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• Jin, Q
• Lavery, JE.
• Fang, S
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Algorithms 2013, 6(1), 12-28; doi:10.3390/a6010012

Article

ℓ¹ Major Component Detection and Analysis (ℓ¹ MCDA): Foundations in Two Dimensions

Ye Tian ^1,2,* , Qingwei Jin ^1,3, John E. Lavery ^1,4 and Shu-Cherng Fang ¹

¹ Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC27695-7906, USA ² School of Business Administration, Southwestern University of Finance and Economics, Chengdu,610074, China ³ Department of Management Science and Engineering, Zhejiang University, Hangzhou, 310058, China ⁴ Mathematical Sciences Division and Computing Sciences Division, Army Research Office, Army Research Laboratory, P.O. Box 12211, Research Triangle Park, NC 27709-2211, USA

* Author to whom correspondence should be addressed.

Received: 5 October 2012; in revised form: 3 January 2013 / Accepted: 7 January 2013 / Published: 17 January 2013

Download PDF Full-Text [513 KB, uploaded 17 January 2013 09:40 CET]

Abstract: Principal Component Analysis (PCA) is widely used for identifying the major components of statistically distributed point clouds. Robust versions of PCA, often based in part on the ℓ¹ norm (rather than the ℓ2 norm), are increasingly used, especially for point clouds with many outliers. Neither standard PCA nor robust PCAs can provide, without additional assumptions, reliable information for outlier-rich point clouds and for distributions with several main directions (spokes). We carry out a fundamental and complete reformulation of the PCA approach in a framework based exclusively on the ℓ¹ norm and heavy-tailed distributions. The ℓ¹ Major Component Detection and Analysis (ℓ¹ MCDA) that we propose can determine the main directions and the radial extent of 2D data from single or multiple superimposed Gaussian or heavy-tailed distributions without and with patterned artificial outliers (clutter). In nearly all cases in the computational results, 2D ℓ¹ MCDA has accuracy superior to that of standard PCA and of two robust PCAs, namely, the projection-pursuit method of Croux and Ruiz-Gazen and the ℓ¹ factorization method of Ke and Kanade. (Standard PCA is, of course, superior to ℓ¹ MCDA for Gaussian-distributed point clouds.) The computing time of ℓ¹ MCDA is competitive with the computing times of the two robust PCAs.

Keywords: heavy-tailed distribution; ℓ¹; ℓ²; major component; multivariate statistics; outliers; principal component analysis; 2D

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