PRINCIPAL SUPPORT VECTOR MACHINES FOR LINEAR AND NONLINEAR SUFFICIENT DIMENSION REDUCTION
We introduce a principal support vector machine (PSVM) approach that can be used for both linear and nonlinear sufficient dimension reduction. The basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separat...
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| Veröffentlicht in: | The Annals of statistics 2011-12, Vol.39 (6), p.3182-3210 |
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| container_title | The Annals of statistics |
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| creator | Li, Bing Artemiou, Andreas Li, Lexin |
| description | We introduce a principal support vector machine (PSVM) approach that can be used for both linear and nonlinear sufficient dimension reduction. The basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them. These optimal hyperplanes are then aligned by the principal components of their normal vectors. It is proved that the aligned normal vectors provide an unbiased, $\sqrt n $ -consistent, and asymptotically normal estimator of the sufficient dimension reduction space. The method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel Hubert space. In that context, the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors. We compare PSVM with other sufficient dimension reduction methods by simulation and in real data analysis, and through both comparisons firmly establish its practical advantages. |
| doi_str_mv | 10.1214/11-aos932 |
| format | Article |
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The basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them. These optimal hyperplanes are then aligned by the principal components of their normal vectors. It is proved that the aligned normal vectors provide an unbiased, $\sqrt n $ -consistent, and asymptotically normal estimator of the sufficient dimension reduction space. The method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel Hubert space. In that context, the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors. We compare PSVM with other sufficient dimension reduction methods by simulation and in real data analysis, and through both comparisons firmly establish its practical advantages.</description><identifier>ISSN: 0090-5364</identifier><identifier>EISSN: 2168-8966</identifier><identifier>DOI: 10.1214/11-aos932</identifier><identifier>CODEN: ASTSC7</identifier><language>eng</language><publisher>Cleveland, OH: Institute of Mathematical Statistics</publisher><subject>62-09 ; 62G08 ; 62H12 ; Algebra ; Contour regression ; Dimensional analysis ; Dimensionality reduction ; Estimators ; Exact sciences and technology ; Functional analysis ; General topics ; Hilbert spaces ; Hyperplanes ; invariant kernel ; inverse regression ; Linear regression ; Linear transformations ; Mathematical analysis ; Mathematical vectors ; Mathematics ; Multivariate analysis ; Nonlinear equations ; Objective functions ; principal components ; Probability and statistics ; reproducing kernel Hilbert space ; Sample size ; Sciences and techniques of general use ; Statistical analysis ; Statistics ; Studies ; support vector machine ; Support vector machines ; Vowels</subject><ispartof>The Annals of statistics, 2011-12, Vol.39 (6), p.3182-3210</ispartof><rights>Copyright © 2011 Institute of Mathematical Statistics</rights><rights>2015 INIST-CNRS</rights><rights>Copyright Institute of Mathematical Statistics Dec 2011</rights><rights>Copyright 2011 Institute of Mathematical Statistics</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c466t-9adc0d576dbe9b91ba4f547996f019c9d707814fe4580c7180bb4e46cf074f03</citedby><cites>FETCH-LOGICAL-c466t-9adc0d576dbe9b91ba4f547996f019c9d707814fe4580c7180bb4e46cf074f03</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/41713612$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/41713612$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,780,784,803,832,885,926,27924,27925,58017,58021,58250,58254</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=25655114$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Li, Bing</creatorcontrib><creatorcontrib>Artemiou, Andreas</creatorcontrib><creatorcontrib>Li, Lexin</creatorcontrib><title>PRINCIPAL SUPPORT VECTOR MACHINES FOR LINEAR AND NONLINEAR SUFFICIENT DIMENSION REDUCTION</title><title>The Annals of statistics</title><description>We introduce a principal support vector machine (PSVM) approach that can be used for both linear and nonlinear sufficient dimension reduction. The basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them. These optimal hyperplanes are then aligned by the principal components of their normal vectors. It is proved that the aligned normal vectors provide an unbiased, $\sqrt n $ -consistent, and asymptotically normal estimator of the sufficient dimension reduction space. The method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel Hubert space. In that context, the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors. We compare PSVM with other sufficient dimension reduction methods by simulation and in real data analysis, and through both comparisons firmly establish its practical advantages.