individual difference scaling

The purpose of individual difference scaling is to represent objects, whose dissimilarities are given, as points in a metrical space. The distances in the space should be in accordance with the dissimilarities as well as is possible. Besides the configuration a Salience matrix is calculated.

The basic Euclidean model is:

δijk ≈ (∑s=1..r wks(xisxjs)2)1/2

Here δijk is the (known) dissimilarity between objects i and j, as measured on data source k. The x's are the coordinates of the objects in an r-dimensional space and the w's are weights or saliences. Because straight minimization of the expression above is difficult, one applies transformations on this expression. Squaring both sides gives the model:

δ2ijk ≈ ∑s=1..r wks(xisxjs)2

and the corresponding least squares loss function:

k=1..numberOfSourcesi=1..numberOfPointsj=1..numberOfPoints (δ2ijkd2ijk)2

This loss function is minimized in the (ratio scale option of the) ALSCAL program of Takane, Young & de Leeuw (1976).

The transformation used by Carroll & Chang (1970) in the INDSCAL model, transforms the data from each source into scalar products of vectors. For the dissimilarities:

βijk = –{ δ2ijkδ2i.kδ2.jk + δ2..k } / 2,

where dots replacing indices indicate averaging over the range of that index. In the same way for the distances:

zijk = –{ d2ijkd2i.kd2.jk + d2..k } / 2.
βijkzijk = ∑s=1..numberOfDimensions wks xis xjs

Translated into matrix algebra, the equation above translates to:

BkZk = X Wk X′,

where X is a numberOfPoints × numberOfDimensions configuration matrix, Wk, a non-negative numberOfDimensions × numberOfDimensions matrix with weights, and Bk the kth slab of βijk.

This translates to the following INDSCAL loss function:

 f(X, W1,..., WnumberOfSources) = ∑k=1..numberOfSources | Bk – XWkX′ |2