Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-14T01:38:38.816Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Least-Squares Unidimensional Scaling: Improved Branch-and-Bound Procedures and Comparison to Dynamic Programming

Published online by Cambridge University Press:  01 January 2025

Michael J. Brusco and
Stephanie Stahl
Michael J. Brusco*
Affiliation:
Florida State University, Tallahassee, FL, USA
Stephanie Stahl
Affiliation:
Tallahassee, FL, USA
*
Requests for reprints should be sent to: Michael J. Brusco, Department of Marketing, College of Business, Florida State University, Tallahassee, FL 32306-1110, USA. E-mail: mbrusco@cob.fsu.edu
Get accessRights & Permissions [Opens in a new window]

Abstract

There are two well-known methods for obtaining a guaranteed globally optimal solution to the problem of least-squares unidimensional scaling of a symmetric dissimilarity matrix: (a) dynamic programming, and (b) branch-and-bound. Dynamic programming is generally more efficient than branch-and-bound, but the former is limited to matrices with approximately 26 or fewer objects because of computer memory limitations. We present some new branch-and-bound procedures that improve computational efficiency, and enable guaranteed globally optimal solutions to be obtained for matrices with up to 35 objects. Experimental tests were conducted to compare the relative performances of the new procedures, a previously published branch-and-bound algorithm, and a dynamic programming solution strategy. These experiments, which included both synthetic and empirical dissimilarity matrices, yielded the following findings: (a) the new branch-and-bound procedures were often drastically more efficient than the previously published branch-and-bound algorithm, (b) when computationally feasible, the dynamic programming approach was more efficient than each of the branch-and-bound procedures, and (c) the new branch-and-bound procedures require minimal computer memory and can provide optimal solutions for matrices that are too large for dynamic programming implementation.

Keywords

combinatorial data analysisleast-squares unidimensional scalingbranch-and-bounddynamic programming

Information

Type
Original Paper
Information
Psychometrika , Volume 70 , Issue 2 , June 2005, pp. 253 - 270
Copyright
Copyright © 2005 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The authors gratefully acknowledge the helpful comments of three anonymous reviewers and the Editor. We especially thank Larry Hubert and one of the reviewers for providing us with the MATLAB files for optimal and heuristic least-squares unidimensional scaling methods.

