Optimal Experimental Design: A Concise Introduction for Researchers (Lecture Notes in Statistics)

✍ Scribed by Jesús López-Fidalgo

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 228
• Category: Library

✦ Synopsis

This textbook provides a concise introduction to optimal experimental design and efficiently prepares the reader for research in the area. It presents the common concepts and techniques for linear and nonlinear models as well as Bayesian optimal designs. The last two chapters are devoted to particular themes of interest, including recent developments and hot topics in optimal experimental design, and real-world applications. Numerous examples and exercises are included, some of them with solutions or hints, as well as references to the existing software for computing designs. The book is primarily intended for graduate students and young researchers in statistics and applied mathematics who are new to the field of optimal experimental design. Given the applications and the way concepts and results are introduced, parts of the text will also appeal to engineers and other applied researchers.

✦ Table of Contents

Preface
Acknowledgments
Contents
List of Symbols
1 Motivating Introduction
1.1 What Is a Statistical Model?
1.2 Importance of Designing an Experiment
Basic Principles for an Experimental Design
1.3 The Statistical Procedure from the Beginning to the End
1.4 Criticisms and Rejoinders
1.4.1 Model Dependence
1.4.2 Information Matrix for Nonlinear Models
1.4.3 Criterion Selection
1.4.4 Controversy of Exact Versus Approximate Designs
1.4.5 Frequently, Optimal Designs Demand ExtremeConditions
1.4.6 Difficult Computation
1.4.7 Scale Problem
2 Optimal Design Theory for Linear Models
2.1 The Linear Model
2.2 From Exact to Approximate Designs
2.3 The Information Matrix
2.4 Optimality Criteria
2.4.1 General Properties for Global Criteria
2.4.2 Efficiency
2.4.3 D-optimality
2.4.4 G-optimality
2.4.5 A-optimality
2.4.6 L-optimality
2.4.7 E-optimality
2.4.8 MV-optimality
2.4.9 I-optimality
2.4.10 p-optimality
2.4.11 Ds-optimality
2.4.12 c-optimality
2.5 Elfving Graphical Procedure for c-Optimality
2.5.1 Elfving's Procedure in Practice
2.5.2 Computing Tangential Points
2.5.3 Procedure for Three or More Parameters
2.6 The Equivalence Theorem
2.6.1 Equivalence Theorem for Differentiable Criteria
2.6.2 Efficiency Bounds
D-optimality
Ds-optimality
L-optimality
E-optimality
D-optimal Design
A-optimal Design
E-optimal Design
c-Optimal Design
2.7 Algorithms
2.7.1 Fedorov–Wynn Algorithm
2.7.2 Multiplicative Algorithm for D- and G-optimality for a Finite Design Space
2.7.3 Recursive Computation of the Determinant and the Inverse of a Matrix
2.8 Summary of the Chapter
2.9 Exercises
3 Designing for Nonlinear Models
3.1 What Is a Statistical Model? (Revisited)
3.2 Uncorrelated Observations
3.3 Fisher Information
3.4 Equivalence Theorem for Nonlinear Models
3.5 Natural Exponential Family
3.6 General Exponential Family
3.7 Sensitivity Analysis
3.8 Illustrative Example
3.9 Polynomial Models and Chebyshev systems
3.10 Correlated Observations and Spatiotemporal Models
3.10.1 Outline
3.10.2 Typical Models for the Covariance Structure
3.10.3 Coordinate Descent Algorithm
3.10.4 Examples
3.11 Exercises
4 Bayesian Optimal Designs
4.1 Introduction
4.2 Linear Models
4.2.1 Criteria Derived from Particular Utility Functions
4.2.2 Unknown Variance
4.3 Nonlinear Models
4.3.1 Shannon Information and Bayesian D-optimality
4.3.2 Bayesian c-Optimality
4.3.3 Bayesian L-optimality
4.3.4 Extended Bayesian Criteria
4.4 Equivalence Theorem
4.4.1 D-optimality
4.4.2 A-optimality
4.5 Examples
4.5.1 Simple Linear Regression
4.5.2 A Nonlinear Model
4.6 Exercises
5 Hot Topics
5.1 Introduction
5.2 Computer Experiments
5.2.1 Design Criteria
5.2.2 Including the Computation Time in the Model
5.3 Active Learning
5.4 Personalized Medicine
5.4.1 Static Planning of the Treatment
5.4.2 Dynamic Planning of the Treatment
5.5 Model Selection: Discrimination Between Rival Models
5.5.1 Bayesian Paradigm
5.5.2 Correlated Observations
5.5.3 Computing Optimal Designs
5.6 Meta-heuristic Algorithm
5.6.1 Particle Swarm Optimization (PSO)
5.6.2 Genetic Algorithm
6 Real Case Examples
6.1 Introduction
6.2 Mixture of Distributions: Engine Emissions
Example with a Mixture of Two Normal Distributions
6.3 Correlated Observations and Spatial Models: Radiation Retention in the Human Body
6.4 Marginally Restricted Designs: Uranium Densification
6.5 Marginally Restricted with Correlated Observations: Irish Wind Data
6.6 Conditionally Restricted Designs: Prediction in Thoracic Surgery with Logistic Regression
6.7 Potential Censoring in the Design: The Complex Reality of Using an Optimal Design in Thoracic Surgery
6.8 Censoring in the Response
Appendices
A Some Mathematical Concepts and Properties
B Linear Models
B.1 Useful Results for Nonlinear Models
B.2 Confidence Regions
C Analysis of the Variance and Classical Experimental Design
D Response Surface Analysis Through an Illustrative Example
E Regression and Correlation
F Probability Measure Theory
F.1 Measure
F.2 Lebesgue Measure and Integration
F.3 Integration with Respect to a Measure
F.4 Probability Measure
F.5 Convergences of Random Variables
G Convex Theory
Bibliography
Index