A simulated annealing heuristic for unidimensional and multidimensional (city-block) scalling of symmetric proximity

Journal of Classification

Volumn 18, Issue 1, 2001, Pages 3-33

  Brusco, Michael J ^a  

^a FLORIDA STATE UNIVERSITY   (United States)

Author keywords

City block distance; Combinatorial programming; Metric multidimensional scaling; Simulated annealing; Unidimensional scaling

Indexed keywords

EID: 0035650086     PISSN: 01764268     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00357-0003-4     Document Type: Article

References (44)

1. ARABIE, P. (1973), "Concerning Monte Carlo Evaluations of Nonmetric Multidimensional Scaling Algorithms," Psychometrika, 38, 607-608.
2. ARABIE, P. (1991), "Was Euclid an Unnecessarily Sophisticated Psychologist?" Psychometrika, 56, 567-587.
3. ARMSTRONG, R. D., FROME, E. L., and KUNG, D. S. (1979), "A Revised Simplex Algorithm for the Absolute Deviation Curve Fitting Problem," Communications in Statistics - Simulation and Computation, B8, 175-190.
4. 1 Linear Approximation," SIAM Journal on Numerical Analysis, 10, 839-848.
5. 1 Linear Approximation with Linear Constraints," SIAM Journal on Numerical Analysis, 15, 603-611.
6. 1 Linear Approximation Problem," ACM Transactions on Mathematical Software, 6, 231-235.
7. BORG, I., and GROENEN, P. (1997), Modern Multidimensional Scaling: Theory and Applications, New York: Springer-Verlag.
8. BRUSCO, M. J. (1999), "Morph-Based Local Search Heuristics for Large-Scale Combinatorial Data Analysis," Journal of Classification, 16, 163-180.
9. BRUSCO, M. J., and 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.
10. CARROLL, J. D., and ARABIE, P. (1980), "Multidimensional Scaling," Annual Review of Psychology, 31, 607-649.
11. CARROLL, J. D., and ARABIE, P. (1998), "Multidimensional Scaling," in Management, Judgment, and Decision Making, Ed., M. H. Birnbaum, San Diego: Academic Press, pp. 179-250.
12. CHEN, G. (2000). Metric Two-Way Multidimensional Scaling and Circular Unidimensional Scaling: Mixed Integer Programming Approaches, Doctoral Dissertation, Department of Management Science and Information Systems, Rutgers University, Newark, New Jersey.
13. DEFAYS, D. (1978), "A Short Note on a Method of Seriation," British Journal of Mathematical and Statistical Psychology, 31, 49-53.
14. DE LEEUW, J., and HEISER, W. J. (1977), "Convergence of Correction-Matrix Algorithms for Multidimensional Scaling," in Geometric Representations of Relational Data: Readings in Multidimensional Scaling, Ed., J.C. Lingoes, Ann Arbor, Michigan: Mathesis Press, 735-752.
15. DE LEEUW, J., and HEISER, W. J. (1980), "Multidimensional Scaling with Restrictions on the Configuration of Points," in Multivariate Analysis, Ed., P.R. Krishnaiah, Amsterdam: North-Holland, Vol. V, 501-522.
16. DE SOETE, G., HUBERT, L., and ARABIE, P. (1988), "The Comparative Performance of Simulated Annealing on Two Problems of Combinatorial Data Analysis," in Data Analysis and Informatics (Vol. 5), Ed. E. Diday, Amsterdam: North Holland, pp. 489-496.
17. GROENEN, P. J. F. (1993), The Majorization Approach to Multidimensional Scaling: some Problems and Extensions, Leiden, The Netherlands: DSWO Press.
18. GROENEN, P. J. F., and HEISER, W. J. (1996), "The Tunneling Method for Global Optimization in Multidimensional Scaling," Psychometrika, 61, 529-550.
19. GROENEN, P. J. F., HEISER, W. J., and MEULMAN, J. J. (1998), "City-Block Scaling: Smoothing Strategies for Avoiding Local Minima," in Classification, Data Analysis, and Data Highways, Eds., I. Balderjahn, R. Mathur, and M. Schader, Berlin: Springer, 46-53.
