Interpolation and Extrapolation Optimal Designs 1 E-Book

Giorgio Celant

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 320

Veröffentlichungsjahr: 2016

Beschreibung

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.

Leseprobe

Table of Contents

Cover

Title

Copyright

Preface

Introduction

I.1. The scope of this book

I.2. A generic case: the Hoel and Levine extrapolation scheme and the uniform interpolation design of Guest

I.3. Extrapolation design in non standard cases, algorithms

I.4. Uniform approximation of functions, an outlook

I.5. A general bibliography

PART 1: Elements from Approximation Theory

1 Uniform Approximation

1.1. Canonical polynomials and uniform approximation

1.2. Existence of the best approximation

1.3. Characterization and uniqueness of the best approximation

2 Convergence Rates for the Uniform Approximation and Algorithms

2.1. Introduction

2.2. The Borel–Chebyshev theorem and standard functions

2.3. Convergence of the minimax approximation

2.4. Proof of the de la Vallée Poussin theorem

2.5. The Yevgeny Yakovlevich Remez algorithm

3 Constrained Polynomial Approximation

3.1. Introduction and examples

3.2. Lagrange polynomial interpolation

3.3. The interpolation error

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

3.5. Convergence of the interpolation approximation

3.6. Runge phenomenon and lack of convergence

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

3.8. Numerical instability

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

PART 2: Optimal Designs for Polynomial Models

4 Interpolation and Extrapolation Designs for the Polynomial Regression

4.1. Definition of the model and of the estimators

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

4.3. An application of the Hoel–Levine design

4.4. Multivariate optimal designs: a special case

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

5.1. Introduction

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

5.3. A constrained Borel–Chebyshev theorem

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

5.5. Identification of the polynomial which characterizes the optimal nodes

5.6. The optimal design in favorable cases

5.7. The optimal design in the general case

5.8. Spruill theorem: the optimal design

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

6.1. Introduction

6.2. The errors that cannot be avoided

6.3. Control of the relative errors

6.4. Randomness

6.5. Some inequalities for the derivatives of polynomials

6.6. Concentration inequalities

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

PART 3: Mathematical Material

Appendix 1: Normed Linear Spaces

A1.1. General notions

A1.2. Compatibility between the topological and the linear structure in linear spaces

A1.3. A characterization of the sup norm of a continuous function defined on a compact set

Appendix 2: Chebyshev Polynomials

Appendix 3: Some Useful Inequalities for Polynomials

A3.1. Bounds on the derivatives of polynomials on compact sets

Bibliography

Index

End User License Agreement

Guide

Cover

Table of Contents

Begin Reading

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Series EditorNikolaos Limnios

Interpolation and Extrapolation Optimal Designs 1

Polynomial Regression and Approximation Theory

Giorgio Celant

Michel Broniatowski

First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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John Wiley & Sons, Inc.

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© ISTE Ltd 2016

The rights of Giorgio Celant and Michel Broniatowski to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016933881

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-995-3

Preface

This book is the first of a series of three which cover a part of the field called optimal designs, in the context of interpolation and extrapolation. This area has been studied extensively, due to its numerous applications in engineering, physics, chemistry, and more generally, in all fields where experiments can be planned according to some expected accuracy on the resulting conclusions, under operational constraints.

The context of the present volume, which has been considered, is very specific by choice. Indeed reducing the model to the case where the observations are real numbers, hence unidimensional, and the environmental variable being unidimensional too, most of the concepts gain in clarity; also the tools leading to optimal solutions are quite accessible, although they require some technicalities. This choice will help the reader to consider more general models, keeping in mind the basic ingredients which are developed in more involved situations, at the cost of some additional regularity assumptions. This is a well-known way to proceed in science.

The focus of this book is statistics. The contents of the book are mostly real analysis and approximation of functions. This duality in the arguments is not surprising, and is a constant in most advanced fields in statistics, and also in various other disciplines. It happens, and this is a major fact in the present field of statistics, that optimal designs are obtained as special problems in the theory of functional approximation. So, those two fields, namely statistics and numerical analysis, meet in the present setting.

The framework of the present book is thus one of the classical real analysis, of the basic tools in algebra together with standard basic tools in statistics. The reader may also be interested in the chronological aspect (or historical aspect) of the development, as the authors have been. Although of statistical concern, the main arguments used in this book stem from the theory of the uniform approximation of functions. This argument has a long and interesting history, from the pioneering works of Lagrange and Legendre, followed by the contributions of Chebyshev and Markov, continuing through famous results by Lebesgue; the results obtained by Bernstein and Vitali provide sharp and interesting insights into the properties of polynomials, and are of interest in the accuracy of the approximation of functions. Borel provided a final description to the approximation results due to Chebyshev. Erdös gave a strong improvement to Bernstein’s contribution in the rates of approximation. The reader will find those elements in the core of the present volume and in the Appendices.

Most optimal designs do not result as analytic solutions for approximation problems. Algorithmic solutions have been developed over the years: the optimal designs are obtained through algorithms which were developed in the field of the theory of the uniform approximation of functions, and are nowadays, important tools in numerical analysis. Henceforth, those algorithms have also been studied by statisticians; such is the case for the Remez algorithm, and to its extension by de Boor and Rice to constrained cases. This volume presents these tools in the statistical context.

The choice of the statistical context is restricted to the regular one, in the sense that all random variables which describe the variability of the inputs are supposed to be independent, essentially with the same distribution, with finite variance, hence allowing the least mean square paradigm in the field of linear models. This is the basic framework. The companion volumes consider nonlinear models, heteroscedastic models, models with dependence in the errors, etc.

This book results from our teaching, both in the University of Padova and in University Pierre and Marie Curie (Sorbonne University) in Paris. This corresponds to a one semester course in statistics or in applied mathematics.

The authors express their gratitude to their families and friends, who provided inestimable support during the completion of this work.

Giorgio CELANT, Padova

Michel BRONIATOWSKI, Paris

February 2016

Introduction

I.1. The scope of this book

This book is devoted to the construction of optimal designs for the estimation of a regression function, either in interpolation or in extrapolation. This deserves some definition.

We assume the following model to hold. A random variable Y(x) is observed whose expectation depends on some environment x, through a function f,

The random measurement Y (x) differs from its expectation by an additive random error term, say ε, with zero mean. We assume that the distribution of ε may depend on the environment x and that ε has a finite variance; in this book, this variance is assumed to be constant with respect to x. In our context, little is known by the experimenter: the function f is unknown, as is the distribution of the random error ε. However, in this volume the function will be assumed to belong to a known class of functions, so that no misspecification issues can occur. This choice is, therefore, not a matter of convenience, but a basic assumption.