Optimal Design and Related Areas in Optimization and Statistics

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Luc Pronzato, Anatoly Zhigljavsky
Springer Science & Business Media, Jul 25, 2010 - Mathematics - 224 pages
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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 “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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Contents

WIterations and Ripples Therefrom
1
13 Derivatives and Optimality Conditions
2
14 Algorithms
3
15 A SteepestAscent Algorithm
9
References
11
Studying Convergence of Gradient Algorithms Via Optimal Experimental Design Theory
13
22 Renormalized Version of Gradient Algorithms
14
23 A Multiplicative Algorithm for Optimal Design
16
55 Indicator Function for Complex Coded Designs
118
56 Indicator Function vs Grobner Basis
121
57 Mixture Designs
126
58 Conclusions
130
References
131
The Geometry of Causal Probability Trees that are Algebraically Constrained
133
62 Manifest Probilities and Solution Spaces
137
63 Expressing Causal Effects Through Algebra
139

which Correspond to a Given Gradient Algorithm
18
25 Optimum Design Gives the Worst Rate of Convergence
19
26 Some Special Cases
20
27 The SteepestDescent Algorithm with Relaxation
23
28 SquareRoot Algorithm
30
29 AOptimality
32
210 αRoot Algorithm and Comparisons
33
References
36
A DynamicalSystem Analysis of the Optimum sGradient Algorithm
39
for the Minimization of a Quadratic Function
40
33 Asymptotic Behaviour of the Optimum sGradient Algorithm in Rd
52
34 The Optimum 2Gradient Algorithm in Rd
55
35 Switching Algorithms
66
References
79
Bivariate Dependence Orderings for Unordered Categorical Variables
81
42 Dependence Orderings for Two Nominal Variables
83
43 InterRaters Agreement for Categorical Classifications
90
44 Conclusions and Further Research
94
References
95
Methods in Algebraic Statistics for the Design of Experiments
97
52 Background
98
53 Generalized Confounding and Polynomial Algebra
102
54 Models and Monomials
113
64 From Models to Causal ACTs to Analysis
142
65 Equivalent Causal ACTs
147
66 Conclusions
151
References
154
Bayes Nets of Time Series Stochastic Realizations and Projections
155
Stochastic Realization and Conditional Independence
160
73 LCOLCI Time Series
163
74 TDAG as Generalized Time
165
References
166
Asymptotic Normality of Nonlinear Least Squares under Singular Experimental Designs
167
82 The Convergence of the Design Sequence to a Design Measure
171
83 Consistency of Estimators
176
84 On the Geometry of the Model Under the Design Measure
179
85 The Regular Asymptotic Normality of h0N
182
86 Estimation of a Multidimensional Function H0
186
References
190
Robust Estimators in Nonlinear Regression Models with LongRange Dependence
192
92 Main Results
196
93 Auxiliary Assertions
207
94 Proofs
215
References
217
Index
222
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