Least Squares Data Fitting with Applications
Published on 12. March 2013
328 pages
978-1-4214-0858-3 (ISBN)
Description
A lucid explanation of the intricacies of both simple and complex least squares methods.
As one of the classical statistical regression techniques, and often the first to be taught to new students, least squares fitting can be a very effective tool in data analysis. Given measured data, we establish a relationship between independent and dependent variables so that we can use the data predictively. The main concern of Least Squares Data Fitting with Applications is how to do this on a computer with efficient and robust computational methods for linear and nonlinear relationships. The presentation also establishes a link between the statistical setting and the computational issues.
In a number of applications, the accuracy and efficiency of the least squares fit is central, and Per Christian Hansen, Víctor Pereyra, and Godela Scherer survey modern computational methods and illustrate them in fields ranging from engineering and environmental sciences to geophysics. Anyone working with problems of linear and nonlinear least squares fitting will find this book invaluable as a hands-on guide, with accessible text and carefully explained problems.
Included are
an overview of computational methods together with their properties and advantages
topics from statistical regression analysis that help readers to understand and evaluate the computed solutions
many examples that illustrate the techniques and algorithms
Least Squares Data Fitting with Applications can be used as a textbook for advanced undergraduate or graduate courses and professionals in the sciences and in engineering.
As one of the classical statistical regression techniques, and often the first to be taught to new students, least squares fitting can be a very effective tool in data analysis. Given measured data, we establish a relationship between independent and dependent variables so that we can use the data predictively. The main concern of Least Squares Data Fitting with Applications is how to do this on a computer with efficient and robust computational methods for linear and nonlinear relationships. The presentation also establishes a link between the statistical setting and the computational issues.
In a number of applications, the accuracy and efficiency of the least squares fit is central, and Per Christian Hansen, Víctor Pereyra, and Godela Scherer survey modern computational methods and illustrate them in fields ranging from engineering and environmental sciences to geophysics. Anyone working with problems of linear and nonlinear least squares fitting will find this book invaluable as a hands-on guide, with accessible text and carefully explained problems.
Included are
an overview of computational methods together with their properties and advantages
topics from statistical regression analysis that help readers to understand and evaluate the computed solutions
many examples that illustrate the techniques and algorithms
Least Squares Data Fitting with Applications can be used as a textbook for advanced undergraduate or graduate courses and professionals in the sciences and in engineering.
More details
Language
English
Place of publication
Baltimore
United States
Target group
Professional and scholarly
Illustrations
8 s/w Zeichnungen, 46 Kurvendiagramme, 9 s/w Photographien bzw. Rasterbilder
9 Halftones, black and white; 46 Graphs; 8 Line drawings, black and white
File size
15,16 MB
ISBN-13
978-1-4214-0858-3 (9781421408583)
DOI
10.1353/book.21076
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
Per Christian Hansen | Victor Pereyra | Godela Scherer
Least Squares Data Fitting with Applications
Book
03/2013
Johns Hopkins University Press
€96.20
Article not available at the moment
Persons
Per Christian Hansen is a professor of scientific computing at the Technical University of Denmark. Víctor Pereyra is a consulting professor of energy resources engineering at Stanford University and was a principal at Weidlinger Associates, Los Altos, California. Godela Scherer is a visiting research fellow in the Department of Mathematics at the University of Reading, United Kingdom, and a professor of scientific computing at the Universidad Simón Bolívar, Venezuela.
Content
Preface
Symbols and Acronyms
Chapter 1. The Linear Data Fitting Problem
1.1. Parameter estimation, data approximation
1.2. Formulation of the data fitting problem
1.3. Maximum likelihood estimation
1.4. The residuals and their properties
1.5. Robust regression
Chapter 2. The Linear Least Squares Problem
2.1. Linear least squares problem formulation
2.2. The QR factorization and its role
2.3. Permuted QR factorization
Chapter 3. Analysis of Least Squares Problems
3.1. The pseudoinverse
3.2. The singular value decomposition
3.3. Generalized singular value decomposition
3.4. Condition number and column scaling
3.5. Perturbation analysis
Chapter 4. Direct Methods for Full-Rank Problems
4.1. Normal equations
4.2. LU factorization
4.3. QR factorization
4.4. Modifying least squares problems
4.5. Iterative refinement
4.6. Stability and condition number estimation
4.7. Comparison of the methods
Chapter 5. Direct Methods for Rank-Deficient Problems
5.1. Numerical rank
5.2. Peters-Wilkinson LU factorization
