Degenerating Families of Dendrograms

Abstract

Abstract. Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist p-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the p-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are p-adic parameter spaces of (families of) dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.

Fast, Linear Time Hierarchical Clustering using the Baire Metric

“…A geometric foundation for ultrametric structures is presented in Bradley [3]. Starting from the point of view that a dendrogram, or ranked or unranked, binary or more general m-way, tree, is an object in a p-adic geometry, it is noted that: "The consequence of using p-adic methods is the shift of focus from imposing a hierarchic structure on data to finding a p-adic encoding which reveals the inherent hierarchies.…”

On p-adic classification

“…In contrast to the Archimedean situation, it is uniquely determined by the data (cf. [4,5]). We view D(X) as a rooted metric tree.…”

“…In [4], the use of more general p-adic numbers for encoding hierarchical data was advocated in order to be able to include the case of non-binary dendrograms into the scheme without having to resort to a larger prime number p. This was applied in [5] to the special case of data consisting in words over a given alphabet and where proximity of words is defined by the length of the common initial part. There, an agglomerative hierarchic p-adic clustering algorithm was described.…”

An ultrametric interpretation of building related event data

“…(2001) to image segmentation, where it did prove more efficient than its classical counterpart. In Bradley (2007), p -adic geometry is discussed as a framework for data classification. A further p -adic hierarchic classification algorithm is developed in Bradley (2009a) with a view on applications to texts and time series analysis.…”