</description><subject>62-09</subject><subject>62G08</subject><subject>62H12</subject><subject>Algebra</subject><subject>Contour regression</subject><subject>Dimensional analysis</subject><subject>Dimensionality reduction</subject><subject>Estimators</subject><subject>Exact sciences and technology</subject><subject>Functional analysis</subject><subject>General topics</subject><subject>Hilbert spaces</subject><subject>Hyperplanes</subject><subject>invariant kernel</subject><subject>inverse regression</subject><subject>Linear regression</subject><subject>Linear transformations</subject><subject>Mathematical analysis</subject><subject>Mathematical vectors</subject><subject>Mathematics</subject><subject>Multivariate analysis</subject><subject>Nonlinear equations</subject><subject>Objective functions</subject><subject>principal components</subject><subject>Probability and statistics</subject><subject>reproducing kernel Hilbert space</subject><subject>Sample size</subject><subject>Sciences and techniques of general use</subject><subject>Statistical analysis</subject><subject>Statistics</subject><subject>Studies</subject><subject>support vector machine</subject><subject>Support vector machines</subject><subject>Vowels</subject><issn>0090-5364</issn><issn>2168-8966</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><recordid>eNo9kMtKw0AUhgdRsF4WPoAQBBcuonMyt8zOkKYaiElJUsHVMJkk0FKNZtqFb-9IQ1fn9p3_HH6EbgA_QgD0CcDXg5UkOEGzAHjoh5LzUzTDWGKfEU7P0YW1G4wxk5TM0MeyTPM4XUaZV62Wy6KsvfckrovSe4vi1zRPKm_hisxlUelF-dzLi3yqqtVikcZpktfePH1L8iotcq9M5qu4dtkVOuv11nbXU7xE9SKp41c_K17SOMp8Qznf-VK3BrdM8LbpZCOh0bRnVEjJewzSyFZgEQLtO8pCbASEuGloR7npsaA9Jpfo-SD7PQ6bzuy6vdmuW_U9rj_1-KsGvVbxKpu6U3AOKSAESxZyIZzE3VHiZ9_ZndoM-_HLPa2kpDyAQBAHPRwgMw7Wjl1_PAFY_VuvAFRUVM56x95Pgtoave1H_WXW9rgQMM4YAHXc7YHb2N0wHucUBBAOAfkDa7GFCQ</recordid><startdate>20111201</startdate><enddate>20111201</enddate><creator>Li, Bing</creator><creator>Artemiou, Andreas</creator><creator>Li, Lexin</creator><general>Institute of Mathematical Statistics</general><general>The Institute of Mathematical Statistics</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope></search><sort><creationdate>20111201</creationdate><title>PRINCIPAL SUPPORT VECTOR MACHINES FOR LINEAR AND NONLINEAR SUFFICIENT DIMENSION REDUCTION</title><author>Li, Bing ; Artemiou, Andreas ; Li, Lexin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c466t-9adc0d576dbe9b91ba4f547996f019c9d707814fe4580c7180bb4e46cf074f03</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><topic>62-09</topic><topic>62G08</topic><topic>62H12</topic><topic>Algebra</topic><topic>Contour regression</topic><topic>Dimensional analysis</topic><topic>Dimensionality reduction</topic><topic>Estimators</topic><topic>Exact sciences and technology</topic><topic>Functional analysis</topic><topic>General topics</topic><topic>Hilbert spaces</topic><topic>Hyperplanes</topic><topic>invariant kernel</topic><topic>inverse regression</topic><topic>Linear regression</topic><topic>Linear transformations</topic><topic>Mathematical analysis</topic><topic>Mathematical vectors</topic><topic>Mathematics</topic><topic>Multivariate analysis</topic><topic>Nonlinear equations</topic><topic>Objective functions</topic><topic>principal components</topic><topic>Probability and statistics</topic><topic>reproducing kernel Hilbert space</topic><topic>Sample size</topic><topic>Sciences and techniques of general use</topic><topic>Statistical analysis</topic><topic>Statistics</topic><topic>Studies</topic><topic>support vector machine</topic><topic>Support vector machines</topic><topic>Vowels</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Li, Bing</creatorcontrib><creatorcontrib>Artemiou, Andreas</creatorcontrib><creatorcontrib>Li, Lexin</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><jtitle>The Annals of statistics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Li, Bing</au><au>Artemiou, Andreas</au><au>Li, Lexin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>PRINCIPAL SUPPORT VECTOR MACHINES FOR LINEAR AND NONLINEAR SUFFICIENT DIMENSION REDUCTION</atitle><jtitle>The Annals of statistics</jtitle><date>2011-12-01</date><risdate>2011</risdate><volume>39</volume><issue>6</issue><spage>3182</spage><epage>3210</epage><pages>3182-3210</pages><issn>0090-5364</issn><eissn>2168-8966</eissn><coden>ASTSC7</coden><abstract>We introduce a principal support vector machine (PSVM) approach that can be used for both linear and nonlinear sufficient dimension reduction. The basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them. These optimal hyperplanes are then aligned by the principal components of their normal vectors. It is proved that the aligned normal vectors provide an unbiased, $\sqrt n $ -consistent, and asymptotically normal estimator of the sufficient dimension reduction space. The method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel Hubert space. In that context, the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors. We compare PSVM with other sufficient dimension reduction methods by simulation and in real data analysis, and through both comparisons firmly establish its practical advantages.</abstract><cop>Cleveland, OH</cop><pub>Institute of Mathematical Statistics</pub><doi>10.1214/11-aos932</doi><tpages>29</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | The Annals of statistics, 2011-12, Vol.39 (6), p.3182-3210 |
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| source | JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete |
| subjects | 62-09 62G08 62H12 Algebra Contour regression Dimensional analysis Dimensionality reduction Estimators Exact sciences and technology Functional analysis General topics Hilbert spaces Hyperplanes invariant kernel inverse regression Linear regression Linear transformations Mathematical analysis Mathematical vectors Mathematics Multivariate analysis Nonlinear equations Objective functions principal components Probability and statistics reproducing kernel Hilbert space Sample size Sciences and techniques of general use Statistical analysis Statistics Studies support vector machine Support vector machines Vowels |
| title | PRINCIPAL SUPPORT VECTOR MACHINES FOR LINEAR AND NONLINEAR SUFFICIENT DIMENSION REDUCTION |
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