×

Nonlinearity effects in multidimensional scaling. (English) Zbl 1122.62061

Summary: When multidimensional scaling of \(n\) cases is derived from dissimilarities that are functions of \(p\) basic continuous variables, the question arises of how to relate the values of the variables to the configuration of \(n\) points. We provide a methodology based on nonlinear biplots that expresses nonlinearity in two ways: (i) each variable is represented by a nonlinear trajectory and (ii) each trajectory is calibrated by an irregular scale. Methods for computing, calibrating and interpreting these trajectories are given and exemplified. Not only are the tools of immediate practical utility but the methodology established assists in a critical appraisal of the consequences of using nonlinear measures in a variety of multidimensional scaling methods.

MSC:

62H99 Multivariate analysis
91C15 One- and multidimensional scaling in the social and behavioral sciences
62A09 Graphical methods in statistics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Clark, P.J., An extension of the coefficient of divergence for use with multiple characters, Copeia, 2, 61-64, (1952)
[2] Eckart, C.; Young, G., The approximation of one matrix by another of lower rank, Psychometrika, 1, 211-218, (1936) · JFM 62.1075.02
[3] Gabriel, K.R., The biplot graphical display of matrices with applications to principal components analysis, Biometrika, 58, 453-467, (1971) · Zbl 0228.62034
[4] Gower, J.C., Some distance properties of latent root and vector methods used in multivariate analysis, Biometrika, 53, 325-338, (1964) · Zbl 0192.26003
[5] Gower, J.C., Recent advances in biplot methodology, (), 295-325 · Zbl 0810.62003
[6] Gower, J.C., A general theory of biplots, (), 283-303
[7] Gower, J.C.; Hand, D.J., Biplots, (1996), Chapman & Hall London · Zbl 0867.62053
[8] Gower, J.C.; Harding, S.A., Nonlinear biplots, Biometrika, 75, 445-455, (1988) · Zbl 0654.62047
[9] Gower, J.C.; Legendre, P., Metric and Euclidean properties of dissimilarity coefficients, J. classification, 3, 5-48, (1986) · Zbl 0592.62048
[10] Gower, J.C.; Meulman, J.J.; Arnold, G.M., Non-metric linear biplots, J. classification, 16, 181-196, (1999) · Zbl 0939.62058
[11] Kruskal, W.J.; Wish, M., Multidimensional scaling, sage university papers on quantitative applications in the social sciences, number 07-011, (1978), Sage Publications Newbury Park, CA
[12] Schoenberg, I.J., Remarks to maurice Fréchet’s article “sur la définition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de hilbert”, Ann. math., 36, 724-732, (1935) · Zbl 0012.30703
[13] W. Stanley, M. Miller, Measuring technological change in jet fighter aircraft, Report No. R-2249-AF, Rand Corporation, Santa Monica, CA, 1979.
[14] Torgerson, W.S., Theory and methods of scaling, (1955), Wiley New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.