{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T05:05:25Z","timestamp":1791263125026,"version":"4.1.0"},"reference-count":36,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T00:00:00Z","timestamp":1601683200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,8,12]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>In many dimension reduction problems in statistics and machine learning, such as in principal component analysis, canonical correlation analysis, independent component analysis and sufficient dimension reduction, it is important to determine the dimension of the reduced predictor, which often amounts to estimating the rank of a matrix. This problem is called order determination. In this article, we propose a novel and highly effective order-determination method based on the idea of predictor augmentation. We show that if the predictor is augmented by an artificially generated random vector, then the parts of the eigenvectors of the matrix induced by the augmentation display a pattern that reveals information about the order to be determined. This information, when combined with the information provided by the eigenvalues of the matrix, greatly enhances the accuracy of order determination.<\/jats:p>","DOI":"10.1093\/biomet\/asaa077","type":"journal-article","created":{"date-parts":[[2020,9,18]],"date-time":"2020-09-18T11:34:00Z","timestamp":1600428840000},"page":"557-574","source":"Crossref","is-referenced-by-count":25,"title":["On order determination by predictor augmentation"],"prefix":"10.1093","volume":"108","author":[{"given":"Wei","family":"Luo","sequence":"first","affiliation":[{"name":"Center for Data Science, Zhejiang University, 866 Yuhangtang Road, Hangzhou 310058, China weiluo@zju.edu.cn"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bing","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Statistics, The Pennsylvania State University, 326 Thomas Building, University Park, Pennsylvania 16802, U.S.A. bing@stat.psu.edu"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,10,3]]},"reference":[{"key":"2021081218511445100_B1","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1111\/1468-0262.00273","article-title":"Determining the number of factors in approximate factor models","volume":"70","author":"Bai,","year":"2002","journal-title":"Econometrica"},{"key":"2021081218511445100_B2","doi-asserted-by":"crossref","first-page":"2055","DOI":"10.1214\/15-AOS1337","article-title":"Controlling the false discovery rate via knockoffs","volume":"43","author":"Barber,","year":"2015","journal-title":"Ann. 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