Fast Robust Regression Algorithms for Problems with Toeplitz Structure (Articolo in rivista)

Type
Label
  • Fast Robust Regression Algorithms for Problems with Toeplitz Structure (Articolo in rivista) (literal)
Anno
  • 2007-01-01T00:00:00+01:00 (literal)
Alternative label
  • Mastronardi N., O'Leary D.P. (2007)
    Fast Robust Regression Algorithms for Problems with Toeplitz Structure
    in Computational statistics & data analysis (Print)
    (literal)
Http://www.cnr.it/ontology/cnr/pubblicazioni.owl#autori
  • Mastronardi N., O'Leary D.P. (literal)
Pagina inizio
  • 1119 (literal)
Pagina fine
  • 1131 (literal)
Http://www.cnr.it/ontology/cnr/pubblicazioni.owl#numeroVolume
  • 52 (literal)
Rivista
Note
  • ISI Web of Science (WOS) (literal)
Http://www.cnr.it/ontology/cnr/pubblicazioni.owl#affiliazioni
  • Istituto per le Applicazioni del Calcolo ``M. Picone'', sede di Bari, Consiglio Nazionale delle Ricerche, Via G. Amendola, 122/D, I-70126 Bari, Italy. Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland,College Park, Maryland 20742, USA. (literal)
Titolo
  • Fast Robust Regression Algorithms for Problems with Toeplitz Structure (literal)
Abstract
  • Consider the problem of computing an approximate solution of an overdetermined system of linear equations. The usual approach to the problem is least squares, in which the 2--norm of the residual isminimized. This produces the minimum variance unbiased estimator of the solution when the errors in the observations are independent and normally distributed with mean 0 and constant variance. It is well known, however, that the least squares solution is not robust if outliers occur, i.e., if some of the observations are contaminated by large error. In this case, alternate approaches have been proposed which judge the size of the residual in a way that is less sensitive to these components. These include the Huber M-function, the Talwar function, the logistic function, the Fair function, and the $\ell_1$ norm. In this paper, new algorithms are proposed to compute the solution to these problems efficiently, in particular when the matrix $A$ has small displacement rank. Matrices with small displacement rank include matrices that are Toeplitz, block-Toeplitz, block-Toeplitz with Toeplitz blocks, Toeplitz plus Hankel, and a variety of other forms. For exposition, only Toeplitz matrices are considered here, but the ideas apply to all matrices with small displacement rank. Algorithms are also presented to compute the solution efficiently when a regularization term is included to handle the case when the matrix of the coefficients is ill-conditioned or rank-deficient. The techniques are illustrated on a problem of FIR system identification. (literal)
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