Depth-Based Classification Method Underlain by a Remote Concentration Measure for Processing Asymmetric Data
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DEPTH-BASED CLASSIFICATION METHOD UNDERLAIN BY A REMOTE CONCENTRATION MEASURE FOR PROCESSING ASYMMETRIC DATA
UDC 519.7
O. A. Galkin
Abstract. A depth-based classification method underlain by a remote concentration measure for processing asymmetric data is developed and investigated. The motivation for the construction of the method was the inefficient use of affine invariant classifiers in combination with depth functions vanishing outside the convex hull of data. The idea of the proposed method is to map a remote space using a remote concentration measure, the Stahel–Donoho remoteness measure, and a corrected remoteness measure. Keywords: depth function, remote concentration measure, multi-dimensional classification. INTRODUCTION The problem of potential consequences of emissions and extreme values in solving modern recognition problems requires the search for new nonparametric methods resistant to emissions. In the majority of cases, emissions are admissible elements available from various data sets. In problems of supervised classification, labels of classes of some data elements in a training set can be erroneously assigned. The majority of classification methods are efficient only in applying to data with elliptical symmetry or with a multinormal distribution. The majority of existing methods that allow one to classify asymmetric multidimensional data are implemented based on depth functions. However, such classifiers often have a rather low efficiency since depth functions go to zero outside of the convex hull of data. Taking into account the urgency of the problematic being investigated, this article is devoted to the development and investigation of a new nonparametric classification method that provides the possibility of processing asymmetric multidimensional data. The proposed method belongs to the class of supervised learning and is based on the conception of a remote space. DEFINITION OF DEPTH REGIONS ON FINITE SAMPLES Proceeding from the requirements of a statistical depth function, a halfspace depth function monotonically decreases along the lines outgoing from the center and is affine invariant. Moreover, a halfspace depth function is equal to zero at infinity and reaches its maximum value at the center of symmetry [1]. The halfspace depth function " z Î R p with respect to H X is defined as the minimal probabilistic group contained in a closed half-space with a limit with respect to z , namely,
Fd ( z, H X ) = inf H X { b¢ X ³ b¢ z}, ||b || = 1
where X is a random quantity on R p with a distribution H X . Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2016, pp. 57–66. Original article submitted November 30, 2015. 386
1060-0396/16/5203-0386 ©2016 Springer Science+Business Media New York
A region E j of j-depth is a set of the points whose depth is no less than j , i.e.,
Ej = { z ÎR p } for " j Î[0, 1] and E ( z, H X ) ³ j . Note that the profile of j-depth is the limit of E j . The halfspace
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