Constructing ultrametric and additive trees based on the L1 norm

A heuristic method for identifying and fitting ultrametric and additive trees is presented, based on iteratively re-weighted iterative projection (IRIP) that minimizes a least absolute deviations (L-1) criterion. Examples of ultrametric and additive trees fitted to two extant data sets are given, plus a Monte Carlo analysis to assess the impact of both typical data error and extreme values on fitted trees. Solutions are compared to the least-squares (L-2) approach of Hubert and Arabie (1995a), with results indicating that (with these data) the L-1 and L-2 optimization strategies perform very similarly. A number of observations are made concerning possible uses of an L-1 approach, the nature and number of identified locally optimal solutions, and metric recovery differences between ultrametrics and additive trees.