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.
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Arabie, P., & Hubert, L.J. (1992). Combinatorial data analysis. Annual Review of Psychology, 43, 169–203.
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.
Brusco, M.J. (2002a). A branch-and-bound algorithm for fitting anti-Robinson structures to symmetric dissimilarity matrices. Psychometrika, 67, 459–471.
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.
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.
Brusco, M.J., & Stahl, S. (2001). An interactive approach to multiobjective combinatorial data analysis. Psychometrika, 66, 5–24.
DeCani, J.S. (1972). A branch and bound algorithm for maximum likelihood paired comparison ranking by linear programming. Biometrika, 59, 131–135.
Defays, D. (1978). A short note on a method of seriation. British Journal of Mathematical and Statistical Psychology, 31, 49–53.
de Leeuw, J., & Heiser, W.J. (1977). Convergence of correction-matrix algorithms for multidimensional scaling. In J. C. Lingoes (Ed.), Geometric representations of relational data: Readings in multidimensional scaling (pp. 735–752). Ann Arbor Michigan: Mathesis Press.
De Soete, G., Hubert, L., & Arabie, P. (1988). The comparative performance of simulated annealing on two problems of combinatorial data analysis. In E. Diday (Ed.), Data analysis and informatics, vol. 5 (pp. 489–496). Amsterdam: North Holland.
Flueck, J.A., & Korsh, J.F. (1974). A branch search algorithm for maximum likelihood paired comparison ranking. Biometrika, 61, 621–626.
Groenen, P.J.F. (1993). The Majorization Approach to Multidimensional Scaling: Some Problems and Extensions. Leiden, Netherlands: DSWO Press.
Groenen, P.J.F., & Heiser, W.J. (1996). The tunneling method for global optimization in multidimensional scaling. Psychometrika, 61, 529–550.
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.
Hubert, L.J. (1976). Seriation using asymmetric proximity measures. British Journal of Mathematical and Statistical Psychology, 29, 32–52.
Hubert, L.J., & Arabie, P. (1986). Unidimensional scaling and combinatorial optimization. In J. de Leeuw, W. Heiser, J. Meulman, and F. Critchley (Eds.), Multidimensional Data Analysis (pp. 181–196). Leiden, Netherlands: DSWO Press.
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.
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.
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.
Hubert, L., Arabie, P., & Meulman, J. (2001). Combinatorial Data Analysis: Optimization by Dynamic Programming. Philadelphia: Society for Industrial and Applied Mathematics.
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.
Hubert, L.J., & Baker, F.B. (1977). The comparison and fitting of given classification schemes. Journal of Mathematical Psychology, 16, 233–253.
Hubert, L.J., & Golledge, R.G. (1981). Matrix reorganization and dynamic programming: Applications to paired comparisons and unidimensional seriation. Psychometrika, 46, 429–441.
Lawler, E.L. (1964). A comment on minimum feedback arc sets. IEEE Transactions on Circuit Theory, 11, 296–297.
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.
Morgan, B.J.T., Chambers, S.M., & Morton, J. (1973). Acoustic confusion of digits in memory and recognition. Perception & Psychophysics, 14, 375–383.
Pliner, V. (1996). Metric unidimensional scaling and global optimization. Journal of Classification, 13, 3–18.
Robinson, W.S. (1951). A method for chronologically ordering archaeological deposits. American Antiquity, 16, 293–301.
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.
Rothkopf, E.Z. (1957). A measure of stimulus similarity and errors in some paired-associate learning tasks. Journal of Experimental Psychology, 53, 94-101.
Younger, D.H. (1963). Minimum feedback arc sets for a directed graph. IEEE Transactions on Circuit Theory, 10, 238-245.
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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.
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Brusco, M.J., Stahl, S. Optimal Least-Squares Unidimensional Scaling: Improved Branch-and-Bound Procedures and Comparison to Dynamic Programming. Psychometrika 70, 253–270 (2005). https://doi.org/10.1007/s11336-002-1032-6
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DOI: https://doi.org/10.1007/s11336-002-1032-6