Multidimensional scaling with constrained dimensions: CONSCAL

Abstract

Multidimensional scaling is a technique used to represent pairwise dissimilarities among a set of objects in a Euclidean space of generally low dimensionality. The data determine the number of dimensions to retain, and in the case of the weighted Euclidean model the psychologically meaningful dimensions to interpret. However, it is often the case that it is difficult to determine the appropriate dimensionality based on goodness-of-fit statistics, and it is sometimes difficult to interpret each recovered dimension. Moreover, this problem may occur in situations where a small number of physical parameters may be used to describe the objects. Then it would appear useful to use this information and constrain the dimensions of the distance model to be monotone transformations of these physical dimensions. We therefore propose two models: djk = [(fj - fk)tI(fj - fk)]1/2 [(fj - fk)tA(fj - fk)]1/2, where djk is the distance between stimulus j and stimulus k; there are R physical dimensions; x(r)j is the coordinate of the jth object on the rth dimension; f(r) is the monotone transformation for the rth dimension; fj is the vector of monotone transformations for the jth object, the rth component being f(r)(x(r)j); A is an R x R symmetric matrix with 1s on the diagonal; I is an R x R identity matrix with 1s on the diagonal, 0s elsewhere. The monotone transformations are represented as I-splines. Model one is a special case of model two for uncorrelated dimensions. Solutions may be compared with those obtained in the unconstrained case to choose the best representation of the data. Examples of real and artificial data will be presented.