20. GROENEN, P. J. F., HEISER, W. J., and MEULMAN, J. J. (1999), "Global Optimization in Least-Squares Multidimensional Scaling by Distance Smoothing," Journal of Classification, 16, 225-254.
21. GROENEN, P. J. F., MATHUR, R., and HEISER, W. J. (1995), "The Majorization Approach to Multidimensional Scaling for Minkowski Distances," Journal of Classification, 12, 3-19.
22. HEISER, W.J. (1989), "The City-Block Model for Three-Way Multidimensional Scaling," in Multiway Data Analysis, Ed., R. Coppi and S. Belasco, Amsterdam: North-Holland, 395-404.
23. HELLER, J. (1961), Catch-22: A Novel, New York: Simon and Schuster.
24. HUBERT, L. J. (1987), Assignment Methods in Combinatorial Data Analysis, New York: Marcel Dekker.
25. HUBERT, L. J., and ARABIE, P. (1986), "Unidimensional Scaling and Combinatorial Optimization," in Multidimensional Data Analysis, Ed., J. de Leeuw, W. Heiser, J. Meulman, and F. Critchley, Leiden, The Netherlands: DSWO Press, 181-196.
26. HUBERT, L. J., and ARABIE, P. (1988), "Relying on Necessary Conditions for Optimization: Unidimensional Scaling and Some Extensions," in Classification and Related Methods for Data Analysis, Ed., H.H. Bock, Amsterdam: North Holland, 463-472.
27. HUBERT, L., and 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.
28. HUBERT, L., ARABIE, P., and HESSON-MCINNIS, M. (1992), Multidimensional Scaling in the City-Block Metric: A Combinatorial Approach," Journal of Classification, 9, 211-236.
29. HUBERT, L., ARABIE, P., and MEULMAN J. (1997), "Linear and Circular Unidimensional Scaling for Symmetric Proximity Matrices," British Journal of Mathematical and Statistical Psychology, 50, 253-284.
30. HUBERT, L., ARABIE, P., and MEULMAN J. (2001), Combinatorial Data Analysis: Optimization by Dynamic Programming, Philadelphia: Society for Industrial and Applied Mathematics, in press.
31. HUBERT, L. J., and BUSK, P. (1976), "Normative Location Theory: Placement in Continuous Space," Journal of Mathematical Psychology, 14, 187-210.
32. HUBERT, L. J., and SCHULTZ, J. V. (1976), "Quadratic Assignment as a General Data Analysis Strategy," British Journal of Mathematical and Statistical Psychology, 29, 190-241.
33. KRUSKAL, J. B. (1964), "Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis," Psychometrika, 29, 1-27.
34. LAU, K., LEUNG, P. L., and TSE, K. (1998), "A Nonlinear Programming Approach to Metric Unidimensional Scaling," Journal of Classification, 15, 3-14.
35. LAWSON, C. L., and HANSON, R. J. (1974), Solving Least Squares Problems, Englewood Cliffs, New Jersey: Prentice-Hall.
36. METROPOLIS, N., ROSENBLUTH, A., ROSENBLUTH, M., TELLER, A., and TELLER, E. (1953), "Equation of State Calculations by Fast Computing Machines," Journal of Chemical Physics, 21, 1087-1092.
37. POOLE, K. T. (1990), "Least Squares Metric, Unidimensional Scaling of Multivariate Linear Models," Psychometrika, 55, 123-149.
38. PLINER, V. (1996), "Metric Unidimensional Scaling and Global Optimization," Journal of Classification, 13, 3-18.
39. ROBINSON, W.S. (1951), "A Method for Chronologically Ordering Archaeological Deposits," American Antiquity, 16, 293-301.
40. ROTHKOPF, E. Z. (1957), "A Measure of Stimulus Similarity and Errors in Some Paired-Associate Learning," Journal of Experimental Psychology, 53, 94-101.
41. SHEPARD, R. N. (1963), "Analysis of Proximities as a Technique for the Study of Information Processing in Man," Human Factors, 5, 33-48.
42. SHEPARD, R. N. (1974), "Representation of Structure in Similarity Data: Problems and Prospects," Psychometrika, 39, 373-421.
43. SIMANTIRAKI, E. (1996), "Unidimensional Scaling: A Linear Programming Approach Minimizing Absolute Deviations," Journal of Classification, 13, 19-25.
44. SPÄTH, H. (1992), Mathematical Algorithms for linear Regression, San Diego, CA: Academic Press.