Henry P. Wynn

Screening, predicting, and computer experiments

Many scientific phenomena are now investigated by complex computer models or codes. Given the input values, the code produces one or more outputs via a complex mathematical model. Often the code is expensive to run, and it may be necessary to build a computationally cheaper predictor to enable, for example, optimization of the inputs. If there are many input factors, an initial step in building a predictor is identifying (screening) the active factors. We model the output of the computer code as the realization of a stochastic process. This model has a number of advantages. First, it provides a statistical basis, via the likelihood, for a stepwise algorithm to determine the important factors. Second, it is very flexible, allowing nonlinear and interaction effects to emerge without explicitly modeling such effects. Third, the same data are used for screening and building the predictor, so expensive runs are efficiently used. We illustrate the methodology with two examples, both having 20 input variables. I...

Algebraic Statistics : Computational Commutative Algebra in Statistics

INTRODUCTION History and Motivation Overview Computer Algebra Summary ALGEBRAIC MODELS Models Polynomials and Polynomial Ideals Term-Orderings Division Algorithm All Ideals Are Finitely Generated Varieties and Equations Grobner Bases Properties of Grobner Basis Elimination Theory Polynomial Functions and Quotients by Ideals Hilbert Function Further Topics THE DIRECT THEORY Designs and Design Ideals Computing the Grobner basis of a design Operations with Designs Examples Span of a Design Models and Identifiability Quotients Examples The Fan of an Experimental Design Subsets and Sequential Algorithms Regression Analysis Other Topics TWO-LEVEL DESIGNS. APPLICATION IN LOGIC AND RELIABILITY The binary case: Boolean Representations Reliability: Coherent Systems are Minimal Fan Designs Two Level Factorial Design: Contrasts and Orthogonality PROBABILITY AND STATISTICS Random Variables on a Finite Support Moments Probability Algebraic Representation of Exponentials Generating Functions Generating Functions and Exponential Models Examples and Further Applications Statistical Modelling Likelihoods and Sufficient Statistics A Ring of Random Variables Score Function and Information

Maximum entropy sampling and optimal Bayesian experimental design

When Shannon entropy is used as a criterion in the optimal design of experiments, advantage can be taken of the classical identity representing the joint entropy of parameters and observations as the sum of the marginal entropy of the observations and the preposterior conditional entropy of the parameters. Following previous work in which this idea was used in spatial sampling, the method is applied to standard parameterized Bayesian optimal experimental design. Under suitable conditions, which include non-linear as well as linear regression models, it is shown in a few steps that maximizing the marginal entropy of the sample is equivalent to minimizing the preposterior entropy, the usual Bayesian criterion, thus avoiding the use of conditional distributions. It is shown using this marginal formulation that under normality assumptions every standard model which has a two-point prior distribution on the parameters gives an optimal design supported on a single point. Other results include a new asymptotic formula which applies as the error variance is large and bounds on support size.

Quality through design : experimental design, off-line quality control and Taguchi's contributions

Introduction. 1: Fundamentals of data analysis. 2: Designing experiments. 3: Further design and analysis techniques. 4: Response surface methods and designs. 5: Off-line quality control principles. 6: Simulation and tolerance design. Appendices

Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences.

In this paper methods from differential algebra are used to study the structural identifiability of biological and pharmacokinetics models expressed in state-space form and with a structure given by rational functions. The focus is on the examples presented and on the application of efficient, automatic methods to test for structural identifiability for various input-output experiments. Differential algebra methods are coupled with Gröbner bases, Lie derivatives and the Taylor series expansion in order to obtain efficient algorithms. In particular, an upper bound on the number of derivatives needed for the Taylor series approach for a structural identifiability analysis of rational function models is given.

Algebraic and Geometric Methods in Statistics

This up-to-date account of algebraic statistics and information geometry explores the emerging connections between the two disciplines, demonstrating how they can be used in design of experiments and how they benefit our understanding of statistical models, in particular, exponential models. This book presents a new way of approaching classical statistical problems and raises scientific questions that would never have been considered without the interaction of these two disciplines. Beginning with a brief introduction to each area, using simple illustrative examples, the book then proceeds with a collection of reviews and some new results written by leading researchers in their respective fields. Part III dwells in both classical and quantum information geometry, containing surveys of key results and new material. Finally, Part IV provides examples of the interplay between algebraic statistics and information geometry. Computer code and proofs are also available online, where key examples are developed in further detail.

Achieving Robust Design from Computer Simulations

Abstract Computer simulations are widely used during product development. In particular, computer experiments are often conducted in order to optimize both product and process performance while respecting constraints that may be imposed. Several methods for achieving robust design in this context are described and compared with the aid of a simple example problem. The methods presented compare classical as well as modern approaches and introduce the idea of a ‘stochastic response’ to aid the search for robust solutions. Emphasis is placed on the efficiency of each method with respect to computational cost and the ability to formulate objectives that encapsulate the notion of robustness.

Nonlinear Matroid Optimization and Experimental Design

We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multicriteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.

Assessing Nonsuperiority, Noninferiority, or Equivalence When Comparing Two Regression Models Over a Restricted Covariate Region

In many scientific problems the purpose of the comparison of two regression models, which describe the relationship between a same response variable and several same covariates for two different groups, is to demonstrate that one model is no higher than the other by a negligible amount, or to demonstrate that the models have only negligible differences and so they can be regarded as describing practically the same relationship between the response variable and the covariates. In this article, methods based on one-sided pointwise confidence bands are proposed for assessing the nonsuperiority of one model to the other and for assessing the equivalence of two regression models. Examples from QT/QTc study and from drug stability study are used to illustrate the methods.

mODa 7 — Advances in Model-Oriented Design and Analysis

A Masked Spectral Bound for Maximum-Entropy Sampling.- Some Bayesian Optimum Designs for Response Transformation in Nonlinear Models with Nonconstant Variance.- Extensions of the Ehrenfest Urn Designs for Comparing Two Treatments.- Nonparametric Testing for Main Effects on Inequivalent Designs.- Maximun Optimal Designs for a Compartmental Model.- Optimal Adaptive Designs in Phase III Clinical Trials for Continuous Responses with Covariates.- Optimal Designs for Regression Models with Forced Measurements at Baseline.- Small Size Designs in Nonlinear Models Computed by Stochastic Optimization.- Asymptotic Properties of Biased Coin Designs for Treatment Allocation.- Lower Bounds on Efficiency Ratios Based on Optimal Designs.- On a Functional Approach to Locally Optimal Designs.- Optimal Design Criteria Based on Tolerance Regions.- Simultaneous Choice of Design and Estimator in Nonlinear Regression with Parameterized Variance.- Minimum Entropy Estimation in Semi-Parametric Models: a Candidate for Adaptive Estimation?- Optimal Designs for Contingent Response Models.- Bayesian D-Optimal Designs for Generalized Linear Models with a Varying Dispersion Parameter.- L-optimum Designs in Multi-factor Models with Heteroscedastic Errors.- Multiplicative Algorithms for Constructing Optimizing Distributions: Further Developments.- Locally Optimal Designs for an Extension of the Michaelis-Menten Model.- Asymptotic Properties of Urn Designs for Three-arm Clinical Trials.- T-Optimum Designs for Multiresponse Dynamic Heteroscedastic Models.- Error Transmission in Mixture Experiments.- Maximum Entropy Sampling and General Equivalence Theory.- Towards Identification of Patient Responses to Anesthesia Infusion in Real Time.