Median-Based Classifiers for High-Dimensional Data

Hall, P., Titterington, D. M. and Xue, J. H. (2009) Median-Based Classifiers for High-Dimensional Data. Journal of the American Statistical Association, 104(488), pp. 1597-1608. (doi:10.1198/jasa.2009.tm08107)

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Publisher's URL: http://dx.doi.org/10.1198/jasa.2009.tm08107

Abstract

Conventional distance-based classifiers use standard Euclidean distance, and so can suffer from excessive volatility if vector components have heavy-tailed distributions. This difficulty can be alleviated by replacing the L-2 distance by its L-1 counterpart. For example, the L-1 version of the popular centroid classifier would allocate a new data value to the population to whose centroid it was closest in L-1 terms. However, this approach can lead to inconsistency, because the centroid is defined using L-2, rather than L-1, distance. In particular, by mixing L-1 and L-2 approaches, we produce a classifier that can seriously misidentify data in cases where the means and medians of marginal distributions take different values. These difficulties motivate replacing centroids by medians. However, in the very-high-dimensional settings commonly encountered today, this can be problematic if we attempt to work with a conventional spatial median. Therefore, we suggest using componentwise medians to construct a robust classifier that is relatively insensitive to the difficulties caused by heavy-tailed data and entails straightforward computation. We also consider generalizations and extensions of this approach based on, for example, using data truncation to achieve additional robustness. Using both empirical and theoretical arguments, we explore the properties of these methods, and show that the resulting classifiers can be particularly effective. Supplementary materials are available online

Item Type:Articles
Keywords:CANCER Centroid classifier CLASS PREDICTION Classification Componentwise median Data depth DISCRIMINANT-ANALYSIS Distance-based classifier DISTRIBUTIONS GENE-EXPRESSION GENERALIZATION ERROR High-dimensional data L-1 method Methods MICROARRAY DATA Robust method Sample median SELECTION SHRUNKEN CENTROIDS Spatial median Strength of dependence VALUES
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Titterington, Professor Michael
Authors: Hall, P., Titterington, D. M., and Xue, J. H.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Statistics
Journal Name:Journal of the American Statistical Association
ISSN:0162-1459

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