
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:
Here δ_{ijk} 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 rdimensional 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:
and the corresponding least squares loss function:
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:
where dots replacing indices indicate averaging over the range of that index. In the same way for the distances:
Translated into matrix algebra, the equation above translates to:
where X is a numberOfPoints × numberOfDimensions configuration matrix, W_{k}, a nonnegative numberOfDimensions × numberOfDimensions matrix with weights, and B_{k} the k^{th} slab of β_{ijk}.
This translates to the following INDSCAL loss function:
f(X, W_{1},..., W_{numberOfSources}) = ∑_{k=1..numberOfSources}  B_{k} – XW_{k}X′ ^{2} 
© djmw, May 2, 1997