Interpolation and extrapolation optimal designs. 2, Finite dimensional general models / Giorgio Celant, Michel Broniatowski
- Author:
- Celant, Giorgio
- Additional Titles:
- Finite dimensional general models
- Published:
- London : ISTE, 2016.
- Physical Description:
- 1 online resource
- Additional Creators:
- Broniatowski, Michel
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- Contents:
- Machine generated contents note: ch. 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 -- 1.2.1.Splines -- 1.3.The choice of the norm in order to specify the error -- 1.4.Optimality with respect to a norm -- 1.4.1.Existence of an optimal solution -- 1.4.2.Uniqueness of the optimal solution -- 1.4.3.Examples -- 1.5.Characterizing the optimal solution -- 1.5.1.The Hilbertian case -- 1.5.2.The non-Hilbertian case -- 1.5.3.Optimization, Lp norms and robustness -- ch. 2 Chebyshev Systems -- 2.1.Introduction -- 2.2.From the classical polynomials to the generalized ones -- 2.2.1.Examples of Chebyshev systems -- 2.3.Properties of a Chebyshev system -- 2.3.1.Vector-type properties -- 2.3.2.Chebyshev systems and interpolation -- 2.3.3.Roots of the generalized polynomials -- ch. 3 Uniform Approximations in a Normed Space -- 3.1.Introduction -- 3.2.Characterization of the best uniform approximation in a normed space -- 3.2.1.The Haar--Kolmogorov theorem -- 3.2.2.The generalized Borel -- Chebyshev theorem -- 3.2.3.Oscillation properties of the best uniform approximation -- ch. 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 Vallee Poussin -- 4.5.Algorithm of Remez -- ch. 5 Optimal Extrapolation Design for the Chebyshev Regression -- 5.1.Introduction -- 5.2.The model and Gauss-Markov estimator -- 5.2.1.Description of the dataset -- 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 -- 5.5.1.The support of the optimal design -- 5.5.2.The frequencies of the optimal design -- 5.5.3.Identification of the optimal design -- ch. 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 -- 6.4.1.Matrices of moments and estimable linear forms -- 6.4.2.An alternative form of the lower bound of the variance of the estimator of the c form -- 6.5.Geometric interpretation of estimability: Elfving set -- 6.5.1.Estimable forms and a convex subset of the regression range; the Elfving set -- 6.5.2.Geometry of the Elfving set -- 6.5.3.The relation between cylinders and the variance of the estimator of the c-form -- 6.5.4.Lower bound for the variance -- 6.5.5.The lower bound can be achieved -- 6.6.Elfving theorem -- 6.7.An intuitive approach to Elfving theorem -- 6.8.Extension of Hoel--Levine result: optimal design for a linear c-form -- ch. 7 Special Topics and Extensions -- 7.1.Introduction -- 7.2.The Gauss--Markov theorem in various contexts -- 7.2.1.The Gauss--Markov theorem for linear transformations of the parameter under i.i.d. errors -- 7.2.2.Gauss--Markov theorem for heteroscedastic models with correlation -- 7.2.3.The Gauss--Markov theorem and the Loewner order on quadratic forms -- 7.3.Criterions for optimal designs -- 7.3.1.Introduction -- 7.3.2.Some specific criterions -- 7.4.G--optimal interpolation and extrapolation designs for the Chebyshev regression -- 7.4.1.Criteria for optimality -- 7.4.2.Design with minimal uniform variance for a Chebyshev regression -- 7.5.Some questions pertaining to the model -- 7.5.1.Linear heteroscedastic models -- 7.5.2.Nonlinear models, estimators and optimal designs -- 7.6.Hypotheses pertaining to the regressor -- 7.6.1.Regressor in a linear space with unknown finite dimension -- 7.6.2.An extension to the case of analytic regressors -- 7.6.3.On the choice of the degree of the polynomial model -- 7.7.A Few Questions Pertaining to the Support of the Optimal Design for Extrapolation -- 7.7.1.Preliminary results and notation -- 7.7.2.Optimal designs whose support is the Chebyshev set of points -- 7.8.The proofs of some technical results -- 7.8.1.Proof of proposition 7.1 -- 7.8.2.Proof of theorem 7.17 -- ch. 8 Multivariate Models and Algorithms -- 8.1.Introduction -- 8.2.Multivariate models -- 8.2.1.Notation -- 8.2.2.Predictors and their variance -- 8.2.3.Some construction of multivariate models -- 8.3.Optimality criterions and some optimal designs -- 8.3.1.Criterions for optimality and characterization of the optimal design -- 8.3.2.D--optimality, direct sum and tensor product -- 8.4.Algorithms -- 8.4.1.General aspects -- 8.4.2.Specific algorithms.
- Subject(s):
- ISBN:
- 9781119422365 (electronic bk.)
1119422361 (electronic bk.)
9781119422327 (electronic bk.)
1119422329 (electronic bk.)
9781786300546
1786300540 - Bibliography Note:
- Includes bibliographical references and index.
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