The solution of the metric STRESS and SSTRESS problems in multidimensional scaling using Newton's method

Computational Statistics

Volumn 13, Issue 3, 1998, Pages 369-396

  Kearsley, Anthony J ^a   Tapia, Richard A ^b   Trosset, Michael W ^b  

^a NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY   (United States)

^b RICE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032376620     PISSN: 09434062     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (44)

1. Barzilai, J. and Borwein, J. (1988), Two-point step size gradient methods. IMA Journal of Numerical Analysis, 8, 141-148.
2. Boggs, P. T., Byrd, R. H., Donaldson, J. R., and Schnabel, R. B. (1991), ODRPACK: Software for weighted orthogonal distance regression. ACM Transactions on Mathematical Software, 15, 100-200.
3. Boggs, P. T. and Dennis, J. E. (1976), A stability analysis for perturbed nonlinear iterative methods. Mathematics of Computation, 30, 1-17.
4. Boggs, P. T., Tolle, J. W., and Kearsley, A. J. (1994), A practical algorithm for general large scale nonlinear optimization problems. Technical Report NISTIR 5407, Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD. To appear in SIAM Journal on Optimization.
5. Browne, M. (1987), The Young-Householder algorithm and the least square multidimensional scaling of squared distance. Journal of Classification, 4, 175-190.
6. Crippen, G. M. and Havel, T. F. (1988), Distance Geometry and Molecular Conformation, John Wiley & Sons, New York.
7. de Leeuw, J. (1977), Applications of convex analysis to multidimensional scaling. In Barra, J. R., Brodeau, F., Romier, G., and van Cutsem, B., editors, Recent Developments in Statistics, pages 133-145. North-Holland Publishing Company, Amsterdam.
8. de Leeuw, J. (1984), Differentiability of Kruskal's Stress at a local minimum. Psychometrika, 49, 111-113.
9. de Leeuw, J. (1988), Convergence of the majorization method for multidimensional scaling. Journal of Classification, 5, 163-180.
10. de Leeuw, J. and Heiser, W. (1982), Theory of multidimensional scaling. In Krishnaiah, P. R. and Kanal, I. N., editors, Handbook of Statistics, volume 2, chapter 13, pages 285-316. North-Holland Publishing Company, Amsterdam.
11. Dennis, J. E., Gay, D. M., and Welsch, R. E. (1981), An adaptive nonlinear least-squares algorithm. ACM Transactions on Mathematical Software, 1, 348-368.
12. Dennis, J. E. and Schnabel, R. B. (1983), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, New Jersey.
13. Gilbert, J. C. and LeMaréchal, C. (1989), Some numerical experiments with variable-storage quasi-Newton algorithms. Mathematical Programming, 45, 407-435.
14. Glunt, W., Hayden, T. L., Hong, S., and Wells, J. (1990), An alternating projection algorithm for computing the nearest Euclidean distance matrix. SIAM Journal on Matrix Analysis and Applications, 11, 589-600.
15. Glunt, W., Hayden, T. L., and Liu, W.-M. (1991), The embedding problem for predistance matrices. Bulletin of Mathematical Biology, 53, 769-796.
16. Glunt, W., Hayden, T. L., and Raydan, M. (1993), Molecular conformations from distance matrices. Journal of Computational Chemistry, 14, 114-120.
17. Glunt, W., Hayden, T. L., and Raydan, M. (1994), Preconditioners for distance matrix algorithms. Journal of Computational Chemistry, 15, 227-232.
18. Gower, J. C. (1966), Some distance properties of latent root and vector methods in multivariate analysis. Biometrika, 53, 315-328.
19. Groenen, P. J. F. (1993), The Majorization Approach to Multidimensional Scaling, DSWO Press, Leiden, The Netherlands.
20. Guttman, L. (1968), A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469-506.
21. Havel, T. F. (1991), An evaluation of computational strategies for use in the determination of protein structure from distance constraints obtained by nuclear magnetic resonance. Progress in Biophysics and Molecular Biology, 56, 43-78.
