A Constrained Least Squares Approach to the General Gauss-Markov Linear Model

Kourouklis, Stavros; Paige, Christopher C.

doi:10.1080/01621459.1981.10477694

Cited by 41 publications

(12 citation statements)

References 9 publications

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“…The model (1) was introduced by Paige [10], and computational algorithms can be found in [7,9,11]. A more detailed analysis of ( 1 ) in terms of the generalized SVD is also given by Paige [12], while Björck [1, §23] extended this analysis to the case when both A and B may be rank-deficient.…”

Section: Introductionmentioning

confidence: 99%

Regularization and the General Gauss-Markov Linear Model

Zha¹,

Hansen²

1990

Mathematics of Computation

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Abstract. If the coefficient matrix in the general Gauss-Markov linear model is ill-conditioned, then the solution is very sensitive to perturbations. For such problems, we propose to add Tikhonov regularization to the model, and we show that this actually stabilizes the solution and decreases its variance. We also give a numerically stable algorithm for computing the regularized solution efficiently.

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Section: Introductionmentioning

confidence: 99%

Regularization and the General Gauss-Markov Linear Model

Zha¹,

Hansen²

1990

Mathematics of Computation

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“…When = BB T , the weighted least-squares problem (6) is equivalent to the generalized linear leastsquares problem (Kourouklis and Paige, 1981) …”

Section: Generalized Least Squaresmentioning

confidence: 99%

“…Solving the normal equations on a computer may lead to a loss of information and is known to be unreliable in the sense that small perturbations in the given data can lead to large errors in the solution when using finite precision arithmetic (Golub and Van Loan, 1996). Second, we propose a generalized least-squares (GLS) formulation of the EnKF, relying on the minimization of an alternative functional due to Kourouklis and Paige (1981), and which is similar to Lorenc (2003).…”

Section: Introductionmentioning

confidence: 99%

A robust formulation of the ensemble Kalman filter

Thomas

Hacker

Anderson

2009

Quart J Royal Meteoro Soc

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ABSTRACT:The ensemble Kalman filter (EnKF) can be interpreted in the more general context of linear regression theory. The recursive filter equations are equivalent to the normal equations for a weighted least-squares estimate that minimizes a quadratic functional. Solving the normal equations is numerically unreliable and subject to large errors when the problem is ill-conditioned. A numerically reliable and efficient algorithm is presented, based on the minimization of an alternative functional. The method relies on orthogonal rotations, is highly parallel and does not 'square' matrices in order to compute the analysis update. Computation of eigenvalue and singular-value decompositions is not required. The algorithm is formulated to process observations serially or in batches and therefore easily handles spatially correlated observation errors. Numerical results are presented for existing algorithms with a hierarchy of models characterized by chaotic dynamics. Under a range of conditions, which may include model error and sampling error, the new algorithm achieves the same or lower mean square errors as the serial Potter and ensemble adjustment Kalman filter (EAKF) algorithms. Published in

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“…Subsequent researchers (see Mitra and Bhimasankaram, 1971;McGilchrist and Sandland, 1979;Haslett, 1985;Bhimasankaram et al, 1995;Bhimasankaram and Jammalamadaka, 1994a and Sengupta, 1999) sought to extend this work to data deletion, variable inclusion/exclusion, heteroscedastic and correlated model errors, rank-deficient V, rank-deficient X, inclusion/exclusion of multiple observations or variables, and so on. Another stream of research focussed on numerically stable methods of recursive estimation in the linear model (see, e.g., Chambers, 1975;Gragg et al, 1979;Kourouklis and Paige, 1981;Farebrother, 1988).…”

Section: Introductionmentioning

confidence: 99%