Minimum-variance unbiased estimates for linear regression models can be obtained by solving least- squares problems. The conjugate gradient method can be successfully used in solving the symmetric and positive definite normal equations obtained from these least-squares problems. Taking into account the results of Golub and Meurant (1997, 2009) [10,11], Hestenes and Stiefel (1952) [17], and Strakoš and Tichý (2002) [16], which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced by Arioli (2005) [18] to the normal equations taking into account the statistical properties of the underpinning linear regression problem. Moreover, we show how the energy norm of the error is linked to the χ 2 -distribution and to the Fisher–Snedecor distribution. Finally, we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criteria.

Linear regression models, least-squares problems, normal equations, and stopping criteria for the conjugate gradient method

Arioli M;
2012

Abstract

Minimum-variance unbiased estimates for linear regression models can be obtained by solving least- squares problems. The conjugate gradient method can be successfully used in solving the symmetric and positive definite normal equations obtained from these least-squares problems. Taking into account the results of Golub and Meurant (1997, 2009) [10,11], Hestenes and Stiefel (1952) [17], and Strakoš and Tichý (2002) [16], which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced by Arioli (2005) [18] to the normal equations taking into account the statistical properties of the underpinning linear regression problem. Moreover, we show how the energy norm of the error is linked to the χ 2 -distribution and to the Fisher–Snedecor distribution. Finally, we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criteria.
Linear regression
Least-squares problems
Conjugate gradient Stopping criteria
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12572/6187
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