Studies in the Robustness of Multidimensional Scaling: Perturbational Analysis of Classical Scaling

Summary This series of papers is devoted to the investigation of the extent to which the accuracy of operation of multidimensional scaling methods can be put onto a quantitative footing. This second paper investigates the response of the classical scaling method to small errors by making expansions in powers of the error term, and retaining the lowest non-cancelling power. Effects on individual eigenvalues and eigenvectors are considered, and shown to lead to useful auxiliary techniques for choice of dimensionality or for correction of bias. Procrustes statistics, as developed in the first paper of this series, are used to provide an overall picture of the error response; a perturbation theory for these is worked out, and is shown to lead to an approximate distribution theory. The details are worked out illustratively for a simple type of error structure.