Spline-based nonlinear biplots

  • Patrick J. F. Groenen
  • Niël J. Le Roux
  • Sugnet Gardner-Lubbe
Regular Article


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.


Biplot Multidimensional scaling Principal components analysis Splines 

Mathematics Subject Classification




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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Patrick J. F. Groenen
    • 1
  • Niël J. Le Roux
    • 2
  • Sugnet Gardner-Lubbe
    • 3
  1. 1.Econometric InstituteErasmus UniversityRotterdamThe Netherlands
  2. 2.Department of Statistics and Actuarial ScienceStellenbosch UniversityStellenboschSouth Africa
  3. 3.Department of Statistical SciencesUniversity of Cape TownCape TownSouth Africa

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