In the analysis of the INDSCAL three-way data matrix (numberOfPoints × numberOfDimensions × numberOfSources) we seek to minimize the function:
|f(X, W1,..., WnumberOfSources) = ∑i=1..numberOfSources | Si – XWiX′ |2|
where Si is a known symmetric numberOfPoints × numberOfPoints matrix with scalar products of distances for source i, X is the unknown configuration numberOfPoints × numberOfDimensions matrix, X′ its transpose, and, Wi is the diagonal numberOfDimensions × numberOfDimensions weight matrix for source i. The function above has no analytical solution for X and the Wi. It can be solved, however, by an iterative procedure which Carroll & Chang have christened CANDECOMP (CANonical DECOMPosition). This method minimizes, instead of the function given above, the following function:
where X and Y are both numberOfPoints × numberOfDimensions configuration matrices.
The algorithm proceeds as follows:
1. Initialize the
W matrices and the configuration matrix X. This can for example be done according to a procedure given in Young, Takane & Lewyckyj (1978).
2. An alternating least squares minimization process is started as described that sequentially updates Y, X an W (Carroll & Chang (1970)):
Evaluate the goodness-of-fit criterion and either repeat the minimization sequence (2.1–2.3) or continue.
3. Done: make Y equal to X and solve a last time for the Wi.
Note: during the minimization the following constraints are effective:
© djmw, December 1, 1997