
In the analysis of the INDSCAL threeway data matrix (numberOfPoints × numberOfDimensions × numberOfSources) we seek to minimize the function:
f(X, W_{1},..., W_{numberOfSources}) = ∑_{i=1..numberOfSources}  S_{i} – XW_{i}X′ ^{2} 
where S_{i} 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, W_{i} is the diagonal numberOfDimensions × numberOfDimensions weight matrix for source i. The function above has no analytical solution for X and the W_{i}. 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 goodnessoffit 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 W_{i}.
Note: during the minimization the following constraints are effective:
© djmw, December 1, 1997