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Discriminant Analysis with Singular Covariance Matrices: Methods and Applications to Spectroscopic Data

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  • W. J. Krzanowski
  • P. Jonathan
  • W. V. McCarthy
  • M. R. Thomas

Abstract

Currently popular techniques such as experimental spectroscopy and computer‐aided molecular modelling lead to data having very many variables observed on each of relatively few individuals. A common objective is discrimination between two or more groups, but the direct application of standard discriminant methodology fails because of singularity of covariance matrices. The problem has been circumvented in the past by prior selection of a few transformed variables, using either principal component analysis or partial least squares. Although such selection ensures non‐singularity of matrices, the decision process is arbitrary and valuable information on group structure may be lost. We therefore consider some ways of estimating linear discriminant functions without such prior selection. Several spectroscopic data sets are analysed with each method, and questions of bias of assessment procedures are investigated. All proposed methods seem worthy of consideration in practice.

Suggested Citation

  • W. J. Krzanowski & P. Jonathan & W. V. McCarthy & M. R. Thomas, 1995. "Discriminant Analysis with Singular Covariance Matrices: Methods and Applications to Spectroscopic Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 44(1), pages 101-115, March.
    • Handle: RePEc:bla:jorssc:v:44:y:1995:i:1:p:101-115
    DOI: 10.2307/2986198
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    Cited by:

    1. S. Robin & M. Lecomte & H. Hofte & G. Mouille, 2003. "A procedure for the clustering of cell wall mutants in the model plant Arabidopsis based on Fourier-transform infrared (FT-IR) spectroscopy," Journal of Applied Statistics, Taylor & Francis Journals, vol. 30(6), pages 669-681.
    2. Duintjer Tebbens, Jurjen & Schlesinger, Pavel, 2007. "Improving implementation of linear discriminant analysis for the high dimension/small sample size problem," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 423-437, September.
    3. Lei-Hong Zhang & Li-Zhi Liao & Michael K. Ng, 2013. "Superlinear Convergence of a General Algorithm for the Generalized Foley–Sammon Discriminant Analysis," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 853-865, June.
    4. W. J. Krzanowski, 1999. "Antedependence models in the analysis of multi-group high-dimensional data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 26(1), pages 59-67.
    5. Bouveyron, C. & Girard, S. & Schmid, C., 2007. "High-dimensional data clustering," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 502-519, September.
    6. John Gower & Casper Albers, 2011. "Between-Group Metrics," Journal of Classification, Springer;The Classification Society, vol. 28(3), pages 315-326, October.
    7. Nickolay T. Trendafilov & Tsegay Gebrehiwot Gebru, 2016. "Recipes for sparse LDA of horizontal data," METRON, Springer;Sapienza Università di Roma, vol. 74(2), pages 207-221, August.
    8. Brendan P. W. Ames & Mingyi Hong, 2016. "Alternating direction method of multipliers for penalized zero-variance discriminant analysis," Computational Optimization and Applications, Springer, vol. 64(3), pages 725-754, July.
    9. Albers, C.J. & Critchley, F. & Gower, J.C., 2011. "Applications of quadratic minimisation problems in statistics," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 714-722, March.
    10. Mkhadri, A. & Celeux, G. & Nasroallah, A., 1997. "Regularization in discriminant analysis: an overview," Computational Statistics & Data Analysis, Elsevier, vol. 23(3), pages 403-423, January.
    11. Casper Albers & John Gower, 2014. "Canonical Analysis: Ranks, Ratios and Fits," Journal of Classification, Springer;The Classification Society, vol. 31(1), pages 2-27, April.

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