The differentiation of pseudo-inverses and non-linear least squares problems whose variables separate.

Abstract

For given data (t, Yi), l, , m, we consider the least squares fit ofnonlinear models of the form It is shown that by defining the matrix {(0t)}i, qgj(0t; ti), and the modified functional r2(0t (lY O(0t)/(0t)yl)22, it is possible to optimize first with respect to the parameters 0t, and then to obtain, a posteriori, the optimal parameters . The matrix (0t) is the Moore-Penrose generalized inverse of O(t). We develop formulas for the Fr6chet derivative of orthogonal projectors associated with and also for /(0t), under the hypothesis that O(0t) is of constant (though not necessarily full) rank. Detailed algorithms are presented which make extensive use ofwell-known reliable linear least squares techniques, and numerical results and comparisons are given. These results are generalizations of those of H. D. Scolnik (20) and Guttman, Pereyra and Scolnik (9).

Keywords

MathematicsLeast-squares function approximationRank (graph theory)Matrix (chemical analysis)InverseA priori and a posterioriConstant (computer programming)Applied mathematicsLinear least squaresGeneralized inverseCalculus (dental)CombinatoricsAlgorithmLinear modelComputer scienceStatisticsGeometry

Related Publications

The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate

Gene H. Golub , Víctor Pereyra

For given data $(t_i ,y_i ),i = 1, \cdots ,m$, we consider the least squares fit of nonlinear models of the form \[ \eta ({\bf a},{\boldsymbol \alpha} ;t) = \sum _{j = 1}^n {a_j...

1973 SIAM Journal on Numerical Analysis 1394 citations

Generalized least squares estimators in the analysis of covariance structures

Michael W. Browne

Let S represent the usual unbiased estimator of a covariance matrix. ?o, whose elements are functions of a parameter vector ?o:?o = ?(?o). A generalized least squares (G.L.S.) e...

1974 South African Statistical Journal 336 citations

The Collinearity Problem in Linear Regression. The Partial Least Squares (PLS) Approach to Generalized Inverses

Svante Wold , Axel Ruhe , Herman Wold +1 more

The use of partial least squares (PLS) for handling collinearities among the independent variables X in multiple regression is discussed. Consecutive estimates $({\text{rank }}1...

1984 SIAM Journal on Scientific and Statis... 2462 citations

Applied Linear Regression

Richard J. Beckman , Sanford Weisberg

Preface.1 Scatterplots and Regression.1.1 Scatterplots.1.2 Mean Functions.1.3 Variance Functions.1.4 Summary Graph.1.5 Tools for Looking at Scatterplots.1.5.1 Size.1.5.2 Transfo...

1987 Technometrics 2855 citations

Response Surface Methodology for Optimizing Hyper Parameters

Irina Czogiel , Karsten Luebke , Claus Weihs

The performance of an algorithm often largely depends on some hyper parameter which should be optimized before its usage. Since most conventional optimization methods suffer fro...

2006 Technische Universität Dortmund Eldor... 10 citations

Publication Info

Year: 1973
Type: article
Pages: 181-201
Citations: 734
Access: Closed

External Links

Citation Metrics

734

OpenAlex

Cite This

APA Style

                            
                                    G. H. GOLUBf, 
                                
                                    Víctor Pereyra
                                
                            (1973). 
                            The differentiation of pseudo-inverses and non-linear least squares problems whose variables separate.. 
                            
                            , 181-201.