{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T15:41:52Z","timestamp":1740152512252,"version":"3.37.3"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2014,8,19]],"date-time":"2014-08-19T00:00:00Z","timestamp":1408406400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/100000183","name":"US Army Research Office","doi-asserted-by":"crossref","award":["W911NF-04-D-0003"],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000001","name":"US NSF","doi-asserted-by":"publisher","award":["DMI-0553310"],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"Based on the recent development of two dimensional \u21131 major component detection and analysis (\u21131 MCDA), we develop a scalable \u21131 MCDA in the n-dimensional space to identify the major directions of star-shaped heavy-tailed statistical distributions with irregularly positioned \u201cspokes\u201d and \u201cclutters\u201d. In order to achieve robustness and efficiency, the proposed \u21131 MCDA in n-dimensional space adopts a two-level median fit process in a local neighbor of a given direction in each iteration. Computational results indicate that in terms of accuracy \u21131 MCDA is competitive with two well-known PCAs when there is only one major direction in the data, and \u21131 MCDA can further determine multiple major directions of the n-dimensional data from superimposed Gaussians or heavy-tailed distributions without and with patterned artificial outliers. With the ability to recover complex spoke structures with heavy-tailed noise and clutter in the data, \u21131 MCDA has potential to generate better semantics than other methods.<\/jats:p>","DOI":"10.3390\/a7030429","type":"journal-article","created":{"date-parts":[[2014,8,19]],"date-time":"2014-08-19T14:40:39Z","timestamp":1408459239000},"page":"429-443","source":"Crossref","is-referenced-by-count":2,"title":["\u21131 Major Component Detection and Analysis (\u21131 MCDA) in Three and Higher Dimensional Spaces"],"prefix":"10.3390","volume":"7","author":[{"given":"Zhibin","family":"Deng","sequence":"first","affiliation":[{"name":"School of Management, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA"}]},{"given":"John","family":"Lavery","sequence":"additional","affiliation":[{"name":"Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, 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, NC 27695-7906, USA"}]},{"given":"Jian","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA"}]}],"member":"1968","published-online":{"date-parts":[[2014,8,19]]},"reference":[{"key":"ref_1","unstructured":"Jolliffe, I.T. (2002). Principal Component Analysis, Springer. [2nd ed.]."},{"key":"ref_2","unstructured":"Cand\u00e8s, E.J., Li, X., Ma, Y., and Wright, J. (2009). Robust Principal Component Analysis, Department of Statistics, Stanford University. Technical Report No. 13."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1441","DOI":"10.1016\/j.csda.2005.01.009","article-title":"L1-Norm projection pursuit principal component analysis","volume":"50","author":"Choulakian","year":"2006","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1080\/00401706.2012.727746","article-title":"Robust sparse principal component analysis","volume":"55","author":"Croux","year":"2013","journal-title":"Technometrics"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.jmva.2004.08.002","article-title":"High breakdown estimators for principal components: The projection-pursuit approach revisited","volume":"95","author":"Croux","year":"2005","journal-title":"J. Multivar. Anal."},{"key":"ref_6","first-page":"739","article-title":"Robust L1 norm factorization in the presence of outliers and missing data by alternative convex programming","volume":"1","author":"Ke","year":"2005","journal-title":"IEEE Conf. Comput. Vis. Pattern Recognit."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1672","DOI":"10.1109\/TPAMI.2008.114","article-title":"Principal component analysis based on L1-norm maximization","volume":"30","author":"Kwak","year":"2008","journal-title":"IEEE Trans. Pattern Anal."},{"key":"ref_8","unstructured":"Lin, Z., Chen, M., Wu, L., and Ma, Y. The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrix. Available online: http:\/\/arxiv.org\/abs\/1009.5055."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1109\/TIT.2012.2212415","article-title":"Outlier-robust PCA: the high-dimensional case","volume":"59","author":"Xu","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/j.sigpro.2005.07.012","article-title":"Sparse approximations in signal and image processing","volume":"86","author":"Gribonval","year":"2006","journal-title":"Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1137\/090775397","article-title":"An unconstrained \u2113q minimization with 0 < q \u2264 1 for sparse solution of under-determined linear systems","volume":"21","author":"Lai","year":"2010","journal-title":"SIAM J. Optim."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1109\/MSP.2007.914731","article-title":"An introduction to compressive sampling","volume":"25","author":"Wakin","year":"2008","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1109\/LSP.2007.898300","article-title":"Exact reconstruction of sparse signals via nonconvex minimization","volume":"14","author":"Chartrand","year":"2007","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cagd.2010.10.002","article-title":"Fast \n \n \n L\n 1\n\t\t\tk\n \n\t\t \n\t\t C\n\t\t\tk\n\t\t \n \n polynomial spline interpolation algorithm with shape-preserving properties","volume":"28","author":"Auquiert","year":"2011","journal-title":"Comput. Aided Geom. Des."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"311","DOI":"10.3390\/a3030311","article-title":"Univariate cubic L1 interpolating splines: Spline functional, window size and analysis-based algorithm","volume":"3","author":"Yu","year":"2010","journal-title":"Algorithms"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"12","DOI":"10.3390\/a6010012","article-title":"\u21131 major component detection and analysis (\u21131 MCDA): Foundations in two dimensions","volume":"6","author":"Tian","year":"2013","journal-title":"Algorithms"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.csda.2012.11.007","article-title":"A pure L1-norm principal component analysis","volume":"61","author":"Brooks","year":"2013","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_18","unstructured":"Deng, Z., and Luo, J. FANGroup-Fuzzy And Neural Group at North Carolina State University. Available online: http:\/\/www.ise.ncsu.edu\/fangroup\/index.htm."},{"key":"ref_19","unstructured":"(2012). MATLAB Release 2012a, The MathWorks Inc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1145\/355744.355745","article-title":"An algorithm for finding best matches in logarithmic expected time","volume":"3","author":"Friedman","year":"1997","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_21","unstructured":"Filzmozer, P., Fritz, H., and Kalcher, K. pcaPP: Robust PCA by Projection Pursuit. Available online: http:\/\/cran.r-project.org\/web\/packages\/pcaPP\/index.html."},{"key":"ref_22","unstructured":"R Development Core Team (2011). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_23","unstructured":"IBM ILOG CPLEX Optimization. Available online: http:\/\/www-01.ibm.com\/software\/commerce\/ optimization\/cplex-optimizer\/."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/7\/3\/429\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,14]],"date-time":"2025-01-14T15:51:57Z","timestamp":1736869917000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/7\/3\/429"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,8,19]]},"references-count":23,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2014,9]]}},"alternative-id":["a7030429"],"URL":"https:\/\/doi.org\/10.3390\/a7030429","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2014,8,19]]}}}