Optimal scaling by alternating length-constrained nonnegative least squares, with application to distance-based analysis

Psychometrika

Volumn 65, Issue 4, 2000, Pages 511-524

  Groenen, Patrick J F ^a   Van Os, Bart Jan ^b   Meulman, Jacqueline J ^b  

^a LEIDEN UNIVERSITY   (Netherlands)

^b NONE

Author keywords

Alternating least squares; Constrained optimization; Distance based (principal components) analysis; Length constraints; Multidimensional scaling; Nonnegative least squares; Optimal scaling

Indexed keywords

EID: 0034561624     PISSN: 00333123     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02296341     Document Type: Article

References (26)

1. Borg, I., & Groenen, P.J.F. (1997). Modern multidimensional scaling: Theory and applications. New York: Springer.
2. Breiman, L., & Friedman, J.H. (1985). Estimating optimal transformations for multiple regression and correlation. Journal of the American Statistical Association, 80, 580-619.
3. Bro, R., & De Jong, S. (1997). A fast non-negativity-constrained least squares algorithm. Journal of Chemometrics, 11, 393-401.
4. De Boor, C. (1978). A practical guide to splines. New York: Springer.
5. de Leeuw, J. (1977). Normalized cone regression. Leiden, The Netherlands: Department of Data Theory.
6. de Leeuw, J., & Heiser, W.J. (1977). Convergence of correction-matrix algorithms for multidimensional scaling. In J.C. Lingoes, E.E. Roskam, & I. Borg (Eds.), Geometric representations of relational data (pp. 735-752). Ann Arbor, MI: Mathesis Press.
7. Dinkelbach, W. (1967). On nonlinear fractional programming. Managment Science, 13, 492-498.
8. Gifi, A. (1990). Nonlinear multivariate analysis. Chichester: Wiley.
9. Gower, J.C., & Hand, D.J. (1996). Biplots. London: Chapman and Hall.
10. Groenen, P.J.F., Commandeur, J.J.F., & Meulman, J.J. (1997). PIONEER: A program for distance-based multivariate analysis. In W. Bandilla & F. Faulbaum (Eds.), Softstat'97: Advances in statistical software (pp. 83-90). Stuttgart: Lucius & Lucius.
11. Heiser, W.J. (1985). Multidimensional scaling by optimizing goodness-of-fit to a smooth hypothesis (Tech. Rep. RR-85-07). Leiden, The Netherlands: Leiden University, Department of Data Theory.
12. Jones, L.K., & Seaton, L. (Eds.). (1994). Math77: Reference manual. Sterling, VA: Language Systems Coorporation.
13. Kiers, H. A.L. (1995). Maximization of sums of quotients of quadratic forms and some generalizations. Psychometrika, 60, 221-245.
14. Kruskal, J.B. (1964a). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-27.
15. Kruskal, J.B. (1964b). Nonmetric multidimensional scaling: A numerical method, Psychometrika, 29, 115-129.
16. Kruskal, J.B., & Carroll, J.D. (1969). Geometrical models and badness-of-fit functions. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. 2, pp. 639-671). New York: Academic Press.
17. Krzanowski, W.J., & Marriott, F.H.C. (1994). Multivariate analysis. Part 1: Distributions, ordination and inference. London: Wiley.
18. Lawson, C.L., & Hanson, R.J. (1974). Solving least squares problems. Englewood Cliffs, NJ: Prentice Hall.
19. Meulman, J.J. (1986). A distance approach to nonlinear multivariate analysis. Leiden, The Netherlands: DSWO Press.
20. Meulman, J.J. (1992). The integration of multidimensional scaling and multivariate analysis with optimal transformations. Psychometrika, 57, 539-565.
21. Meyer, R. (1997). Nonlinear eigenvector algorithms for local optimization in multivariate data analysis. Linear Algebra and its Applications, 264, 225-246.
22. Ramsay, J.O. (1988). Monotone regression splines in action. Statistical Science, 3, 425-461.
23. Rockafellar, R.T. (1970). Convex analysis. Princeton, NJ: Princeton University Press.
24. Verboon, P. (1994). A robust approach to nonlinear multivariate analysis. Leiden, The Netherlands: DSWO Press.
25. Watson, G.A. (1985). On a class of algorithms for total approximation. Journal of Approximation Theory, 45, 219-231.
26. Young, F.W., de Leeuw, J., & Takane, Y. (1976). Regression with qualitative and quantitative variables: An alternating least squares approach. Psychometrika, 41, 505-529.