Skip to main content
Log in

Spline-based nonlinear biplots

  • Regular Article
  • Published:
Advances in Data Analysis and Classification Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. If there are \(k\) parameters in the loss function to be minimized, then the Nelder–Mead algorithm requires a simplex of \(k +1\) vectors of test parameters to be set up. The main idea is to remove the test vector with the worst fit. Then the algorithm tests if this point when mirrored in the centroid of the \(k\) remaining simplex vectors has a better function value. If so, then the new point is kept, if not, then the simplex is shrunken. For more details on this heuristic method we refer to Nelder and Mead (1965).

References

  • Borg I, Groenen PJF (2005) Modern multidimensional scaling. Springer Science + Business Media Inc, New York

    MATH  Google Scholar 

  • De Boor C (1978) A practical guide to splines. Springer-Verlag, New York

    Book  MATH  Google Scholar 

  • Gifi A (1990) Nonlinear multivariate analysis. Wiley, Chichester

    MATH  Google Scholar 

  • Gower JC (1966) Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53:588–589

    MathSciNet  Google Scholar 

  • Gower JC (1982) Euclidean distance geometry. Math Sci 7:1–14

    MATH  MathSciNet  Google Scholar 

  • Gower JC, Hand DJ (1996) Biplots. Chapman and Hall, London

    MATH  Google Scholar 

  • Gower JC, Legendre P (1986) Metric and Euclidean properties of dissimilarity coefficients. J Classif 3:5–48

    Article  MATH  MathSciNet  Google Scholar 

  • Gower JC, Ngouenet RF (2005) Nonlinearity effects in multidimensional scaling. J Multivar Anal 94: 344–365

  • Gower JC, Meulman JJ, Arnold GM (1999) Nonmetric linear biplots. J Classif 16:181–196

    Article  MATH  Google Scholar 

  • Gower JC, Lubbe S, Le Roux NJ (2011) Understanding biplots. Wiley, Chichester

    Book  Google Scholar 

  • Hand DJ, Daly F, Lunn AD, McConway KJ, Ostrowski E (1994) A handbook of small data sets. Chapman & Hall, London

    Book  Google Scholar 

  • Hastie T, Stuetzle W (1989) Principal curves. J Am Stat Assoc 84:502–516

    Article  MATH  MathSciNet  Google Scholar 

  • Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning, 2nd edn. Springer, New York

    Book  MATH  Google Scholar 

  • Jolliffe IT (2002) Principal component analysis, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Kruskal JB, Wish MW (1978) Multidimensional scaling. Sage Publications, Beverley Hills

    Google Scholar 

  • Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313

    Article  MATH  Google Scholar 

  • Vapnik V (1996) The nature of statistical learning theory. Springer, New York

    Google Scholar 

Download references

Acknowledgments

We would like to thank Anthony la Grange for his valuable remarks and input for this project. We are also grateful to the remarks of the reviewers who have improved the quality of this manuscript. This work is based upon research supported by the National Research Foundation of South Africa. Any opinions, findings and conclusions, or recommendations expressed in this material are those of the authors and therefore the NRF does not accept any liability in regard thereof.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Niël J. Le Roux.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Groenen, P.J.F., Le Roux, N.J. & Gardner-Lubbe, S. Spline-based nonlinear biplots. Adv Data Anal Classif 9, 219–238 (2015). https://doi.org/10.1007/s11634-014-0179-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11634-014-0179-1

Keywords

Mathematics Subject Classification

Navigation