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Dealing with Distances and Transformations for Fuzzy C-Means Clustering of Compositional Data

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Abstract

Clustering techniques are based upon a dissimilarity or distance measure between objects and clusters. This paper focuses on the simplex space, whose elements—compositions—are subject to non-negativity and constant-sum constraints. Any data analysis involving compositions should fulfill two main principles: scale invariance and subcompositional coherence. Among fuzzy clustering methods, the FCM algorithm is broadly applied in a variety of fields, but it is not well-behaved when dealing with compositions. Here, the adequacy of different dissimilarities in the simplex, together with the behavior of the common log-ratio transformations, is discussed in the basis of compositional principles. As a result, a well-founded strategy for FCM clustering of compositions is suggested. Theoretical findings are accompanied by numerical evidence, and a detailed account of our proposal is provided. Finally, a case study is illustrated using a nutritional data set known in the clustering literature.

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Authors and Affiliations

  1. Biomathematics and Statistics Scotland, JCMB, The King’s Buildings, Edinburgh, EH9 3JZ, UK

    Javier Palarea-Albaladejo

  2. Universitat de Girona, Girona, Spain

    Josep Antoni Martín-Fernández

  3. Universidad Católica San Antonio, Murcia, Spain

    Jesús A. Soto

Authors
  1. Javier Palarea-Albaladejo
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  2. Josep Antoni Martín-Fernández
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  3. Jesús A. Soto
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Corresponding author

Correspondence to Javier Palarea-Albaladejo.

Additional information

This research has been supported by the Scottish Government, the Spanish Ministry of Science and Innovation under the project “CODA-RSS” Ref. MTM2009-13272; and by the Agència de Gestió d’Ajuts Universitaris i de Recerca of the Generalitat de Catalunya under the project Ref: 2009SGR424. We are in debt with the editor and the referees for their helpful comments and suggestions on an earlier version of this paper.

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Palarea-Albaladejo, J., Martín-Fernández, J.A. & Soto, J.A. Dealing with Distances and Transformations for Fuzzy C-Means Clustering of Compositional Data. J Classif 29, 144–169 (2012). https://doi.org/10.1007/s00357-012-9105-4

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  • DOI: https://doi.org/10.1007/s00357-012-9105-4

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