8. Ultrametrics and Additive Trees for Two-Mode (Rectangular) Proximity Data
Society for Industrial and Applied Mathematics eBooks · 2006
Thus far in Part II, the proximity data considered for obtaining some type of structure, such as an ultrametric or an additive tree, have been assumed to be on one intact set of objects, S = {O1, …, On}, and complete in the sense that proximity values are present between all object pairs. Just as linear unidimensional scaling (LUS) was generalized for two-mode proximity data in Chapter 4, suppose now that the available proximity data are two-mode and between two distinct object sets, SA = {O1A, …, OnaA} and SB = {O1B, …, OnbB}, containing na and nb objects, respectively, given by an na × nb proximity matrix Q = {qrs}. Again, we assume that the entries in Q are keyed as dissimilarities, and a joint structural representation is desired for the set SA ∪ SB. Conditions have been proposed in the literature for when the entries in a matrix fitted to Q characterize an ultrametric or an additive tree representation. In particular, suppose an na × nb matrix F = {frs} is fitted to Q through least-squares subject to the constraints that follow: Ultrametric (Furnas, 1980): for all distinct object quadruples, OrA, OsA, OrB, OsB, where OrA, OsA ∈ SA and OrB, OsB ∈ SB, and considering the entries in F corresponding to the pairs (OrA, OrB), (OrA, OsB), (OsA OrB), and (OsA, OsB), say frArB, frAsB, fsArB, fsAsB, respectively, the largest two must be equal.