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Recipes for sparse LDA of horizontal data

Author

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  • Nickolay T. Trendafilov

    (The Open University)

  • Tsegay Gebrehiwot Gebru

    (The Open University)

Abstract

Many important modern applications require analyzing data with more variables than observations, called for short horizontal. In such situation the classical Fisher’s linear discriminant analysis (LDA) does not possess solution because the within-group scatter matrix is singular. Moreover, the number of the variables is usually huge and the classical type of solutions (discriminant functions) are difficult to interpret as they involve all available variables. Nowadays, the aim is to develop fast and reliable algorithms for sparse LDA of horizontal data. The resulting discriminant functions depend on very few original variables, which facilitates their interpretation. The main theoretical and numerical challenge is how to cope with the singularity of the within-group scatter matrix. This work aims at classifying the existing approaches according to the way they tackle this singularity issue, and suggest new ones.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metron:v:74:y:2016:i:2:d:10.1007_s40300-016-0093-8
    DOI: 10.1007/s40300-016-0093-8
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    1. Timmerman, Marieke E. & Ceulemans, Eva & Kiers, Henk A.L. & Vichi, Maurizio, 2010. "Factorial and reduced K-means reconsidered," Computational Statistics & Data Analysis, Elsevier, vol. 54(7), pages 1858-1871, July.
    2. J. C. Gower & W. J. Krzanowski, 1999. "Analysis of distance for structured multivariate data and extensions to multivariate analysis of variance," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 48(4), pages 505-519.
    3. 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.
    4. Vichi, Maurizio & Saporta, Gilbert, 2009. "Clustering and disjoint principal component analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3194-3208, June.
    5. W. J. Krzanowski, 1984. "Principal Component Analysis in the Presence of Group Structure," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 33(2), pages 164-168, June.
    6. W. Krzanowski, 1990. "Between-group analysis with heterogeneous covariance matrices: The common principal component model," Journal of Classification, Springer;The Classification Society, vol. 7(1), pages 81-98, March.
    7. St鰨ane Mussard & Luc Savard, 2012. "The Gini multi-decomposition and the role of Gini's transvariation: application to partial trade liberalization in the Philippines," Applied Economics, Taylor & Francis Journals, vol. 44(10), pages 1235-1249, April.
    8. Trendafilov, Nickolay T. & Jolliffe, Ian T., 2007. "DALASS: Variable selection in discriminant analysis via the LASSO," Computational Statistics & Data Analysis, Elsevier, vol. 51(8), pages 3718-3736, May.
    9. Ning Hao & Bin Dong & Jianqing Fan, 2015. "Sparsifying the Fisher linear discriminant by rotation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(4), pages 827-851, September.
    10. Bernhard Flury & Martin Schmid & A. Narayanan, 1994. "Error rates in quadratic discrimination with constraints on the covariance matrices," Journal of Classification, Springer;The Classification Society, vol. 11(1), pages 101-120, March.
    11. 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.
    12. Jianqing Fan & Yang Feng & Xin Tong, 2012. "A road to classification in high dimensional space: the regularized optimal affine discriminant," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(4), pages 745-771, September.
    13. Trendafilov, Nickolay T. & Vines, Karen, 2009. "Simple and interpretable discrimination," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 979-989, February.
    14. 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.
    15. Daniela Bragoli & Piero Ganugi & Giancarlo Ianulardo, 2013. "Gini’s transvariation analysis: an application on financial crises in developing countries," Empirica, Springer;Austrian Institute for Economic Research;Austrian Economic Association, vol. 40(1), pages 153-174, February.
    16. Clemens Hage & Martin Kleinsteuber, 2014. "Robust PCA and subspace tracking from incomplete observations using $$\ell _0$$ ℓ 0 -surrogates," Computational Statistics, Springer, vol. 29(3), pages 467-487, June.
    17. Dhillon, Inderjit S. & Modha, Dharmendra S. & Spangler, W. Scott, 2002. "Class visualization of high-dimensional data with applications," Computational Statistics & Data Analysis, Elsevier, vol. 41(1), pages 59-90, November.
    18. Nickolay Trendafilov, 2014. "From simple structure to sparse components: a review," Computational Statistics, Springer, vol. 29(3), pages 431-454, June.
    19. Angela Montanari, 2004. "Linear Discriminant Analysis and Transvariation," Journal of Classification, Springer;The Classification Society, vol. 21(1), pages 71-88, March.
    20. Trendafilov, Nickolay T. & Jolliffe, Ian T., 2006. "Projected gradient approach to the numerical solution of the SCoTLASS," Computational Statistics & Data Analysis, Elsevier, vol. 50(1), pages 242-253, January.
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