Abstract
This paper establishes a general framework for metric scaling of any distance measure between individuals based on a rectangular individuals-by-variables data matrix. The method allows visualization of both individuals and variables as well as preserving all the good properties of principal axis methods such as principal components and correspondence analysis, based on the singular-value decomposition, including the decomposition of variance into components along principal axes which provide the numerical diagnostics known as contributions. The idea is inspired from the chi-square distance in correspondence analysis which weights each coordinate by an amount calculated from the margins of the data table. In weighted metric multidimensional scaling (WMDS) we allow these weights to be unknown parameters which are estimated from the data to maximize the fit to the original distances. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing a matrix and displaying its rows and columns in biplots.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
BARTELS, R.H., GOLUB, G.H. & SAUNDERS, M.A. (1970): Numerical techniques in mathematical programming. In Nonlinear Programming (eds J.B. Rosen, O.L. Mangasarian & K. Ritter), London: Academic Press, pp. 123–176.
FIELD, J.G., CLARKE, K.R. & WARWICK, R.M. (1982): A practical strategy for analysing multispecies distribution patterns. Marine Ecology Progress Series, 8, 37–52.
GABRIEL, K. R. (1971): The biplot-graphic display of matrices with applications to principal component analysis. Biometrika, 58, 453–467.
GOWER, J.C. & LEGENDRE, P. (1986): Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification, 3, 5–48.
GREENACRE, M.J. (1984): Theory and Applications of Correspondence Analysis. London: Academic Press.
RIOS, M., VILLAROYA, A. & OLLER, J.M. (1994): Intrinsic data analysis: a method for the simultaneous representation of populations and variables. Research report 160, Department of Statistics, University of Barcelona.
VIVES, S. & VILLAROYA, A. (1996): La combinació de tècniques de geometria diferencial amb anàlisi multivariant clàssica: una aplicació a la caracterització de les comarques catalanes. Qüestiio, 20, 449–482.
S-PLUS (1999). S-PLUS 2000 Guide to Statistics, Volume 1, Mathsoft, Seattle, WA.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin · Heidelberg
About this paper
Cite this paper
Greenacre, M. (2005). Weighted Metric Multidimensional Scaling. In: Bock, HH., et al. New Developments in Classification and Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27373-5_17
Download citation
DOI: https://doi.org/10.1007/3-540-27373-5_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23809-6
Online ISBN: 978-3-540-27373-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)