A note on order statistics-based parametric pattern classification. Issue 1 (January 2015)

• Record Type: Journal Article
• Authors: Hu, Lixia
• Is Part Of: Pattern recognition. Volume 48:Issue 1(2015:Jan.)
• Page Start: 43
• Page End: 49
• Publication Date: 2015-01
• Subjects: Pattern perception -- Periodicals; Perception des structures -- Périodiques; Patroonherkenning; 006.4
• DOI: 10.1016/j.patcog.2014.07.021
• Languages: English
• ISSNs: 0031-3203

Abstract

<abstract abstract-type="author" id="ab0005"> <title id="sect0005">Abstract</title> <sec> <p id="sp0045">Recently, a novel classification paradigm is proposed, named Classification by Moments of Order Statistics (CMOS), which is shown to attain the optimal Bayesian bound for symmetric distributions and a near-optimal accuracy for asymmetric distributions <xref id="crs0005" rid="bib13 bib9">[13, 9]</xref>. However, in the process of deriving the order statistics-based classification scheme, the authors use a plausible relation "<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh206f3r8j" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si0011.gif" overflow="scroll" id="d13e915" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi><mml:mo>[</mml:mo><mml:mi>Φ</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>, </mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo><mml:mo>]</mml:mo><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mo>⟹</mml:mo><mml:mi>E</mml:mi><mml:mo>[</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>, </mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>]</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>Φ</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:math></alternatives></inline-formula>", where <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh206fb5xm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si0012.gif" overflow="scroll" id="d13e1006" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Φ</mml:mi></mml:math></alternatives></inline-formula> is the cumulative distribution function of random variable <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh206f8hxt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si0013.gif" overflow="scroll" id="d13e1010" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>, and <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh206f7zgw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si0014.gif" overflow="scroll" id="d13e1014" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>, </mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> is the <italic>k</italic>-th order statistics of a sample of size <italic>n</italic> from <italic>X</italic>. Therefore, the new approach actually should be viewed as the classification scheme based on the percentiles of distribution, instead of the so-called order statistics-based classification. In this paper, we will build the CMOS using 2-OS criteria in its true sense. Furthermore, we show that the order statistics-based classification reaches the optimal Bayesian bound for symmetric distributions, and compare the accuracy of CMOS, Bayesian classification, median-based classifier and percentiles-based classification for non-symmetric distributions. The theoretical results are verified by rigorous experiments as well.</p> </sec> </abstract>