A least squares algorithm for fitting additive trees to proximity data

Additive trees have proved to be a valuable alternative to multidimensional scaling for representing proximity data [Sattath & Tversky, 1977]. Especially when the stimuli are conceptual rather than perceptual, an additive tree representation can be very useful [cf. Pruzansky, Tversky, & Carroll, 1982]. A tree is a connected graph where every pair of nodes is connected by a unique path. In an additive tree a nonnegative weight is attached to each link such that the distance between any pair of nodes is the sum of the weights associated with the links that connect the two nodes. Alternative names for additive trees are path length trees [Carroll, 1976] and weighted free trees [Cunningham, 1978]. Let ~ be a nonnegative symmetric dissimilarity measure defined on a finite set of n objects. Then the n objects can be represented by the n terminal nodes of an additive tree whenever A = {bo l i < j} satisfies