22. Hebden, M. D. (1973), An algorithm for minimization using exact second derivatives. Report TP515, Atomic Energy Research Establishment, Harwell, England.
23. Heiser, W. J. and de Leeuw, J. (1986), SMACOF-I. Technical Report UG-86-02, Department of Data Theory, University of Leiden, Leiden, The Netherlands.
24. Hoare, M. R. and Pal, P. (1971), Physical cluster mechanics: Statics and energy surfaces for monatomic systems. Advances in Physics, 20, 161-196.
25. Kruskal, J. B. (1964a), Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-27.
26. Kruskal, J. B. (1964b), Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 28-42.
27. Kruskal, J. B. (1977), Multidimensional scaling and other methods for discovering structure. In Enslein, K., Ralston, A., and Wilf, H. S., editors, Statistical Methods for Digital Computers, chapter 12, pages 296-339. John Wiley & Sons, New York. Volume III of Mathematical Methods for Digital Computers.
28. Kruskal, J. B. (1977), Multidimensional scaling and other methods for discovering structure. In Enslein, K., Ralston, A., and Wilf, H. S., editors, Statistical Methods for Digital Computers, chapter 12, pages 296-339. John Wiley & Sons, New York. Volume III of Mathematical Methods for Digital Computers.
29. Kruskal, J. B. and Wish, M. (1978), Multidimensional Scaling, Sage Publications, Beverly Hills. Sage University Paper series on Quantitative Applications in the Social Sciences, 07-001.
30. Liu, D. C. and Nocedal, J. (1989), On the limited memory BFGS method for large scale optimization. Mathematical Programming, 45, 503-528.
31. McFarlane, M. and Young, F. W. (1994), Graphical sensitivity analysis for multidimensional scaling. Journal of Computational and Graphical Statistics, 3, 23-33.
32. Moré, J. J. (1977), The Levenberg-Marquardt algorithm: Implementation and theory. In Watson, G. A., editor, Numerical Analysis, pages 105-116. Springer Verlag, Berlin. Lecture Notes in Mathematics, 630.
33. Moré, J. J. (1977), The Levenberg-Marquardt algorithm: Implementation and theory. In Watson, G. A., editor, Numerical Analysis, pages 105-116. Springer Verlag, Berlin. Lecture Notes in Mathematics, 630.
34. Moré, J. J., Garbow, B. S., and Hillstrom, K. E. (1980), User's guide for MINPACK-1. Technical Report ANL 80-74, Argonne National Laboratories.
35. Northby, J. A. (1987), Structure and binding of Lennard-Jones clusters: 13 ≤ n ≤ 147. Journal of Chemical Physics, 87, 6166-6177.
36. Powell, M. J. D. (1970), A hybrid method for nonlinear equations. In Rabinowitz, P., editor, Numerical Methods for Nonlinear Algebraic Equations, pages 87-114. Gordon and Breach, London.
37. Raydan, M. (1993), On the Barzilai and Borwein choice of a steplength for the gradient method. IMA Journal of Numerical Analysis, 13, 321-326.
38. Sorensen, D. C. (1991), Implicit application of polynomial filters in a k-step Arnoldi method. SIAM Journal on Matrix Analysis, 13, 357-385.
39. Takane, Y., Young, F. W., and de Leeuw, J. (1977), Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 42, 7-67.
40. Tarazaga, P. and Trosset, M. W. (1993), An optimization problem on subsets of the symmetric postive semidefinite matrices. Journal of Optimization Theory and Applications, 79, 513-524.
41. Torgerson, W. S. (1952), Multidimensional scaling: I. Theory and method. Psychometrika, 17, 401-419.
42. Torgerson, W. S. (1958), Theory and Methods of Scaling, John Wiley & Sons, New York.
43. Wagenaar, W. A. and Padmos, P. (1971), Quantitative interpretation of Stress in Kruskal's method multidimensional scaling technique. British Journal of Mathematical and Statistical Psychology, 24, 101-110.
44. Young, G. and Householder, A. S. (1938), Discussion of a set of points in terms of their mutual distances. Psychometrika, 3, 19-22.