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