On possible psychophysical maps: I. Quadratic transformations

Abstract

A certain class of quadratic transformations can be used to map multidimensional physical spaces into multidimensional perceptual spaces; their numerical representations change under various coordinate transformations in both spaces.where E denotes the physical. experimentally controllable space that is represented by a (possibly bounded) region in Re x Re = Re' • and s eRe' denotes the subjective space. The second pair. a,s, can be interpreted as log transforms of area (hw) and shape (h /w), Krantz and Tversky found decomposability violated for both Equations I and 2. This means that neither pair can give rise to Minkowsky metrics monotone with the observed dissimilarities. Krantz and Tversky then processed the observed dissimilarities with two multidimensional scaling programs. They accepted a fit of the INDSCAL model (1. D. Carroll & Chang. 1970) in two dimensions. but without testing it with any stringency. If the assumptions of the model are met. then the coordinate axes in S. that is. the "psychological dimensions ." are uniquely identified by the data.1 The main result of this scaling analysis was a distortion of the design configuration that seemed to corroborate the resultsThe problem of relating two or more multidimensional spaces arises both in scaling and in factor analysis. The technology for fitting such transformations optimally in some sense (usually a least squares sense) is reasonably well understood for the group of affine transformations and its subgroups : similarities. rigid motions, and rotations ("Procrustes" problems, e.g

Subadditive concatenation in dissimilarity judgments

“…This line of reasoning has been expanded by Beals, Krantz, and Tversky (1968), Hyman and Well (1967), Krantz and Tversky (1975), Shepard (1962), and Wender (1969Wender ( , 1971, among others. The Krantz and Tversky study, in particular, has spawned numerous follow-up studies (Borg & Leutner, 1983;Noma & Johnson, 1977;Schonemann, 1977Schonemann, , 1979Schonemann, , 1980Schonemann, , 1982Schonemann, , 1983Schonemann & Borg, 1981;Schonemann & Kienapple, 1984;Takane, 1981; Copyright 1985 Psychonomic Society, Inc. is segmentally additive, otherwise it will be positive. If it is negative, the triangle inequality is violated.…”

“…However, we should not lose sight of the fact that the intervening monotone transformations are completely ad hoc, psychologically unexplained, and tests, whereas in the second part they argue for an A/S interaction on the basis of an MDS analysis that rests on Euclidean distances and hence on decomposability. (2) Although this seeming contradiction can be resolved by applying a nonlinear, instead of an affine, transformation to the objective H/W system to restore decomposability (Schonemann, 1977(Schonemann, , 1979(Schonemann, , 1980, Krantz and Tversky's claim that the underlying dimensions were A/S, rather than HIW, is more difficult to justify. Horan's model rests on two fairly strong conditions, a common-space condition and a diagonality condition.…”