{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:30:31Z","timestamp":1760243431972,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2013,1,17]],"date-time":"2013-01-17T00:00:00Z","timestamp":1358380800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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 \u21131 norm (rather than the \u21132 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 \u21131 norm and heavy-tailed distributions. The \u21131 Major Component Detection and Analysis (\u21131 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 \u21131 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 \u21131 factorization method of Ke and Kanade. (Standard PCA is, of course, superior to \u21131 MCDA for Gaussian-distributed point clouds.) The computing time of \u21131 MCDA is competitive with the computing times of the two robust PCAs.<\/jats:p>","DOI":"10.3390\/a6010012","type":"journal-article","created":{"date-parts":[[2013,1,17]],"date-time":"2013-01-17T11:25:50Z","timestamp":1358421950000},"page":"12-28","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["\u21131 Major Component Detection and Analysis (\u21131 MCDA): Foundations in Two Dimensions"],"prefix":"10.3390","volume":"6","author":[{"given":"Ye","family":"Tian","sequence":"first","affiliation":[{"name":"Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC27695-7906, USA"},{"name":"School of Business Administration, Southwestern University of Finance and Economics, Chengdu,610074, China"}]},{"given":"Qingwei","family":"Jin","sequence":"additional","affiliation":[{"name":"Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC27695-7906, USA"},{"name":"Department of Management Science and Engineering, Zhejiang University, Hangzhou, 310058, China"}]},{"given":"John","family":"Lavery","sequence":"additional","affiliation":[{"name":"Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC27695-7906, USA"},{"name":"Mathematical Sciences Division and Computing Sciences Division, Army Research Office, Army Research Laboratory, P.O. Box 12211, Research Triangle Park, NC 27709-2211, USA"}]},{"given":"Shu-Cherng","family":"Fang","sequence":"additional","affiliation":[{"name":"Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC27695-7906, USA"}]}],"member":"1968","published-online":{"date-parts":[[2013,1,17]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Gorban, A., Kegl, B., Wunsch, D., and Zinovyev, A. (2007). Principal Manifolds for Data Visualisation and Dimension Reduction, Springer.","key":"ref_1","DOI":"10.1007\/978-3-540-73750-6"},{"unstructured":"Jolliffe, I.T. (2002). Principal Component Analysis, Springer. [2nd ed.].","key":"ref_2"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3706","DOI":"10.1016\/j.csda.2009.03.014","article-title":"Robust probabilistic PCA with missing data and contribution analysis for outlier detection","volume":"53","author":"Chen","year":"2009","journal-title":"Comput. Stat. 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