Summary
Local search algorithms in Multidimensional Scaling (MDS), based on gradients or subgradients, often get stuck at local minima of STRESS, particularly if the underlying dissimilarity matrix is far from being Euclidean. However, in order to remove ambiguity from the model building process, it is of paramount interest to fit a suggested model best to a given data set. Hence, finding the global minimum of STRESS is very important for applications of MDS.
In this paper a hybrid iteration scheme is suggested consisting of a local optimization phase and a genetic type global optimization step. Local search is based on the simple and fast majorization approach. Extensive numerical testing shows that the presented method has a high success probability and clearly outperforms simple random multistart.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
COX, T. F. and COX M. A. A. (1994): Multidimensional Scaling. Chapman and Hall, London.
DE LEEUW, J. (1988): Convergence of the majorization method of multidimensional scaling. Journal of Classification, 5, 163–180.
DE LEEUW, J. and HEISER, W. (1980): Multidimensional scaling with restrictions on the configuration. In: P.R. Krishnaiah (ed.): Multivariate Analysis, Vol. V. North-Holland, Amsterdam, 501–522.
DE LEEUW, J. and STOOP, I. (1984): Upper bounds of Kruskal’s stress. Psychometrika 49, 391–402.
GLUNT, W.; HAYDEN, T. L. and RAYDAN, M. (1993): Molecular conformations from distance matrices. Journal of Computational Chemistry, 14, 114–120.
GOLDBERG, D. E. (1989): Genetic Algorithms. Addison-Wesley, Reading, Mass.
GREEN, F. J.; CARMONE, F. J. and SMITH, S. M. (1989): Multidimensional Scaling: Concepts and Applications. Allyn and Bacon, Boston.
GROENEN, P. J. F. (1993): The Majorization Approach to Multidimensional Scaling, Some Problems and Extensions. DSWO-Press, Leiden.
GROENEN, P. J. F.; MATHAR, R. and HEISER, W. (1995): The majorization approach to multidimensional scaling for Minkowski distances. Journal of Classification, 12, 3–19.
HELLEBRANDT, M.; MATHAR, R. (1996): Estimating position and velocity of mobiles in celular radio networks. To appear: IEEE Transactions on Vehicular Technology.
KEARSLEY, A. J.; TAPIA, R. A. and TROSSET, M.W. (1994): The solution of the metric stress and sstress problems in multidimensional scaling using Newton’s method. Technical Report, Dept. of Computational and Applied Mathematics, Rice University Houston, Texas, TR94-44.
MATHAR, R. and GROENEN, P. (1991): Algorithms in convex analysis applied to multidimensional scaling. In: E. Diday and Y. Lechevallier (eds.): Symbolic-Numeric Data Analysis and Learning. Nova Science Publishers, Commack, NY, 45–56.
MATHAR, R. and MEYER, R. (1994): Algorithms in convex analysis to fit linp-distance matrices. Journal of Multivariate Analysis, 51, 102–120.
MATHAR, R. and ZILINSKAS, A. (1993): On global optimization in two-dimensional scaling. Acta Applicandae Mathematicae, 33, 109–118.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mathar, R. (1997). A Hybrid Global Optimization Algorithm for Multidimensional Scaling. In: Klar, R., Opitz, O. (eds) Classification and Knowledge Organization. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59051-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-59051-1_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62981-8
Online ISBN: 978-3-642-59051-1
eBook Packages: Springer Book Archive