Measuring diversity from dissimilarities with Rao's quadratic entropy: are any dissimilarities suitable?

Rao has developed quadratic entropy to measure diversity in a set of entities divided up among a fixed set of categories. This index depends on a chosen matrix of dissimilarities among categories and a frequency distribution of these categories. With certain choices of dissimilarities, this index could be maximized over all frequency distributions by eliminating several categories. This unexpected result is radically opposite to those obtained with usual diversity indices. We demonstrate that the elimination of categories to maximize the quadratic entropy depends on mathematical properties of the chosen dissimilarities. In particular, when quadratic entropy is applied to ultrametric dissimilarities, all categories are retained in order to reach its maximal value. Three examples, varying from simple one-dimensional to ultrametric dissimilarity matrices, are provided. We conclude that, as far as diversity measurement is concerned, quadratic entropy is most relevant when applied to ultrametric dissimilarities.     

New combinatorial clustering methods

Sixteen clustering methods are compatible with the general recurrence equation of combinatorial SAHN (sequential, agglomerative, hierarchical and nonoverlapping) classificatory strategies. These are subdivided into two classes: the d-SAHN methods seek for minimal between-cluster distances the h-SAHN strategies for maximal within-cluster homogeneity. The parameters and some basic features of all combinatorial methods are listed to allow comparisons between these two families of clustering procedures. Interest is centred on the h-SAHN techniques; the derivation of updating parameters is presented and the monotonicity properties are examined. Three new strategies are described, a weighted and an unweighted variant of the minimization of the increase of average distance within clusters and a homogeneity-optimizing flexible method. The performance of d- and h-SAHN techniques is compared using field data from the rock grassland communities of the Sashegy Nature Reserve, Budapest, Hungary.Abbreviations CP = Closest pair - RNN = Reciprocal nearest neighbor - SAHN = Sequential, agglomerative, hierarchical and nonoverlapping