References

Arabie, P., & Hubert, L.J. (1992). Combinatorial data analysis. Annual Review of Psychology, 43, 169–203.CrossRefGoogle Scholar
Baker, F.B., & Hubert, L.J. (1977). Applications of combinatorial programming to data analysis: Seriation using asymmetric proximity measures. British Journal of Mathematical and Statistical Psychology, 30, 154–164.CrossRefGoogle Scholar
Brusco, M.J. (2002a). A branch-and-bound algorithm for fitting anti-Robinson structures to symmetric dissimilarity matrices. Psychometrika, 67, 459–471.CrossRefGoogle Scholar
Brusco, M.J. (2002b). Identifying a reordering of the rows and columns of multiple proximity matrices using multiobjective programming. Journal of Mathematical Psychology, 46, 731–745.10.1006/jmps.2002.1423CrossRefGoogle Scholar
Brusco, M.J., & Stahl, S. (2000). Using quadratic assignment methods to generate initial permutations for least-squares unidimensional scaling of symmetric proximity matrices. Journal of Classification, 17, 197–223.CrossRefGoogle Scholar
Brusco, M.J., & Stahl, S. (2001). An interactive approach to multiobjective combinatorial data analysis. Psychometrika, 66, 5–24.CrossRefGoogle Scholar
De Cani, J.S. (1972). A branch and bound algorithm for maximum likelihood paired comparison ranking by linear programming. Biometrika, 59, 131–135.CrossRefGoogle Scholar
Defays, D. (1978). A short note on a method of seriation. British Journal of Mathematical and Statistical Psychology, 31, 49–53.CrossRefGoogle Scholar
de Leeuw, J., & Heiser, W.J. (1977). Convergence of correction-matrix algorithms for multidimensional scaling. In Lingoes, J.C. (Eds.), Geometric representations of relational data: Readings in multidimensional scaling (pp. 735–752). Ann Arbor Michigan: Mathesis Press.Google Scholar
De Soete, G., Hubert, L., & Arabie, P. (1988). The comparative performance of simulated annealing on two problems of combinatorial data analysis. In Diday, E. (Eds.), Data analysis and informatics, Vol. 5 (pp. 489–496). (5th ed.). North Holland: Amsterdam.Google Scholar
Flueck, J.A., & Korsh, J.F. (1974). A branch search algorithm for maximum likelihood paired comparison ranking. Biometrika, 61, 621–626.CrossRefGoogle Scholar
Groenen, P.J.F. (1993). The Majorization Approach to Multidimensional Scaling: Some Problems and Extensions. Leiden, Netherlands: DSWO Press.Google Scholar
Groenen, P.J.F., & Heiser, W.J. (1996). The tunneling method for global optimization in multidimensional scaling. Psychometrika, 61, 529–550.CrossRefGoogle Scholar
Groenen, P.J.F., Heiser, W.J., & Meulman, J.J. (1999). Global optimization of least-squares multidimensional scaling by distance smoothing. Journal of Classification, 16, 225–254.CrossRefGoogle Scholar
Hubert, L.J. (1976). Seriation using asymmetric proximity measures. British Journal of Mathematical and Statistical Psychology, 29, 32–52.10.1111/j.2044-8317.1976.tb00701.xCrossRefGoogle Scholar
Hubert, L.J., & Arabie, P. (1986). Unidimensional scaling and combinatorial optimization. In de Leeuw, J., Heiser, W., Meulman, J., & Critchley, F. (Eds.), Multidimensional Data Analysis. (pp. 181–196). Leiden Netherlands: DSWO Press.Google Scholar
Hubert, L., Arabie, P. (1994). The analysis of proximity matrices through sums of matrices having (anti-) Robinson forms. British Journal of Mathematical and Statistical Psychology, 47, 1–40.10.1111/j.2044-8317.1994.tb01023.xCrossRefGoogle Scholar
Hubert, L., Arabie, P., & Meulman, J. (1997). Linear and circular unidimensional scaling for symmetric proximity matrices. British Journal of Mathematical and Statistical Psychology, 50, 253–284.10.1111/j.2044-8317.1997.tb01145.xCrossRefGoogle Scholar
Hubert, L., Arabie, P., & Meulman, J. (1998). Graph-theoretic representations for proximity matrices through strongly anti-Robinson or circular strongly anti-Robinson matrices. Psychometrika, 63, 341–358.CrossRefGoogle Scholar
Hubert, L., Arabie, P., & Meulman, J. (2001). Combinatorial Data Analysis: Optimization by Dynamic Programming. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Hubert, L., Arabie, P., & Meulman, J. (2002). Linear unidimensional scaling in the L2-Norm: Basic optimization methods using MATLAB. Journal of Classification, 19, 303–328.10.1007/s00357-001-0047-5CrossRefGoogle Scholar
Hubert, L.J., & Baker, F.B. (1977). The comparison and fitting of given classification schemes. Journal of Mathematical Psychology, 16, 233–253.10.1016/0022-2496(77)90054-2CrossRefGoogle Scholar
Hubert, L.J., & Golledge, R.G. (1981). Matrix reorganization and dynamic programming: Applications to paired comparisons and unidimensional seriation. Psychometrika, 46, 429–441.10.1007/BF02293800CrossRefGoogle Scholar
Lawler, E.L. (1964). A comment on minimum feedback arc sets. IEEE Transactions on Circuit Theory, 11, 296–297.CrossRefGoogle Scholar
Manning, S.K., & Shofner, E. (1991). Similarity ratings and confusability of lipread consonants compared with similarity ratings of auditory and orthographic stimuli. American Journal of Psychology, 104, 587–604.CrossRefGoogle ScholarPubMed
Morgan, B.J.T., Chambers, S.M., & Morton, J. (1973). Acoustic confusion of digits in memory and recognition. Perception & Psychophysics, 14, 375–383.CrossRefGoogle Scholar
Pliner, V. (1996). Metric unidimensional scaling and global optimization. Journal of Classification, 13, 3–18.10.1007/BF01202579CrossRefGoogle Scholar
Robinson, W.S. (1951). A method for chronologically ordering archaeological deposits. American Antiquity, 16, 293–301.CrossRefGoogle Scholar
Ross, B.H., & Murphy, G.L. (1999). Food for thought: Cross-classification and category organization in a complex real-world domain. Cognitive Psychology, 38, 495–553.CrossRefGoogle Scholar
Rothkopf, E.Z. (1957). A measure of stimulus similarity and errors in some paired-associate learning tasks. Journal of Experimental Psychology, 53, 94–101.CrossRefGoogle ScholarPubMed
Younger, D.H. (1963). Minimum feedback arc sets for a directed graph. IEEE Transactions on Circuit Theory, 10, 238–245.10.1109/TCT.1963.1082116CrossRefGoogle Scholar