Interpolation and Extrapolation Optimal Designs 2 Finite Dimensional General Models 1st Edition by Giorgio Celant, Michel Broniatowski 1786300540 9781786300546

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ISBN 10: 1786300540

ISBN 13: 9781786300546

Author: Giorgio Celant, Michel Broniatowski

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. Non-linear 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 self-consistent.

Table of contents:

1 Approximation of Continuous Functions in Normed Spaces

1.1. Introduction

1.2. Some remarks on the meaning of the word “simple”. Choosing the approximation

1.3. The choice of the norm in order to specify the error

1.4. Optimality with respect to a norm

1.5. Characterizing the optimal solution

2 Chebyshev Systems

2.1. Introduction

2.2. From the classical polynomials to the generalized ones

2.3. Properties of a Chebyshev system

3 Uniform Approximations in a Normed Space

3.1. Introduction

3.2. Characterization of the best uniform approximation in a normed space

4 Calculation of the Best Uniform Approximation in a Chebyshev System

4.1. Some preliminary results

4.2. Functional continuity of the approximation scheme

4.3. Property of the uniform approximation on a finite collection of points in [a, b]

4.4. Algorithm of de la Vallée Poussin

4.5. Algorithm of Remez

5 Optimal Extrapolation Design for the Chebyshev Regression

5.1. Introduction

5.2. The model and Gauss-Markov estimator

5.3. An expression of the extrapolated value through an orthogonalization procedure

5.4. The Gauss-Markov estimator of the extrapolated value

5.5. The Optimal extrapolation design for the Chebyshev regression

6 Optimal Design for Linear Forms of the Parameters in a Chebyshev Regression

6.1. Outlook and notations

6.2. Matrix of moments

6.3. Estimable forms

6.4. Matrix of moments and Gauss-Markov estimators of a linear form

6.5. Geometric interpretation of estimability: Elfving set

6.6. Elfving theorem

6.7. An intuitive approach to the Elfving theorem

6.8. Extension of Hoel–Levine result: optimal design for a linear c-form

7 Special Topics and Extensions

7.1. Introduction

7.2. The Gauss–Markov theorem in various contexts

7.3. Criterions for optimal designs

7.4. G–optimal interpolation and extrapolation designs for the Chebyshev regression

7.5. Some questions pertaining to the model

7.6. Hypotheses pertaining to the regressor

7.7. A few questions pertaining to the support of the optimal design for extrapolation

7.8. The proofs of some technical results

8 Multivariate Models and Algorithms

8.1. Introduction

8.2. Multivariate models

8.3. Optimality criterions and some optimal designs

8.4. Algorithms

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Tags: Giorgio Celant, Michel Broniatowski, Interpolation, Optimal