Weighted Metric Multidimensional Scaling

UPF Economics and Business Working Paper 777

13 Pages Posted: 16 Nov 2005

See all articles by Michael Greenacre

Michael Greenacre

Universitat Pompeu Fabra - Faculty of Economic and Business Sciences

Date Written: September 2003

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.

Keywords: Biplot, correspondence analysis, distance, multidimensional scaling, singular-value decomposition

JEL Classification: C19, C88

Suggested Citation

Greenacre, Michael John, Weighted Metric Multidimensional Scaling (September 2003). UPF Economics and Business Working Paper 777, Available at SSRN: https://ssrn.com/abstract=848606 or http://dx.doi.org/10.2139/ssrn.848606

Michael John Greenacre (Contact Author)

Universitat Pompeu Fabra - Faculty of Economic and Business Sciences ( email )

Ramon Trias Fargas 25-27
Barcelona, 08005
Spain
34 93 542 25 51 (Phone)
34 93 542 17 46 (Fax)

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