Unitarily invariant errors-in-variables estimation
Linear relations, containing measurement errors in the input and output data, are considered. Parameters of these errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input–output disturbances, i.e., penalizing the orthogonal squared misfit. This appro...
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| Published in | Statistical papers (Berlin, Germany) Vol. 57; no. 4; pp. 1041 - 1057 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2016
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0932-5026 1613-9798 |
| DOI | 10.1007/s00362-016-0800-9 |
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| Abstract | Linear relations, containing measurement errors in the input and output data, are considered. Parameters of these errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input–output disturbances, i.e., penalizing the orthogonal squared misfit. This approach corresponds to minimizing the Frobenius norm of the error matrix. An extension of the traditional TLS estimator in the EIV model—the EIV estimator—is proposed in the way that a general unitarily invariant norm of the error matrix is minimized. Such an estimator is highly non-linear. Regardless of the chosen unitarily invariant matrix norm, the corresponding EIV estimator is shown to coincide with the TLS estimator. Its existence and uniqueness is discussed. Moreover, the EIV estimator is proved to be scale invariant, interchange, direction, and rotation equivariant. |
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| AbstractList | Linear relations, containing measurement errors in the input and output data, are considered. Parameters of these errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input–output disturbances, i.e., penalizing the orthogonal squared misfit. This approach corresponds to minimizing the Frobenius norm of the error matrix. An extension of the traditional TLS estimator in the EIV model—the EIV estimator—is proposed in the way that a general unitarily invariant norm of the error matrix is minimized. Such an estimator is highly non-linear. Regardless of the chosen unitarily invariant matrix norm, the corresponding EIV estimator is shown to coincide with the TLS estimator. Its existence and uniqueness is discussed. Moreover, the EIV estimator is proved to be scale invariant, interchange, direction, and rotation equivariant. Linear relations, containing measurement errors in the input and output data, are considered. Parameters of these errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input-output disturbances, i.e., penalizing the orthogonal squared misfit. This approach corresponds to minimizing the Frobenius norm of the error matrix. An extension of the traditional TLS estimator in the EIV model-the EIV estimator-is proposed in the way that a general unitarily invariant norm of the error matrix is minimized. Such an estimator is highly non-linear. Regardless of the chosen unitarily invariant matrix norm, the corresponding EIV estimator is shown to coincide with the TLS estimator. Its existence and uniqueness is discussed. Moreover, the EIV estimator is proved to be scale invariant, interchange, direction, and rotation equivariant. |
| Author | Pešta, Michal |
| Author_xml | – sequence: 1 givenname: Michal surname: Pesta fullname: Pesta, Michal |
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| CitedBy_id | crossref_primary_10_1007_s00362_017_0939_z crossref_primary_10_3390_math9010089 crossref_primary_10_1002_acs_2751 |
| Cites_doi | 10.1137/1.9781611971002 10.1007/BF02288367 10.1080/01621459.1959.10501505 10.1016/0024-3795(72)90013-4 10.1007/978-1-4612-0653-8 10.1080/02331888.2012.658806 10.1214/aos/1176342472 10.1080/03610928208828287 10.1214/aos/1176345330 10.1093/biomet/60.3.525 10.21236/ADA037530 10.1093/qmath/11.1.50 10.1137/0717073 10.1007/s001840300272 |
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| DOI | 10.1007/s00362-016-0800-9 |