5.3. QR factorization with column permutations
5.4. UTV and VSV decompositions
5.5. Bidiagonalization
5.6. SVD computations
Chapter 6. Methods for Large-Scale Problems
6.1. Iterative versus direct methods
6.2. Classical stationary methods
6.3. Non-stationary methods, Krylov methods
6.4. Practicalities: preconditioning and stopping criteria
6.5. Block methods
Chapter 7. Additional Topics in Least Squares
7.1. Constrained linear least squares problems
7.2. Missing data problems
7.3. Total least squares (TLS)
7.4. Convex optimization
7.5. Compressed sensing
Chapter 8. Nonlinear Least Squares Problems
8.1. Introduction
8.2. Unconstrained problems
8.3. Optimality conditions for constrained problems
8.4. Separable nonlinear least squares problems
8.5. Multiobjective optimization
Chapter 9. Algorithms for Solving Nonlinear LSQ Problems
9.1. Newton's method
9.2. The Gauss-Newton method
9.3. The Levenberg-Marquardt method
9.4. Additional considerations and software
9.5. Iteratively reweighted LSQ algorithms for robust data fitting problems
9.6. Variable projection algorithm
9.7. Block methods for large-scale problems
Chapter 10. Ill-Conditioned Problems
10.1. Characterization
10.2. Regularization methods
10.3. Parameter selection techniques
10.4. Extensions of Tikhonov regularization
10.5. Ill-conditioned NLLSQ problems
Chapter 11. Linear Least Squares Applications
11.1. Splines in approximation
11.2. Global temperatures data fitting
11.3. Geological surface modeling
Chapter 12. Nonlinear Least Squares Applications
12.1. Neural networks training
12.2. Response surfaces, surrogates or proxies
12.3. Optimal design of a supersonic aircraft
12.4. NMR spectroscopy
12.5. Piezoelectric crystal identification
12.6. Travel time inversion of seismic data
Appendix A: Sensitivity Analysis
A.1. Floating-point arithmetic
A.2. Stability, conditioning and accuracy
Appendix B: Linear Algebra Background
B.1. Norms
B.2. Condition number
B.3. Orthogonality
B.4. Some additional matrix properties
Appendix C: Advanced Calculus Background
C.1. Convergence rates
C.2. Multivariable calculus
Appendix D: Statistics
D.1. Definitions
D.2. Hypothesis testing
References
Index
Symbols and Acronyms
Chapter 1. The Linear Data Fitting Problem
1.1. Parameter estimation, data approximation
1.2. Formulation of the data fitting problem
1.3. Maximum likelihood estimation
1.4. The residuals and their properties
1.5. Robust regression
Chapter 2. The Linear Least Squares Problem
2.1. Linear least squares problem formulation
2.2. The QR factorization and its role
2.3. Permuted QR factorization
Chapter 3. Analysis of Least Squares Problems
3.1. The pseudoinverse
3.2. The singular value decomposition
3.3. Generalized singular value decomposition
3.4. Condition number and column scaling
3.5. Perturbation analysis
Chapter 4. Direct Methods for Full-Rank Problems
4.1. Normal equations
4.2. LU factorization
4.3. QR factorization
4.4. Modifying least squares problems
4.5. Iterative refinement
4.6. Stability and condition number estimation
4.7. Comparison of the methods
Chapter 5. Direct Methods for Rank-Deficient Problems
5.1. Numerical rank
5.2. Peters-Wilkinson LU factorization
5.3. QR factorization with column permutations
5.4. UTV and VSV decompositions
5.5. Bidiagonalization
5.6. SVD computations
Chapter 6. Methods for Large-Scale Problems
6.1. Iterative versus direct methods
6.2. Classical stationary methods
6.3. Non-stationary methods, Krylov methods
6.4. Practicalities: preconditioning and stopping criteria
6.5. Block methods
Chapter 7. Additional Topics in Least Squares
7.1. Constrained linear least squares problems
7.2. Missing data problems
7.3. Total least squares (TLS)
7.4. Convex optimization
7.5. Compressed sensing
Chapter 8. Nonlinear Least Squares Problems
8.1. Introduction
8.2. Unconstrained problems
8.3. Optimality conditions for constrained problems
8.4. Separable nonlinear least squares problems
8.5. Multiobjective optimization
Chapter 9. Algorithms for Solving Nonlinear LSQ Problems
9.1. Newton's method
9.2. The Gauss-Newton method
9.3. The Levenberg-Marquardt method
9.4. Additional considerations and software
9.5. Iteratively reweighted LSQ algorithms for robust data fitting problems
9.6. Variable projection algorithm
9.7. Block methods for large-scale problems
Chapter 10. Ill-Conditioned Problems
10.1. Characterization
10.2. Regularization methods
10.3. Parameter selection techniques
10.4. Extensions of Tikhonov regularization
10.5. Ill-conditioned NLLSQ problems
Chapter 11. Linear Least Squares Applications
11.1. Splines in approximation
11.2. Global temperatures data fitting
11.3. Geological surface modeling
Chapter 12. Nonlinear Least Squares Applications
12.1. Neural networks training
12.2. Response surfaces, surrogates or proxies
12.3. Optimal design of a supersonic aircraft
12.4. NMR spectroscopy
12.5. Piezoelectric crystal identification
12.6. Travel time inversion of seismic data
Appendix A: Sensitivity Analysis
A.1. Floating-point arithmetic
A.2. Stability, conditioning and accuracy
Appendix B: Linear Algebra Background
B.1. Norms
B.2. Condition number
B.3. Orthogonality
B.4. Some additional matrix properties
Appendix C: Advanced Calculus Background
C.1. Convergence rates
C.2. Multivariable calculus
Appendix D: Statistics
D.1. Definitions
D.2. Hypothesis testing
References
Index
