[9]

Peter H. Schonemann

On metric multidimensional unfolding

Psychometrika, 1970, 35, 349-366

The problem of locating two sets of points in a joint space, given the Euclidean distances between elements from distinct sets, is solved algebraically. For error free data the solution is exact, for fallible data it has least squares properties.

Clyde Coombs was a Thurstone student who went his own way. He did not use mathematics just to impress the uninitiated, but as a tool for solving psychological problems  which, to me at least,  made some sense.

A case on point is his Unfolding Model. Basically, it postulates that most people choose the next best thing if their "ideal" choice is unavailable. While different people  have different ideal points (points of maximal preference), they alll evaluate the available stimulus choices in terms of their (psychological) distances from their own ideal point. Hence the preference order of a given set of concrete choice objects (e.g., cars, or homes) will be different for different people.

In the unidimensional case this choice model can be visualized as a string with knots on it indicating the available choice points and the ideal points. Each subject picks up the string at his own ideal point. His perference ranking is then proportinal to the order of the choice points on this folded string. The unfolding problem consists in reconstructing the underlying joint scale (i.e., the stretched out knotted string) from several such individual preference orders.

Coombs devised methods for solving this non-metric  unfolding problem in one dimension. When the model is generalized to more than one dimensions, e.g. to a plane of choice points and ideal points the problem becomes more difficult. In this case, one may try to simplify it by introducing some convenient metric. One thus arrives at a metric  multidimensional unfolding problem. The most obvious choice  is the euclidean metric, since it is intuitively well understood, has attractive invariance properties, and interfaces with linear algebra.

The above paper presents an exact algebraic solution of this geometric problem which, technically, is a generalization of the complete Euclidean embedding problem. Psychologists usually credit Torgerson (1958), and occasionally  Young and Housholer (1938) with having solved the complete embedding problem. But in fact it goes back at least to Cayley (On a theorem in the geometry of position, Cambridge Mathematical Journal, 1841, 267-271). See also the Appendix to Schonemann (1970).

To solve the incomplete case, the key idea was to convert a quadratic problem into a linear problem by differencing. Geometrically this amounts to a shift of the origin (see loc. cit.). As a consequence,  the solution is invariant under the addition of  row (or column) constants to the matirx of squared euclidean distances. This fact enabled  Schonemann and Wang (1972) to devise a metric multidimensional preference scaling model which combines features of the Coombs unfolding model with the Bradley-Terry-Luce (BTL) paired comparison model, and which has an exact algebraic solution.