Some Uses of the Farris Transform in Mathematics and Phylogenetics—A Review

Abstract

In 1970, Farris introduced a procedure that can be used to transform a tree metric into an ultra metric. Since its discovery, Farris' procedure has been used extensively within phylogenetics where it has become commonly known as the Farris transform. Remarkably, the Farris transform has not only been rediscovered several times within phylogenetics, but also in other fields. In this paper, we will review some of its various properties and uses.The paper is divided into four parts and, altogether, 12 sections. In the first part, we introduce a standardized scheme for classifying those dissimilarity mappings to which the Farris transform can be applied -a scheme that has evolved over the years, but has apparently not been spelled out before in sufficient detail. In the second part, we will discuss how a straightforward generalization of the Farris transform naturally arises in T-Theory. The third part describes how this generalized Farris transform can be used to approximate dissimilarities by tree metrics. And in the final part, we describe some further, "non-standard" applications of the Farris transform.

On encodings of phylogenetic networks of bounded level

“…Using the well-known Farris transform (see e.g. Semple and Steel 2003, p. 149;Dress et al 2007 for a recent overview) a similarity measure can be canonically transformed into a distance measure D C on X . For a set Y such a measure is defined as a map from Y × Y into the non-negative reals that is symmetric, satisfies the triangle inequality, and vanishes on the main diagonal.…”

Phylogenetic Diversity Over an Abelian Group

“…(see [9] for a recent survey of uses of this transformation, and [6] for further recent applications). For a general abelian group, D x will play the same role as this transform (when G has elements of order 2, the direct group-theoretic analogue of the Farris transform is not well defined).…”

Split Decomposition over an Abelian Group Part 1: Generalities

“…group homomorphism in this case, implying in particular that also P (2) [17] for further references and a recent survey of uses of this transformation and [16] for further recent applications). The results put together in this note can therefore be viewed as pointing towards yet another way of studying the Farris transform and its variants in a more general context.…”