Least Squares Algorithms for Constructing Constrained Ultrametric and Additive Tree Representations of Symmetric Proximity Data

Abstract

A mathematical programming algorithm is developed for fitting ultrametric or additive trees to proximity data where external constraints are imposed on the topology of the tree. The two procedures minimize a least squares loss function. The method is illustrated on both synthetic and real data. A constrained ultrametric tree analysis was performed on similarities between 32 subjects based on preferences for ten odors, while a constrained additive tree analysis was carried out on some proximity data between kinship terms. Finally, some extensions of the methodology to other tree fitting procedures are mentioned.

Keywords: Hierarchical clustering, Path length trees, Mathematical programming, Constrained classification methods