×

Spline-based nonlinear biplots. (English) Zbl 1414.62209

Summary: Biplots are helpful tools to establish the relations between samples and variables in a single plot. Most biplots use a projection interpretation of sample points onto linear lines representing variables. These lines can have marker points to make it easy to find the reconstructed value of the sample point on that variable. For classical multivariate techniques such as principal components analysis, such linear biplots are well established. Other visualization techniques for dimension reduction, such as multidimensional scaling, focus on an often nonlinear mapping in a low dimensional space with emphasis on the representation of the samples. In such cases, the linear biplot can be too restrictive to properly describe the relations between the samples and the variables. In this paper, we propose a simple nonlinear biplot that represents the marker points of a variable on a curved line that is governed by splines. Its main attraction is its simplicity of interpretation: the reconstructed value of a sample point on a variable is the value of the closest marker point on the smooth curved line representing the variable. The proposed spline-based biplot can never lead to a worse overall sample fit of the variable as it contains the linear biplot as a special case.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Borg I, Groenen PJF (2005) Modern multidimensional scaling. Springer Science + Business Media Inc, New York · Zbl 1085.62079
[2] De Boor C (1978) A practical guide to splines. Springer-Verlag, New York · Zbl 0406.41003
[3] Gifi A (1990) Nonlinear multivariate analysis. Wiley, Chichester · Zbl 0697.62048
[4] Gower, JC, Some distance properties of latent root and vector methods used in multivariate analysis, Biometrika, 53, 588-589, (1966)
[5] Gower, JC, Euclidean distance geometry, Math Sci, 7, 1-14, (1982) · Zbl 0492.51017
[6] Gower JC, Hand DJ (1996) Biplots. Chapman and Hall, London · Zbl 0867.62053
[7] Gower, JC; Legendre, P., Metric and Euclidean properties of dissimilarity coefficients, J Classif, 3, 5-48, (1986) · Zbl 0592.62048
[8] Gower JC, Ngouenet RF (2005) Nonlinearity effects in multidimensional scaling. J Multivar Anal 94: 344-365 · Zbl 1122.62061
[9] Gower, JC; Meulman, JJ; Arnold, GM, Nonmetric linear biplots, J Classif, 16, 181-196, (1999) · Zbl 0939.62058
[10] Gower JC, Lubbe S, Le Roux NJ (2011) Understanding biplots. Wiley, Chichester
[11] Hand DJ, Daly F, Lunn AD, McConway KJ, Ostrowski E (1994) A handbook of small data sets. Chapman & Hall, London · Zbl 0949.62500
[12] Hastie, T.; Stuetzle, W., Principal curves, J Am Stat Assoc, 84, 502-516, (1989) · Zbl 0679.62048
[13] Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning, 2nd edn. Springer, New York · Zbl 1273.62005
[14] Jolliffe IT (2002) Principal component analysis, 2nd edn. Springer, New York · Zbl 1011.62064
[15] Kruskal JB, Wish MW (1978) Multidimensional scaling. Sage Publications, Beverley Hills
[16] Nelder, JA; Mead, R., A simplex method for function minimization, Comput J, 7, 308-313, (1965) · Zbl 0229.65053
[17] Vapnik V (1996) The nature of statistical learning theory. Springer, New York · Zbl 0934.62009
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.