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Interpolation and Extrapolation Optimal Designs V1: Polynomial Regression and Approximation Theory

Interpolation and Extrapolation Optimal Designs V1: Polynomial Regression and Approximation Theory

Giorgio Celant, Michel Broniatowski

ISBN: 978-1-119-29227-2

Apr 2016, Wiley-ISTE

254 pages


This book is the first of a series which focuses on the interpolation and extrapolation of optimal designs, an area with significant applications in engineering, physics, chemistry and most experimental fields.

In this volume, the authors emphasize the importance of problems associated with the construction of design. After a brief introduction on how the theory of optimal designs meets the theory of the uniform approximation of functions, the authors introduce the basic elements to design planning and link the statistical theory of optimal design and the theory of the uniform approximation of functions.

The appendices provide the reader with material to accompany the proofs discussed throughout the book.

Preface ix

Introduction  xi

Part 1. Elements from Approximation Theory  1

Chapter 1. Uniform Approximation 3

1.1. Canonical polynomials and uniform approximation  3

1.2. Existence of the best approximation 4

1.3. Characterization and uniqueness of the best approximation  5

1.3.1. Proof of the Borel–Chebyshev theorem  7

1.3.2. Example  13

Chapter 2. Convergence Rates for the Uniform Approximation and Algorithms  15

2.1. Introduction  15

2.2. The Borel–Chebyshev theorem and standard functions  15

2.3. Convergence of the minimax approximation  20

2.3.1. Rate of convergence of the minimax approximation  21

2.4. Proof of the de la Vallée Poussin theorem  24

2.5. The Yevgeny Yakovlevich Remez algorithm  28

2.5.1. The Remez algorithm 29

2.5.2. Convergence of the Remez algorithm  33

Chapter 3. Constrained Polynomial Approximation  43

3.1. Introduction and examples  43

3.2. Lagrange polynomial interpolation 47

3.3. The interpolation error  50

3.3.1. A qualitative result  50

3.3.2. A quantitative result  52

3.4. The role of the nodes and the minimization of the interpolation error  54

3.5. Convergence of the interpolation approximation  56

3.6. Runge phenomenon and lack of convergence  57

3.7. Uniform approximation for C(∞) ([a, b]) functions  62

3.8. Numerical instability 63

3.9. Convergence, choice of the distribution of the nodes, Lagrange interpolation and splines  67

Part 2. Optimal Designs for Polynomial Models  69

Chapter 4. Interpolation and Extrapolation Designs for the Polynomial Regression  71

4.1. Definition of the model and of the estimators  71

4.2. Optimal extrapolation designs: Hoel–Levine or Chebyshev designs 75

4.2.1. Uniform optimal interpolation designs (according to Guest) 85

4.2.2. The interplay between the Hoel–Levine and the Guest designs  95

4.2.3. Confidence bound for interpolation/extrapolation designs 98

4.3. An application of the Hoel–Levine design 100

4.4. Multivariate optimal designs: a special case  103

Chapter 5. An Introduction to Extrapolation Problems Based on Observations on a Collection of Intervals 113

5.1. Introduction  113

5.2. The model, the estimator and the criterion for the choice of the design  119

5.2.1. Criterion for the optimal design 121

5.3. A constrained Borel–Chebyshev theorem  122

5.3.1. Existence of solutions to the Pg−1 (0, 1) problem 122

5.3.2. A qualitative discussion on some constrained Borel–Chebyshev theorem 123

5.3.3. Borel–Chebyshev theorem on [a, b] ∪ [d, e] 125

5.3.4. From the constrained Borel–Chebyshev theorem to the support of the optimal design 126

5.4. Qualitative properties of the polynomial which determines the optimal nodes 127

5.4.1. The linear case 127

5.4.2. The general polynomial case 128

5.5. Identification of the polynomial which characterizes the optimal nodes  130

5.5.1. The differential equation 130

5.5.2. Example  132

5.6. The optimal design in favorable cases  134

5.6.1. Some explicit optimal designs  136

5.7. The optimal design in the general case 137

5.7.1. The extreme points of a linear functional  138

5.7.2. Some results on the representation of the extreme points 138

5.7.3. The specific case of the Dirac functional at point 0  142

5.7.4. Remez algorithm for the extreme polynomial: the optimal design in general cases  145

5.8. Spruill theorem: the optimal design 146

Chapter 6. Instability of the Lagrange Interpolation Scheme With Respect to Measurement Errors 147

6.1. Introduction  147

6.2. The errors that cannot be avoided  147

6.2.1. The role of the errors: interpolation designs with minimal propagation of the errors  150

6.2.2. Optimizing on the nodes 153

6.3. Control of the relative errors 157

6.3.1. Implementation of the Remez algorithm for the relative errors 162

6.4. Randomness  166

6.5. Some inequalities for the derivatives of polynomials  167

6.6. Concentration inequalities  168

6.7. Upper bounds of the extrapolation error due to randomness, and the resulting size of the design for real analytic regression functions 172

6.7.1. Case 1: the range of the observations is bounded 177

6.7.2. Case 2: the range of the observations is unbounded  183

Part 3. Mathematical Material  185

Appendix 1. Normed Linear Spaces  187

Appendix 2. Chebyshev Polynomials 217

Appendix 3. Some Useful Inequalities for Polynomials 221

Bibliography 243

Index  251

"it seems that the book deserves more attention than a typical textbook, due to its particular features. Firstly, the authors are active researchers in the eld. Secondly, they concentrate mainly on characterizing optimal designs analytically. Both in the book and in their research, they put emphasis on optimal experiment design for function approximation or interpolation from observations with random errors"...."The book is rigorously written and it will be useful not only for advanced teaching, but also as a good starting point for further research" Ewaryst Rafaj lowicz, Mathematical Reviews, Sept 2017