The Choice of Initial Configurations in Multidimensional Scaling: Local Minima, Fit, and Interpretability

Ingwer Borg,Patrick Mair

doi:10.17713/ajs.v46i2.561

Abstract

Multidimensional scaling (MDS) algorithms can easily end up in local minima, depending on the starting configuration. This is particularly true for 2-dimensional ordinal MDS. A simulation study shows that there can be many local minima that all have an excellent model fit (i.e., small Stress) even if they do not recover a known latent configuration very well, and even if they differ substantially among each other. MDS programs give the user only one supposedly Stress-optimal solution. We here present a procedure for analyzing all MDS solutions resulting from using a variety of different starting configurations. The solutions are compared in terms of fit and configurational similarity. This allows the MDS user to identify different types of solutions with acceptable Stress, if they exist, and then pick the one that is best interpretable.

Highlights

Multidimensional scaling (MDS) is a statistical method that optimally maps proximity data on pairs of objects into distances among points in a multidimensional space
Even if the solution given by the MDS program is the global minimum solution, an MDS user may be interested to see what other local-minima solutions exist, what their Stress values are, and how similar they are. We study this issue with an artificial data set and describe a systematic approach that can be used by the applied MDS user to answer these questions for his or her data and the particular choice of MDS model
We describe a procedure for generating local minima solutions in MDS beginning with different initial configurations, and for comparing the configurational similarity of these solutions so that the user can identify those solutions he or she wants to check for their interpretability

Summary

Introduction

Multidimensional scaling (MDS) is a statistical method that optimally maps proximity data on pairs of objects (i.e., data expressing the similarity or the dissimilarity of pairs of objects) into distances among points in a multidimensional space. MDS is used for exploring or testing the structure of proximity data. There are many variants of MDS (see Borg and Groenen 2005; Cox and Cox 2000). Two-dimensional ordinal MDS is probably the most popular model. The proximity data—converted first to dissimilarity indices δij in case the proximities are similarity measures—are optimally mapped into Euclidean distances dij(X) among points of a two-dimensional Euclidean space (with the coordinate matrix X). The order of the distances corresponds to the order of the data, and ties in the data can be broken in the distances (“primary approach to ties”)

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Conclusion