Retractions of finite distance functions onto tree metrics
A problem arising in areas of classification, particularly in evolutionary biology, is how to approximate an arbitrary distance function by a tree metric, induced by trees with positively weighted edges, and thereby estimate the underlying tree that generated the data. Such transformations, from distances to tree metrics (and thereby to edge-weighted trees) should have some basic properties such as continuity. It is shown that this is lacking for the commonly used neighbor-joining method which is not a retraction. A continuous transformation, due to Buneman [see \textit{F. Hodson, D. Kendall} and \textit{P. Tautu} (eds.), Mathematics in the archaeological and historical sciences, Edinburgh University Press, Edinburgh, 387-395 (1971)] is refined so as to lead to more informative trees without sacrificing continuity and two simple examples of the use of the refined Buneman map, including an application to biological data are provided. A sufficient condition for both the Buneman construction, and its refinement to correctly recover the underlying tree is also provided.
scientific article; zbMATH DE number 1271074