9. Generalized Biplots

  1. John Gower1,
  2. Sugnet Lubbe2 and
  3. Niël le Roux3

Published Online: 3 NOV 2010

DOI: 10.1002/9780470973196.ch9

Understanding Biplots

Understanding Biplots

How to Cite

Gower, J., Lubbe, S. and Roux, N. l. (2011) Generalized Biplots, in Understanding Biplots, John Wiley & Sons, Ltd, Chichester, UK. doi: 10.1002/9780470973196.ch9

Author Information

  1. 1

    The Open University, UK

  2. 2

    University of Cape Town, South Africa

  3. 3

    University of Stellenbosch, South Africa

Publication History

  1. Published Online: 3 NOV 2010
  2. Published Print: 7 JAN 2011

ISBN Information

Print ISBN: 9780470012550

Online ISBN: 9780470973196



  • category-level points (CLPs);
  • extended matching coefficient (EMC);
  • generalized biplots;
  • inter-sample distances;
  • nonlinear biplot;
  • principal component analysis (PCA);
  • Pythagorean distance


Nonlinear biplots generalize the Principal component analysis (PCA) biplot by providing for distance measures for quantitative variables other than Pythagorean distance; generalized biplots offer a further generalization that allows categorical variables to be included. The reference system in generalized biplots consists of the usual biplot trajectories representing the continuous variables, and a set of category-level points (CLPs) with accompanying nearest-neighbour and prediction regions defined for each categorical variable. Interpolation in the generalized biplot is achieved similarly to the method for the nonlinear biplot. Prediction for continuous variables is performed with either normal or circular projection as exactly as in the case of the nonlinear biplot. Inter-sample distances are calculated using Pythagorean distances for the continuous variables and the extended matching coefficient (EMC) for the categorical variables. The quantitative variables are represented by calibrated linear axes. The main function for constructing generalized biplots is the function Genbipl.

Controlled Vocabulary Terms

principal components analysis; Pythagorean mean