Solving non-uniqueness in agglomerative hierarchical clustering using multidendrograms

In agglomerative hierarchical clustering, pair-group methods suffer from a problem of non-uniqueness when two or more distances between different clusters coincide during the amalgamation process. The traditional approach for solving this drawback has been to take any arbitrary criterion in order to break ties between distances, which results in different hierarchical classifications depending on the criterion followed. In this article we propose a variable-group algorithm that consists in grouping more than two clusters at the same time when ties occur. We give a tree representation for the results of the algorithm, which we call a multidendrogram, as well as a generalization of the Lance and Williams' formula which enables the implementation of the algorithm in a recursive way.Comment: Free Software for Agglomerative Hierarchical Clustering using Multidendrograms available at http://deim.urv.cat/~sgomez/multidendrograms.ph

Morphological phenotypic dispersion of garlic cultivars by cluster analysis and multidimensional scaling

Multivariate techniques have become a useful tool for studying the phenotypic diversity of Germplasm Bank accessions, since they make it possible to combine a variety of different information from these accessions. This study aimed to characterize the phenotypic dispersion of garlic (Allium sativum L.) using two multivariate techniques with different objective functions. Twenty accessions were morphologically characterized for bulb diameter, length, and weight; number of cloves per bulb; number of leaves per plant; and leaf area. Techniques based on generalized quadratic distance of Mahalanobis, UPGMA (Unweighted Pair Group Method with Arithmetic Mean) clustering, and nMDS (nonmetrric MultiDimensional Scaling) were applied and the relative importance of variables quantified. The two multivariate techniques were capable of identifying cultivars with different characteristics, mainly regarding their classification in subgroups of common garlic or noble garlic, according to the number of cloves per bulb. The representation of the phenotypic distance of cultivars by multidimensional scaling was slightly more effective than that with UPGMA clustering