Total Problem of Constructing Linear Regression Using Matrix Correction Methods with Minimax Criterion
A linear problem of regression analysis is considered under the assumption of the presence of noise in the output and input variables. This approximation problem may be interpreted as an improper interpolation problem, for which it is required to correct optimally the positions of the original point...
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| Vydáno v: | Mathematics (Basel) Ročník 11; číslo 3; s. 546 |
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| Jazyk: | angličtina |
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01.01.2023
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| Abstract | A linear problem of regression analysis is considered under the assumption of the presence of noise in the output and input variables. This approximation problem may be interpreted as an improper interpolation problem, for which it is required to correct optimally the positions of the original points in the data space so that they all lie on the same hyperplane. The use of the quadratic approximation criterion for such a problem led to the appearance of the total least squares method. In this paper, we use the minimax criterion to estimate the measure of correction of the initial data. It leads to a nonlinear mathematical programming problem. It is shown that this problem can be reduced to solving a finite number of linear programming problems. However, this number depends exponentially on the number of parameters. Some methods for overcoming this complexity of the problem are proposed. |
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| AbstractList | A linear problem of regression analysis is considered under the assumption of the presence of noise in the output and input variables. This approximation problem may be interpreted as an improper interpolation problem, for which it is required to correct optimally the positions of the original points in the data space so that they all lie on the same hyperplane. The use of the quadratic approximation criterion for such a problem led to the appearance of the total least squares method. In this paper, we use the minimax criterion to estimate the measure of correction of the initial data. It leads to a nonlinear mathematical programming problem. It is shown that this problem can be reduced to solving a finite number of linear programming problems. However, this number depends exponentially on the number of parameters. Some methods for overcoming this complexity of the problem are proposed. |
| Audience | Academic |
| Author | Zolotova, Tatiana Gorelik, Victor |
| Author_xml | – sequence: 1 givenname: Victor orcidid: 0000-0003-2435-0796 surname: Gorelik fullname: Gorelik, Victor – sequence: 2 givenname: Tatiana orcidid: 0000-0001-5185-0687 surname: Zolotova fullname: Zolotova, Tatiana |
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| Cites_doi | 10.1142/9789812385192 10.1016/j.lrp.2012.09.008 10.1016/j.im.2019.05.003 10.1007/s10208-006-0196-8 10.1017/9781108583664 10.1201/9780203748923 10.1137/0717073 10.1080/09585192.2017.1416655 10.1016/j.sigpro.2007.04.004 10.1007/978-3-030-80519-7 10.1007/978-3-030-62867-3_10 10.1137/18M118935X 10.1137/S0895479893258802 10.1039/C8AN00599K 10.1134/S0965542516020081 10.1007/978-3-319-05542-8_15-2 10.1016/j.sigpro.2006.11.004 |
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| References | Rosen (ref_8) 1996; 17 Markovsky (ref_9) 2007; 87 ref_14 Ringle (ref_18) 2020; 31 Benitez (ref_11) 2020; 57 ref_12 Gorelik (ref_2) 2001; 11 ref_22 Gorelik (ref_21) 2020; 12422 Golub (ref_6) 1980; 17 Ahmadi (ref_10) 2019; 3 ref_20 ref_1 ref_3 ref_19 Lee (ref_17) 2018; 143 ref_15 Gorelik (ref_5) 2016; 56 Back (ref_7) 2007; 87 ref_4 Hair (ref_16) 2012; 45 Caponnetto (ref_13) 2007; 7 |
| References_xml | – ident: ref_20 doi: 10.1142/9789812385192 – ident: ref_4 – ident: ref_3 – volume: 45 start-page: 320 year: 2012 ident: ref_16 article-title: The use of partial least squares structural equation modeling in strategic management research: A review of past practices and recommendations for future applications publication-title: Long Range Plan. doi: 10.1016/j.lrp.2012.09.008 – volume: 11 start-page: 1615 year: 2001 ident: ref_2 article-title: Matrix correction of a linear programming problem with inconsistent constraints publication-title: Comput. Math. Math. Phys. – volume: 57 start-page: 103168 year: 2020 ident: ref_11 article-title: How to perform and report an impactful analysis using partial least squares: Guidelines for confirmatory and explanatory IS research publication-title: Inf. Manag. doi: 10.1016/j.im.2019.05.003 – volume: 7 start-page: 331 year: 2007 ident: ref_13 article-title: Optimal rates for the regularized least-squares algorithm publication-title: Found. Comput. Math. doi: 10.1007/s10208-006-0196-8 – ident: ref_12 doi: 10.1017/9781108583664 – ident: ref_14 doi: 10.1201/9780203748923 – volume: 17 start-page: 883 year: 1980 ident: ref_6 article-title: An analysis of the total least squares problem publication-title: SIAM J. Numer. Anal. doi: 10.1137/0717073 – volume: 31 start-page: 1617 year: 2020 ident: ref_18 article-title: Partial least squares structural equation modeling in HRM research publication-title: Int. J. Hum. Resour. Manag. doi: 10.1080/09585192.2017.1416655 – volume: 87 start-page: 2283 year: 2007 ident: ref_9 article-title: Overview of total least-squares methods publication-title: Signal Process. doi: 10.1016/j.sigpro.2007.04.004 – ident: ref_15 doi: 10.1007/978-3-030-80519-7 – volume: 12422 start-page: 122 year: 2020 ident: ref_21 article-title: Method of Parametric Correction in Data Transformation and Approximation Problems publication-title: Lect. Notes Comput. Sci. doi: 10.1007/978-3-030-62867-3_10 – volume: 3 start-page: 193 year: 2019 ident: ref_10 article-title: DSOS and SDSOS optimization: More tractable alternatives to sum of squares and semidefinite optimization publication-title: SIAM J. Appl. Algebra Geom. doi: 10.1137/18M118935X – volume: 17 start-page: 110 year: 1996 ident: ref_8 article-title: Total least norm formulation and solution for strucured problems publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/S0895479893258802 – ident: ref_1 – volume: 143 start-page: 3526 year: 2018 ident: ref_17 article-title: Partial least squares-discriminant analysis (PLS-DA) for classification of high-dimensional (HD) data: A review of contemporary practice strategies and knowledge gaps publication-title: Analyst doi: 10.1039/C8AN00599K – volume: 56 start-page: 200 year: 2016 ident: ref_5 article-title: Solution of the linear regression problem using matrix correction methods in the l1 metric publication-title: Comput. Math. Math. Phys. doi: 10.1134/S0965542516020081 – ident: ref_22 – ident: ref_19 doi: 10.1007/978-3-319-05542-8_15-2 – volume: 87 start-page: 2303 year: 2007 ident: ref_7 article-title: The matrix-restricted total least squares problem publication-title: Signal Process. doi: 10.1016/j.sigpro.2006.11.004 |
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| SubjectTerms | Analysis Approximation Chebyshev approximation Criteria data processing Hyperplanes Hypotheses Interpolation Least squares method Linear equations Linear models (Statistics) Linear programming linear programming problem linear regression Linear regression models Mathematical analysis Mathematical programming Matrices matrix correction Maximum likelihood method Methods minimax criterion Minimax technique Normal distribution Norms Random variables Regression analysis |
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