Interpolation and extrapolation optimal designs
 Responsibility
 Giorgio Celant, Michel Broniatowski.
 Publication
 London, UK : ISTE ; Hoboken, NJ, USA : John Wiley, 2016
 Copyright notice
 ©2016
 Physical description
 volumes ; 25 cm.
 Series
 Mathematics and statistics series (ISTE)
At the library
Science Library (Li and Ma)
Stacks
Call number  Status 

QA281 .C35 2016 V.1  Unknown 
QA281 .C35 2016 V.2  Unknown 
More options
Description
Creators/Contributors
 Author/Creator
 Celant, Giorgio, author.
 Contributor
 Broniatowski, Michel, author.
Contents/Summary
 Bibliography
 Includes bibliographical references and index.
 Contents

 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 BorelChebyshev 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 BorelChebyshev 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 Vallee 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(â ) ([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: HoelLevine or Chebyshev designs
 75 4.2.1. Uniform optimal interpolation designs (according to Guest)
 85 4.2.2. The interplay between the HoelLevine and the Guest designs
 95 4.2.3. Confidence bound for interpolation/extrapolation designs
 98 4.3. An application of the HoelLevine 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 BorelChebyshev theorem
 122 5.3.1. Existence of solutions to the Pgâ
 1 (0, 1) problem
 122 5.3.2. A qualitative discussion on some constrained BorelChebyshev theorem
 123 5.3.3. BorelChebyshev theorem on [a, b] â ª [d, e]
 125 5.3.4. From the constrained BorelChebyshev 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.
 (source: Nielsen Book Data)
 1. Approximation of Continuous Functions in Normed Spaces.
 2. Chebyshev Systems.
 3. Uniform Approximations in a Normed Space.
 4. Calculation of the Best Uniform Approximation in a Chebyshev System.
 5. Optimal Extrapolation Design for the Chebyshev Regression.
 6. Optimal Design for Linear Forms of the Parameters in a Chebyshev Regression.
 7. Special Topics and Extensions.
 8. Multivariate Models and Algorithms.
 (source: Nielsen Book Data)
 Publisher's Summary
 ["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.", {"source"=>"(source: Nielsen Book Data)"}, "9781119292296", "20171002"]
 ["This book considers various extensions of the topics treated in the first volume of this series, in relation to the class of models and the type of criterion for optimality. The regressors are supposed to belong to a generic finite dimensional Haar linear space, which substitutes for the classical polynomial case. The estimation pertains to a general linear form of the coefficients of the model, extending the interpolation and extrapolation framework; the errors in the model may be correlated, and the model may be heteroscedastic. Nonlinear models, as well as multivariate ones, are briefly discussed. The book focuses to a large extent on criteria for optimality, and an entire chapter presents algorithms leading to optimal designs in multivariate models. Elfving s theory and the theorem of equivalence are presented extensively. The volume presents an account of the theory of the approximation of real valued functions, which makes it selfconsistent.", {"source"=>"(source: Nielsen Book Data)"}, "9781786300546", "20171002"]
Subjects
Bibliographic information
 Beginning date
 2016
 Series
 Mathematics and statistics
 ISBN
 9781848219953 print
 1848219954
 9781786300546
 1786300540
 9781119292296 electronic bk.
 1119292298 electronic bk.
 9781119292272 electronic bk.
 1119292271 electronic bk.