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Nontraditional approaches to statistical classification: Some perspectives on L_p-norm methods

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  • Antonie Stam

Abstract

The body of literature on classification methods which estimate boundaries between the groups (classes) by optimizing a function of the L p -norm distances of observations in each group from these boundaries, is maturing fast. The number of published research articles on this topic, especially on mathematical programming (MP) formulations and techniques for L p -norm classification, is now sizable. This paper highlights historical developments that have defined the field, and looks ahead at challenges that may shape new research directions in the next decade. In the first part, the paper summarizes basic concepts and ideas, and briefly reviews past research. Throughout, an attempt is made to integrate a number of the most important L p -norm methods proposed to date within a unified framework, emphasizing their conceptual differences and similarities, rather than focusing on mathematical detail. In the second part, the paper discusses several potential directions for future research in this area. The long-term prospects of L p -norm classification (and discriminant) research may well hinge upon whether or not the channels of communication between on the one hand researchers active in L p -norm classification, who tend to have their roots primarily in the decision sciences, the management sciences, computer science and engineering, and on the other hand practitioners and researchers in the statistical classification community, will be improved. This paper offers potential reasons for the lack of communication between these groups, and suggests ways in which L p -norm research may be strengthened from a statistical viewpoint. The results obtained in L p -norm classification studies are clearly relevant and of importance to all researchers and practitioners active in classification and discriminant analysis. The paper also briefly discusses artificial neural networks, a promising non-traditional method for classification which has recently emerged, and suggests that it may be useful to explore hybrid classification methods that take advantage of the complementary strengths of different methods, e.g., neural network and L p -norm methods. Copyright Kluwer Academic Publishers 1997

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  • Antonie Stam, 1997. "Nontraditional approaches to statistical classification: Some perspectives on L_p-norm methods," Annals of Operations Research, Springer, vol. 74(0), pages 1-36, November.
  • Handle: RePEc:spr:annopr:v:74:y:1997:i:0:p:1-36:10.1023/a:1018958001886
    DOI: 10.1023/A:1018958001886
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    Cited by:

    1. Brusco, Michael J. & Steinley, Douglas, 2011. "Exact and approximate algorithms for variable selection in linear discriminant analysis," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 123-131, January.
    2. Mingue Sun, 2009. "Liquidity Risk and Financial Competition: A Mixed Integer Programming Model for Multiple-Class Discriminant Analysis," Working Papers 0102, College of Business, University of Texas at San Antonio.
    3. Zopounidis, Constantin & Doumpos, Michael, 2002. "Multi-group discrimination using multi-criteria analysis: Illustrations from the field of finance," European Journal of Operational Research, Elsevier, vol. 139(2), pages 371-389, June.
    4. Henze, Norbert & Nikitin, Yakov & Ebner, Bruno, 2009. "Integral distribution-free statistics of Lp-type and their asymptotic comparison," Computational Statistics & Data Analysis, Elsevier, vol. 53(9), pages 3426-3438, July.
    5. J J Glen, 2005. "Mathematical programming models for piecewise-linear discriminant analysis," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 56(3), pages 331-341, March.
    6. Mike G. Tsionas, 2021. "Multi-criteria optimization in regression," Annals of Operations Research, Springer, vol. 306(1), pages 7-25, November.
    7. Chun-Na Li & Pei-Wei Ren & Yan-Ru Guo & Ya-Fen Ye & Yuan-Hai Shao, 2024. "Regularized linear discriminant analysis based on generalized capped $$l_{2,q}$$ l 2 , q -norm," Annals of Operations Research, Springer, vol. 339(3), pages 1433-1459, August.
    8. Pedro Duarte Silva, A., 2017. "Optimization approaches to Supervised Classification," European Journal of Operational Research, Elsevier, vol. 261(2), pages 772-788.
    9. J. J. Glen, 2004. "Dichotomous categorical variable formation in mathematical programming discriminant analysis models," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(4), pages 575-596, June.
    10. Mingue Sun, 2009. "Liquidity Risk and Financial Competition: A Mixed Integer Programming Model for Multiple-Class Discriminant Analysis," Working Papers 0102, College of Business, University of Texas at San Antonio.
    11. Sueyoshi, Toshiyuki, 2004. "Mixed integer programming approach of extended DEA-discriminant analysis," European Journal of Operational Research, Elsevier, vol. 152(1), pages 45-55, January.
    12. Sueyoshi, Toshiyuki, 2006. "DEA-Discriminant Analysis: Methodological comparison among eight discriminant analysis approaches," European Journal of Operational Research, Elsevier, vol. 169(1), pages 247-272, February.
    13. J J Glen, 2008. "An additive utility mixed integer programming model for nonlinear discriminant analysis," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(11), pages 1492-1505, November.
    14. Glen, J.J., 2006. "A comparison of standard and two-stage mathematical programming discriminant analysis methods," European Journal of Operational Research, Elsevier, vol. 171(2), pages 496-515, June.
    15. Zopounidis, Constantin & Doumpos, Michael, 2002. "Multicriteria classification and sorting methods: A literature review," European Journal of Operational Research, Elsevier, vol. 138(2), pages 229-246, April.
    16. Doumpos, Michael & Zopounidis, Constantin, 2004. "A multicriteria classification approach based on pairwise comparisons," European Journal of Operational Research, Elsevier, vol. 158(2), pages 378-389, October.
    17. Lam, Kim Fung & Moy, Jane W., 2003. "A piecewise linear programming approach to the two-group discriminant problem - an adaptation to Fisher's linear discriminant function model," European Journal of Operational Research, Elsevier, vol. 145(2), pages 471-481, March.
    18. K Falangis & J J Glen, 2010. "Heuristics for feature selection in mathematical programming discriminant analysis models," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(5), pages 804-812, May.

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