A Note on Procrustean Fittings of Noisy Configurations

When comparing two or more multidimensional scaling (MDS) configurations, one usually first eliminates meaningless differences by Procrustean transformations. Such fittings lead to a number of unresolved issues such as the typical shrinkage of the fitted configuration relative to the target or how to interpret major similarity measures under various conditions of noise in the data. We here prove that the shrinkage ratio is equivalent to the correlation of the coordinates of the target and the fitted configuration. Thus, in real-life applications, the fitted configuration is always smaller than the target configuration. Both coefficients approach 0 as the noise level goes up. The congruence coefficient of the configurations' distances, in contrast, remains at a high level even in case of pure noise, falsely suggesting that the configurations are somewhat similar. This is important information for the user of Procrustean analyses.