Name
Abstract | ||
|---|---|---|
The problem of computing an approximate solution of an overdetermined system of linear equations is considered. The usual approach to the problem is least squares, in which the 2-norm of the residual is minimized. 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 @?"1 norm. 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, 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. |
Year | DOI | Venue |
|---|---|---|
2007 | 10.1016/j.csda.2007.05.008 | Computational Statistics & Data Analysis |
Keywords | Field | DocType |
robust regression algorithm,talwar function,approximate solution,constant variance,toeplitz block,toeplitz structure,squares solution,fir system identification,toeplitz matrix,small displacement rank,fair function,logistic function,robust regression,minimum variance,overdetermined system,outliers,normal distribution,unbiased estimator,iteratively reweighted least squares,linear equations,least square,system identification | Least squares,Econometrics,Overdetermined system,Matrix (mathematics),Algorithm,Robust regression,Toeplitz matrix,Iteratively reweighted least squares,Statistics,Mathematics,Block matrix,Numerical linear algebra | Journal |
Volume | Issue | ISSN |
52 | 2 | Computational Statistics and Data Analysis |
Citations | PageRank | References |
2 | 0.41 | 9 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Nicola Mastronardi | 1 | 153 | 29.66 |
| O'Leary, Dianne P. | 2 | 1064 | 222.93 |