Two-Way Multidimensional Scaling: A Review

Multidimensional scaling (MDS) is a technique used to extract a set of independent variables from a proximity matrix or matrices. Applications of MDS are found in a wide range of areas, including visualization, pattern analysis, data preprocessing, scale development, cybernetics, and localization. The overall rationale behind the paper is to help share innovations across disciplines. We survey and synthesize MDS methods from the academic areas of psychometrics, statistics, and computing. We define classical MDS and distance-based MDS. We then introduce basic MDS formulations and functions. We survey MDS techniques designed for nonlinear data and describe distance-based MDS in terms of minimizing the energy function in a spring system. We describe completely nonmetric MDS techniques for ordinal input data and describe how MDS solutions can be compared using ordinal neighborhood information. We describe optimization methods for fitting MDS models, covering both continuous optimization techniques and combinatorial techniques. We give several illustrative applications of MDS from the areas of cybernetics, air traffic control, molecular chemistry, robotics, and network localization. We link this work to the techniques described in the previous sections of the paper. We list a wide range of currently available MDS software and discuss possible future work in the area.