Optimal Design and Related Areas in Optimization and Statistics (eBook)

Luc Pronzato, Anatoly Zhigljavsky (Herausgeber)

eBook Download: PDF

2010 | 2009
XV, 224 Seiten
Springer New York (Verlag)
978-0-387-79936-0 (ISBN)

Infos

The present volume is a collective monograph devoted to applications of the optimal design theory in optimization and statistics. The chapters re?ect the topics discussed at the workshop 'W-Optimum Design and Related Statistical Issues' that took place in Juan-les-Pins, France, in May 2005. The title of the workshop was chosen as a light-hearted celebration of the work of Henry Wynn. It was supported by the Laboratoire I3S (CNRS/Universit´ e de Nice, Sophia Antipolis), to which Henry is a frequent visitor. The topics covered partly re?ect the wide spectrum of Henry's research - terests. Algorithms for constructing optimal designs are discussed in Chap. 1, where Henry's contribution to the ?eld is acknowledged. Steepest-ascent - gorithms used to construct optimal designs are very much related to general gradientalgorithmsforconvexoptimization. Inthelasttenyears,asigni?cant part of Henry's research was devoted to the study of the asymptotic prop- ties of such algorithms. This topic is covered by Chaps. 2 and 3. The work by Alessandra Giovagnoli concentrates on the use of majorization and stoch- tic ordering, and Chap. 4 is a hopeful renewal of their collaboration. One of Henry's major recent interests is what is now called algebraic statistics, the application of computational commutative algebra to statistics, and he was partly responsible for introducing the experimental design sub-area, reviewed in Chap. 5. One other sub-area is the application to Bayesian networks and Chap. 6 covers this, with Chap. 7 being strongly related.

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The present volume is a collective monograph devoted to applications of the optimal design theory in optimization and statistics. The chapters re?ect the topics discussed at the workshop "e;W-Optimum Design and Related Statistical Issues"e; that took place in Juan-les-Pins, France, in May 2005. The title of the workshop was chosen as a light-hearted celebration of the work of Henry Wynn. It was supported by the Laboratoire I3S (CNRS/Universit' e de Nice, Sophia Antipolis), to which Henry is a frequent visitor. The topics covered partly re?ect the wide spectrum of Henry's research - terests. Algorithms for constructing optimal designs are discussed in Chap. 1, where Henry's contribution to the ?eld is acknowledged. Steepest-ascent - gorithms used to construct optimal designs are very much related to general gradientalgorithmsforconvexoptimization. Inthelasttenyears,asigni?cant part of Henry's research was devoted to the study of the asymptotic prop- ties of such algorithms. This topic is covered by Chaps. 2 and 3. The work by Alessandra Giovagnoli concentrates on the use of majorization and stoch- tic ordering, and Chap. 4 is a hopeful renewal of their collaboration. One of Henry's major recent interests is what is now called algebraic statistics, the application of computational commutative algebra to statistics, and he was partly responsible for introducing the experimental design sub-area, reviewed in Chap. 5. One other sub-area is the application to Bayesian networks and Chap. 6 covers this, with Chap. 7 being strongly related.

Inhalt

Preface 10
List of Contributors 11
Contents 13
1 W-Iterations and Ripples Therefrom 16
1.1 Introduction 16
1.2 Optimal Design and an Optimization Problem 16
1.3 Derivatives and Optimality Conditions 17
1.4 Algorithms 18
1.5 A Steepest-Ascent Algorithm 24
1.6 Simultaneous Approach to Optimal Weight and Support Point Determination 24
References 26
2 Studying Convergence of Gradient Algorithms Via Optimal Experimental Design Theory 28
2.1 Introduction 28
2.2 Renormalized Version of Gradient Algorithms 29
2.3 A Multiplicative Algorithm for Optimal Design 31
2.4 Constructing Optimality Criteria which Correspond to a Given Gradient Algorithm 33
2.5 Optimum Design Gives the Worst Rate of Convergence 34
2.6 Some Special Cases 35
2.7 The Steepest-Descent Algorithm with Relaxation 38
2.8 Square-Root Algorithm 45
2.9 A-Optimality 47
2.10 a-Root Algorithm and Comparisons 48
References 51
3 A Dynamical-System Analysis of the Optimum s- Gradient Algorithm 54
3.1 Introduction 54
3.2 The Optimum s-Gradient Algorithm for the Minimization of a Quadratic Function 55
3.3 Asymptotic Behaviour of the Optimum s-Gradient Algorithm in Rd 67
3.4 The Optimum 2-Gradient Algorithm in Rd 70
3.5 Switching Algorithms 81
References 94
4 Bivariate Dependence Orderings for Unordered Categorical Variables 96
4.1 Introduction 96
4.2 Dependence Orderings for Two Nominal Variables 98
4.3 Inter-Raters Agreement for Categorical Classifications 105
4.4 Conclusions and Further Research 109
References 110
5 Methods in Algebraic Statistics for the Design of Experiments 112
5.1 Introduction 112
5.2 Background 113
5.3 Generalized Confounding and Polynomial Algebra 117
5.4 Models and Monomials 128
5.5 Indicator Function for Complex Coded Designs 133
5.6 Indicator Function vs. Gröbner Basis 136
5.7 Mixture Designs 141
5.8 Conclusions 145
References 146
6 The Geometry of Causal Probability Trees that are Algebraically Constrained 148
6.1 The Algebra of Probability Trees 148
6.2 Manifest Probabilities and Solution Spaces 152
6.3 Expressing Causal Effects Through Algebra 154
6.4 From Models to Causal ACTs to Analysis 157
6.5 Equivalent Causal ACTs 162
6.6 Conclusions 166
Appendix: Maple Code 167
References 169
7 Bayes Nets of Time Series: Stochastic Realizations and Projections 170
7.1 Bayes Nets and Projections 170
7.2 Time Series: Stochastic Realization and Conditional Independence 175
7.3 LCO/LCI Time Series 178
7.4 TDAG as Generalized Time 180
References 181
8 Asymptotic Normality of Nonlinear Least Squares under Singular Experimental Designs 182
8.1 Introduction 182
8.2 The Convergence of the Design Sequence to a Design Measure 186
8.3 Consistency of Estimators 191
8.4 On the Geometry of the ModelUnder the Design Measure . 194
8.5 The Regular Asymptotic Normality of h(ˆ.N) 197
8.6 Estimation of a Multidimensional Function H(.) 201
Appendix. Proofs of Lemmas 1–3 202
References 205
9 Robust Estimators in Non-linear Regression Models with Long- Range Dependence 208
9.1 Introduction 208
9.2 Main Results 211
9.3 Auxiliary Assertions 222
9.4 Proofs 230
References 232
Index 238

Produkt-Details

• Erscheint lt. Verlag: 25.7.2010
• Reihe/Serie: Springer Optimization and Its Applications
• Reihe/Serie: Springer Optimization and Its Applications
• Zusatzinfo: XV, 224 p. 23 illus.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Programmiersprachen / -werkzeuge
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Planung / Organisation
• Schlagworte: algebraic statistics • Algorithm analysis and problem complexity • algorithms • Bayesian networks • gradient-type algorithms • linear optimization • optimal design theory • Optimization • SOIA
• ISBN-10: 0-387-79936-2 / 0387799362
• ISBN-13: 978-0-387-79936-0 / 9780387799360