vector and with volume, shape and orientation derived by the covariance matrix. To reduce the large number of parameters produced by the covariance matrices, parsimonious parameterizations of the latter were proposed in literature, e.g., the eigen-decomposition and the parsimonious GMMs based on mixtures of probabilistic principal component analyzers and mixtures of factor analyzers. We introduce a new parameterization of a covariance matrix by defining an extended ultrametric covariance matrix and we implement it into a GMM. This structure can be used to describe multidimensional phenomena which are characterized by nested latent concepts having different levels of abstraction, from the most specific to the most general. The proposal is able to pinpoint a hierarchical structure on variables for each component of the GMM, thus identifying a different characterization of a multidimensional phenomenon for each component (cluster, subpopulation) of the mixture. At the same time, it defines a new parsimonious GMM since the ultrametric covariance structure reconstructs the relationships among variables with a limited number of parameters. The proposal is applied on synthetic and real data. On the former it shows good performance in terms of classification when compared to the other existing parameterizations, and on the latter it also provides insight into the hierarchical relationships among the variables for each cluster.

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Exploring Hierarchical Concepts: Theoretical and Application Comparisons

Cited by 2 publications

References 18 publications

Complex Dimensionality Reduction: Ultrametric Models for Mixed-Type Data

Complex Dimensionality Reduction: Ultrametric Models for Mixed-Type Data

Gaussian mixture model with an extended ultrametric covariance structure

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