Abstract
For the problem of metric unidimensional scaling, the number of local minima is estimated. For locating the globally optimal solution we develop an approach, called the “smoothing technique.” Although not guaranteed inevitably to locate the global optimum, the smoothing technique did so in all computational experiments where the global optimum was known.
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BERESNEVA, I. B., MASLIAKOVA, T. V., MININA, T. R., NISANOVA, E. V. and PEREKREST, V. T. (1986),Application of Quasi-Dynamic Programming to Dimensionality Reduction Problems (in Russian), Preprint, Leningrad: Institute of Social and Economic Problems.
DEFAYS, D. (1978), “A Short Note on a Method of Seriation,”British Journal of Mathematical and Statistical Psychology, 31, 49–53.
dE LEEUW, J. (1984), “Differentiability of Kruskal's STRESS at a Local Minimum,”Psychometrika, 49, 111–113.
dE LEEUW, J., and HEISER, W. J. (1977), “Convergence of Correction-Matrix Algorithms for Multidimensional Scaling,” inGeometric Representations of Relational Data: Readings in Multidimensional Scaling, Ed., J. C. Lingoes, Ann Arbor, MI: Mathesis, 735–752.
dE LEEUW, J., and HEISER, W. J. (1980), “Multidimensional Scaling with Restrictions on the Configuration,” inMultivariate Analysis, Ed., P.R. Krishnaiah, Amsterdam: North-Holland, Vol.V, 501–522.
GREEN, P. E., CARMONE, F. J. Jr, and SMITH, S. M. (1989),Multidimensional Scaling, Concepts and Applications, Boston: Allyn and Bacon.
GUTTMAN, L. (1968), “A General Nonmetric Technique for Finding the Smallest Coordinate Space for a Configuration of Points,”Psychometrika, 3, 469–506.
HUBERT, L. J., and ARABIE, P. (1986), “Unidimensional Scaling and Combinatorial Optimization,” inMultidimensional Data Analysis, Ed., J. De Leeuw et al, Leiden, The Netherlands: DSWO Press, 181–196.
HUBERT, L. J., and ARABIE, P. (1988), “Relying on Necessary Conditions for Optimization: Unidimensional Scaling and Some Extensions,” inClassification and Related Methods of Data Analysis, Ed., H. H. Bock, Amsterdam: North-Holland, 463–472.
NOBLE, B. (1969),Applied Linear Algebra, Englewood Cliffs, NJ: Prentice-Hall.
OSLON, A. A. (1984), “One-Dimensional Metric Scaling,”Automation and Remote Control, 45, 783–788.
PLINER, V. (1984), “A Class of Metric Scaling Models,”Automation and Remote Control, 45, 789–794.
PLINER, V. (1986), “The Problem of Multidimensional Metric Scaling,”Automation and Remote Control, 47, 560–567.
PLINER, V., and KHACHATUROVA, T. V. (1984), “Analysis of Dependence of Quantitative Index on Qualitative Factor as a Problem of Unidimensional Representation of Data Structure” (in Russian), inTheoretical and Methodical Problems of Sociological Data Base Creation, Eds., Yu. Zalatorjus, A. Eigirdas, and Yu. Eidukas, Vilnius, Lithuania: Institute of Philosophy, Sociology, and Law, pp. 162–169.
POLYAK, B. T. (1983),Introduction to Optimization (in Russian), Moscow: Nauka.
ROBINSON, W. S. (1951), “A Method for Chronologically Ordering Archaeological Deposits,”American Antiquity, 16, 293–301.
STERNIN, H. (1965),Statistical Methods of Time Sequencing, Department of Statistics Technical Report 112, Stanford University.
YUDIN, D. B. (1965), “Quantitative Analysis of Complex Systems I,”Engineering Cybernetics, No. 1, 1–9.
ZABOLOTNYY, A. M., and PLINER, V. (1986) “Discrete Representations in Metric Scaling Problems,”Soviet Journal of Computer and Systems Sciences, 24, 145–149.
ZAR, J. H. (1984),Biostatistical Analysis, 2nd ed., Englewood Cliffs, NJ: Prentice-Hall.
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Pliner, V. Metric unidimensional scaling and global optimization. Journal of Classification 13, 3–18 (1996). https://doi.org/10.1007/BF01202579
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DOI: https://doi.org/10.1007/BF01202579