Nontraditional Approaches to Statistical Classification: Some Perspectives on Lp-Norm Methods

Stam, A. (1996). Nontraditional Approaches to Statistical Classification: Some Perspectives on Lp-Norm Methods. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-96-128

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Abstract

The body of literature on classification method 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 decision sciences, the management sciences, computer sciences 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 discrimination analysis. The paper also briefly discusses artificial neural networks, a promising nontraditional 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.

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