individual difference scaling

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:

δ_ijk ≈ (∑_s=1..r w_ks(x_is – x_js)²)^1/2

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 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:

δ²_ijk ≈ ∑_s=1..r w_ks(x_is – x_js)²

and the corresponding least squares loss function:

∑_{k=1..numberOfSources} ∑_{i=1..numberOfPoints} ∑_{j=1..numberOfPoints} (δ²_ijk – d²_ijk)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:

β_ijk = –{ δ²_ijk – δ²_i.k – δ²_.jk + δ²_..k } / 2,

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

z_ijk = –{ d²_ijk – d²_i.k – d²_.jk + d²_..k } / 2.

β_ijk ≈ z_ijk = ∑_{s=1..numberOfDimensions} w_ks x_is x_js

Translated into matrix algebra, the equation above translates to:

B_k ≈ Z_k = X W_k X′,

where X is a numberOfPoints × numberOfDimensions configuration matrix, W_k, a non-negative 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, W₁,..., W_{numberOfSources}) = ∑_{k=1..numberOfSources} | B_k – XW_kX′ |²