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| Keywords | Errors-in-variables model 15B51 Total least squares 62F10 Equivariance Invariance Unitarily invariant matrix norm 62H12 62J99 |
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| References | GleserLJWatsonGSEstimation of a linear transformationBiometrika19736052553433943210.1093/biomet/60.3.5250271.62074 Von NeumannJSome matrix inequalities and metrization of matrix-spaceTomsk Univ Rev19371286300 Healy JD (1975) Estimation and tests for unknown linear restrictions in multivariate linear models. PhD thesis, Purdue University, Purdue Van HuffelSVandewalleJThe total least squares problem: computational aspects and analysis1991PhiladelphiaSIAM10.1137/1.97816119710020789.62054 GleserLJEstimation in a multivariate “errors in variables” regression model: large sample resultsAnn Stat19819244460053010.1214/aos/11763453300496.62049 MirskyLSymmetric gauge functions and unitarily invariant normsQ J Math196011505911482110.1093/qmath/11.1.500105.01101 MadanskyAThe fitting of straight lines when both variables are subject to errorJ Am Stat Assoc19595417320510287510.1080/01621459.1959.105015050088.35804 GalloPPConsistency of regression estimates when some variables are subject to errorCommunications in statistics: theory and methods19821197398365546610.1080/036109282088282870515.62064 GolubGHVan LoanCFAn analysis of the total least squares problemSIAM J Numer Anal198017688389359545110.1137/07170730468.65011 PeštaMTotal least squares and bootstrapping with application in calibrationStat: J Theor Appl Stat201347596699110.1080/02331888.2012.65880606231528 Bhatia R (1996) Matrix analysis. Volume 169 of Graduate Texts in Mathematics. Springer, New York KukushAVan HuffelSConsistency of elementwise-weighted total least squares estimator in a multivariate errors-in-variables model AX=B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AX=B$$\end{document}Metrika20045917597204343310.1007/s0018403002721062.62100 Gallo PP (1982b) Properties of estimators in errors-in-variables models. PhD thesis, University of North Carolina, Chapel Hill RaoCRKrishnaiahPRMatrix approximations and reduction of dimensionality in multivariate statistical analysisMultivariate analysis1980AmsterdamNorth-Holland322 OkamotoMDistinctness of the eigenvalues of a quadratic form in a multivariate sampleAnn Stat1973176376533164310.1214/aos/11763424720261.62043 StewartGWSunJMatrix perturbation theory1990San DiegoAcademic Press0706.65013 ThompsonRCPrincipal submatrices IX: interlacing inequalities for singular values of submatricesLinear Algebra Appl1972511229779110.1016/0024-3795(72)90013-40252.15009 EckartCYoungGThe approximation of one matrix by another of lower rankPsychometrika19361321121810.1007/BF0228836762.1075.02 LJ Gleser (800_CR6) 1973; 60 A Madansky (800_CR10) 1959; 54 800_CR4 M Pešta (800_CR13) 2013; 47 800_CR1 GH Golub (800_CR7) 1980; 17 S Huffel Van (800_CR17) 1991 L Mirsky (800_CR11) 1960; 11 CR Rao (800_CR14) 1980 C Eckart (800_CR2) 1936; 1 LJ Gleser (800_CR5) 1981; 9 800_CR8 PP Gallo (800_CR3) 1982; 11 GW Stewart (800_CR15) 1990 M Okamoto (800_CR12) 1973; 1 RC Thompson (800_CR16) 1972; 5 J Neumann Von (800_CR18) 1937; 1 A Kukush (800_CR9) 2004; 59 |
| References_xml | – reference: Gallo PP (1982b) Properties of estimators in errors-in-variables models. PhD thesis, University of North Carolina, Chapel Hill – reference: GolubGHVan LoanCFAn analysis of the total least squares problemSIAM J Numer Anal198017688389359545110.1137/07170730468.65011 – reference: Van HuffelSVandewalleJThe total least squares problem: computational aspects and analysis1991PhiladelphiaSIAM10.1137/1.97816119710020789.62054 – reference: GleserLJWatsonGSEstimation of a linear transformationBiometrika19736052553433943210.1093/biomet/60.3.5250271.62074 – reference: MirskyLSymmetric gauge functions and unitarily invariant normsQ J Math196011505911482110.1093/qmath/11.1.500105.01101 – reference: EckartCYoungGThe approximation of one matrix by another of lower rankPsychometrika19361321121810.1007/BF0228836762.1075.02 – reference: MadanskyAThe fitting of straight lines when both variables are subject to errorJ Am Stat Assoc19595417320510287510.1080/01621459.1959.105015050088.35804 – reference: RaoCRKrishnaiahPRMatrix approximations and reduction of dimensionality in multivariate statistical analysisMultivariate analysis1980AmsterdamNorth-Holland322 – reference: OkamotoMDistinctness of the eigenvalues of a quadratic form in a multivariate sampleAnn Stat1973176376533164310.1214/aos/11763424720261.62043 – reference: PeštaMTotal least squares and bootstrapping with application in calibrationStat: J Theor Appl Stat201347596699110.1080/02331888.2012.65880606231528 – reference: KukushAVan HuffelSConsistency of elementwise-weighted total least squares estimator in a multivariate errors-in-variables model AX=B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AX=B$$\end{document}Metrika20045917597204343310.1007/s0018403002721062.62100 – reference: Bhatia R (1996) Matrix analysis. Volume 169 of Graduate Texts in Mathematics. Springer, New York – reference: StewartGWSunJMatrix perturbation theory1990San DiegoAcademic Press0706.65013 – reference: Von NeumannJSome matrix inequalities and metrization of matrix-spaceTomsk Univ Rev19371286300 – reference: GalloPPConsistency of regression estimates when some variables are subject to errorCommunications in statistics: theory and methods19821197398365546610.1080/036109282088282870515.62064 – reference: ThompsonRCPrincipal submatrices IX: interlacing inequalities for singular values of submatricesLinear Algebra Appl1972511229779110.1016/0024-3795(72)90013-40252.15009 – reference: Healy JD (1975) Estimation and tests for unknown linear restrictions in multivariate linear models. PhD thesis, Purdue University, Purdue – reference: GleserLJEstimation in a multivariate “errors in variables” regression model: large sample resultsAnn Stat19819244460053010.1214/aos/11763453300496.62049 – volume-title: The total least squares problem: computational aspects and analysis year: 1991 ident: 800_CR17 doi: 10.1137/1.9781611971002 – volume: 1 start-page: 211 issue: 3 year: 1936 ident: 800_CR2 publication-title: Psychometrika doi: 10.1007/BF02288367 – volume: 54 start-page: 173 year: 1959 ident: 800_CR10 publication-title: J Am Stat Assoc doi: 10.1080/01621459.1959.10501505 – volume: 5 start-page: 1 year: 1972 ident: 800_CR16 publication-title: Linear Algebra Appl doi: 10.1016/0024-3795(72)90013-4 – ident: 800_CR1 doi: 10.1007/978-1-4612-0653-8 – volume: 47 start-page: 966 issue: 5 year: 2013 ident: 800_CR13 publication-title: Stat: J Theor Appl Stat doi: 10.1080/02331888.2012.658806 – volume: 1 start-page: 763 year: 1973 ident: 800_CR12 publication-title: Ann Stat doi: 10.1214/aos/1176342472 – volume-title: Matrix perturbation theory year: 1990 ident: 800_CR15 – volume: 11 start-page: 973 year: 1982 ident: 800_CR3 publication-title: Communications in statistics: theory and methods doi: 10.1080/03610928208828287 – volume: 9 start-page: 24 year: 1981 ident: 800_CR5 publication-title: Ann Stat doi: 10.1214/aos/1176345330 – ident: 800_CR4 – volume: 60 start-page: 525 year: 1973 ident: 800_CR6 publication-title: Biometrika doi: 10.1093/biomet/60.3.525 – start-page: 3 volume-title: Multivariate analysis year: 1980 ident: 800_CR14 – volume: 1 start-page: 286 year: 1937 ident: 800_CR18 publication-title: Tomsk Univ Rev – ident: 800_CR8 doi: 10.21236/ADA037530 – volume: 11 start-page: 50 year: 1960 ident: 800_CR11 publication-title: Q J Math doi: 10.1093/qmath/11.1.50 – volume: 17 start-page: 883 issue: 6 year: 1980 ident: 800_CR7 publication-title: SIAM J Numer Anal doi: 10.1137/0717073 – volume: 59 start-page: 75 issue: 1 year: 2004 ident: 800_CR9 publication-title: Metrika doi: 10.1007/s001840300272 |
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