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

• Published in: Journal of classification Vol. 29; no. 2; pp. 144 - 169
• Main Authors: Palarea-Albaladejo, Javier, Martín-Fernández, Josep Antoni, Soto, Jesús A.
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.07.2012; Springer Nature B.V
• Subjects: Algorithms; Bioinformatics; Cluster analysis; Clustering; Data analysis; Marketing; Mathematics and Statistics; Pattern Recognition; Psychometrics; Signal,Image and Speech Processing; Statistical Theory and Methods; Statistics; FCM; Simplex space; Fuzzy clustering; Closed data; Aitchison distance; Compositional data
• ISSN: 0176-4268; 1432-1343
• DOI: 10.1007/s00357-012-9105-4

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.