Least Squares Multidimensional Scaling with Transformed Distances
- 281 Downloads
We consider a general least squares loss function for multidimensional scaling. Special cases of this loss function are STRESS, S-STRESS, and MULTISCALE. Several analytic results are presented. In particular, we present the gradient and Hessian, and look at the differentiability at a local minimum. We also consider fulldimensional scaling and indicate when a global minimum can be obtained. Furthermore, we treat the problem of inverse multidimensional scaling, where the aim is to find those dissimilarity matrices for which a fixed configuration is a stationary point.
Unable to display preview. Download preview PDF.
- CRITCHLEY, F. (1986): Dimensionality theorems in multidimensional scaling and hierarchical cluster analysis. In: E. Diday, Y. Escoufier, L. Lebart, J. Lepage, Y. Schektman, and R. Tomassone (eds.), Informatics, IV, North-Holland, Amsterdam, 45–70.Google Scholar
- DE LEEUW, J., and GROENEN, P.J.F. (1993): Inverse scaling. Tech. rep. 144, UCLA Statistics Series, Interdivisonal Program in Statistics, UCLA, Los Angeles, California.Google Scholar
- DE LEEUW, J., and HEISER, W. J. (1980): Multidimensional scaling with restrictions on the configuration. In: Krishnaiah, P. (ed.), Multivariate Analysis, volume V. North Holland, Amsterdam, 501–522.Google Scholar
- GLUNT, W., HAYDEN, T., and LIU, W.-M. (1991): The embedding problem for predistance matrices. Bulletin of Mathematical Biology, 53, 769–796.Google Scholar
- GROENEN, P. J.F., MATHAR, R., and HEISER, W. J. (1992): The majorization approach to multidimensional scaling for Minkowski distances. Tech. rep. RR- 92–11, Department of Data Theory, Leiden.Google Scholar
- KRUSKAL, J. B., YOUNG, F. W., and SEERY, J. (1977): How to use kyst-2, a very flexible program to do multidimensional scaling. Tech. rep. AT&T Bell Laboratories, Murray Hill, New Jersey.Google Scholar
- STOOP, I., HEISER, W.J., and DE LEEUW, J. (1981): How to use smacof-I A. Tech. rep. Department of Data Theory, Leiden.Google Scholar