Data Fitting Problems with Bounded Uncertainties in the Data

An analysis of a class of data fitting problems, where the data uncertainties are subject to known bounds, is given in a very general setting. It is shown how such problems can be posed in a computationally convenient form, and the connection with other more conventional data fitting problems is examined. The problems have attracted interest so far in the special case when the underlying norm is the least squares norm. Here the special structure can be exploited to computational advantage, and we include some observations which contribute to algorithmic development for this particular case. We also consider some variants of the main problems and show how these too can be posed in a form which facilitates their numerical solution.

[1]  Ali H. Sayed,et al.  The Degenerate Bounded Errors-in-Variables Model , 2001, SIAM J. Matrix Anal. Appl..

[2]  Pramod P. Khargonekar,et al.  FILTERING AND SMOOTHING IN AN H" SETTING , 1991 .

[3]  Sabine Van Huffel,et al.  The total least squares problem , 1993 .

[4]  G. Golub,et al.  Parameter Estimation in the Presence of Bounded Data Uncertainties , 1998, SIAM J. Matrix Anal. Appl..

[5]  A.H. Sayed,et al.  Estimation in the presence of multiple sources of uncertainties with applications , 1998, Conference Record of Thirty-Second Asilomar Conference on Signals, Systems and Computers (Cat. No.98CH36284).

[6]  N. Draper,et al.  Applied Regression Analysis , 1966 .

[7]  Laurent El Ghaoui,et al.  Robust Solutions to Least-Squares Problems with Uncertain Data , 1997, SIAM J. Matrix Anal. Appl..

[8]  Sabine Van Huffel,et al.  Recent advances in total least squares techniques and errors-in-variables modeling , 1997 .

[9]  M. R. Osborne,et al.  An Analysis of the Total Approximation Problem in Separable Norms, and an Algorithm for the Total $l_1 $ Problem , 1985 .

[10]  Sabine Van Huffel,et al.  Total least squares problem - computational aspects and analysis , 1991, Frontiers in applied mathematics.

[11]  Ming Gu,et al.  An Efficient Algorithm for a Bounded Errors-in-Variables Model , 1999, SIAM J. Matrix Anal. Appl..

[12]  G. A. Watson,et al.  Choice of norms for data fitting and function approximation , 1998, Acta Numerica.

[13]  S. Chandrasekaran,et al.  Parameter estimation in the presence of bounded modeling errors , 1997, IEEE Signal Processing Letters.

[14]  J. W. Gorman,et al.  Fitting Equations to Data. , 1973 .

[15]  G. Golub,et al.  Efficient algorithms for least squares type problems with bounded uncertainties , 1997 .

[16]  T. Kailath,et al.  Inertia conditions for the minimization of quadratic forms in indefinite metric spaces , 1996 .

[17]  Ali H. Sayed,et al.  Recursive linear estimation in Krein spaces. I. Theory , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.