Statistical Algorithms for Optimal Experimental Design with Correlated Observations

This research is in three parts with different although related objectives. The first part developed an efficient, modified simulated annealing algorithm to solve the D-optimal (determinant maximization) design problem for 2-way polynomial regression with correlated observations. Much of the previous work in D-optimal design for regression models with correlated errors focused on polynomial models with a single predictor variable, in large part because of the intractability of an analytic solution. In this research, we present an improved simulated annealing algorithm, providing practical approaches to specifications of the annealing cooling parameters, thresholds and search neighborhoods for the perturbation scheme, which finds approximate D-optimal designs for 2-way polynomial regression for a variety of specific correlation structures with a given correlation coefficient. Results in each correlated-errors case are compared with the best design selected from the class of designs that are known to be D-optimal in the uncorrelated case: annealing results had generally higher D-efficiency than the best comparison design, especially when the correlation parameter was well away from 0. The second research objective, using Balanced Incomplete Block Designs (BIBDs), wasto construct weakly universal optimal block designs for the nearest neighbor correlation structure and multiple block sizes, for the hub correlation structure with any block size, and for circulant correlation with odd block size. We also constructed approximately weakly universal optimal block designs for the block-structured correlation. Lastly, we developed an improved Particle Swarm Optimization(PSO) algorithm with time varying parameters, and solved D-optimal design for linear regression with it. Then based on that improved algorithm, we combined the non-linear regression problem and decision making, and developed a nested PSO algorithm that finds (nearly) optimal experimental designs with each of the pessimistic criterion, index of optimism criterion, and regret criterion for the Michaelis-Menten model and logistic regression model

(Dt,C) Optimal run orders.

Cost considerations have rarely been taken into account in optimum design theory. A few authors consider measurement costs, i.e. the costs associated with a particular factor level combination. A second cost approach results from the fact that it is often expensive to change factor levels from one observation to another. We refer to these costs as transition costs. In view of cost minimization, one should minimize the number of factor level changes. However, there is a substantial likelihood that there is some time order dependence in the results. Consequently, when considering both time order dependence and transition costs, an optimal ordering is not easy to find. There is precious little in the literature on how to select good time order sequences for arbitrary design problems and up to now, no thorough analysis of both costs is found in the literature. For arbitrary design problems, our proposed design algorithm incorporates cost considerations in optimum design construction and enables one to compute cost-efficient run orders that are optimally balanced for time trends. The results show that cost considerations in the construction of trend-resistant run orders entail considerable reductions in the total cost of an experiment and imply a large increase in the amount of information per unit cost.Optimal; Run orders;

A comparison of general-purpose optimization algorithms forfinding optimal approximate experimental designs

Several common general purpose optimization algorithms are compared for findingA- and D-optimal designs for different types of statistical models of varying complexity,including high dimensional models with five and more factors. The algorithms of interestinclude exact methods, such as the interior point method, the Nelder–Mead method, theactive set method, the sequential quadratic programming, and metaheuristic algorithms,such as particle swarm optimization, simulated annealing and genetic algorithms.Several simulations are performed, which provide general recommendations on theutility and performance of each method, including hybridized versions of metaheuristicalgorithms for finding optimal experimental designs. A key result is that general-purposeoptimization algorithms, both exact methods and metaheuristic algorithms, perform wellfor finding optimal approximate experimental designs

Trend-resistant and cost-efficient cross-over designs for mixed models.

A mixed model approach is used to construct optimal cross-over designs. In a cross-over experiment the same subject is tested at different points in time. Consider as an example an experiment to investigate the influence of physical attributes of the work environment such as luminance, ambient temperature and relative humidity on human performance of acceptance inspection in quality assurance. In a mixed model context, the subject effects are assumed to be independent and normally distributed. Besides the induction of correlated observations within the same inspector, the mixed model approach also enables one to specify the covariance structure of the inspection data. Here, several covariance structures are considered either depending on the time variable or not. Unfortunately, a serious drawback of the inspection experiment is that the results may be influenced by an unknown time trend because of inspector fatigue due to monotony of the inspection task. In other circumstances, time trend effects can be caused by learning effects of the test subjects in behavioural and life sciences, heating or aging of material in prototype experiments, etc. An algorithm is presented to construct cross-over designs that are optimally balanced for time trend effects. The costs for using the subjects and for altering the factor levels between consecutive observations can also be taken into account. A number of examples illustrate utility of the outlined design methodology.Optimal; Models; Model;

Optimal design for correlated processes with input-dependent noise

Optimal design for parameter estimation in Gaussian process regression models with input-dependent noise is examined. The motivation stems from the area of computer experiments, where computationally demanding simulators are approximated using Gaussian process emulators to act as statistical surrogates. In the case of stochastic simulators, which produce a random output for a given set of model inputs, repeated evaluations are useful, supporting the use of replicate observations in the experimental design. The findings are also applicable to the wider context of experimental design for Gaussian process regression and kriging. Designs are proposed with the aim of minimising the variance of the Gaussian process parameter estimates. A heteroscedastic Gaussian process model is presented which allows for an experimental design technique based on an extension of Fisher information to heteroscedastic models. It is empirically shown that the error of the approximation of the parameter variance by the inverse of the Fisher information is reduced as the number of replicated points is increased. Through a series of simulation experiments on both synthetic data and a systems biology stochastic simulator, optimal designs with replicate observations are shown to outperform space-filling designs both with and without replicate observations. Guidance is provided on best practice for optimal experimental design for stochastic response models

Statistical Testing of Optimality Conditions in Multiresponse Simulation-Based Optimization (Replaced by Discussion Paper 2007-45)

This paper derives a novel procedure for testing the Karush-Kuhn-Tucker (KKT) first-order optimality conditions in models with multiple random responses.Such models arise in simulation-based optimization with multivariate outputs.This paper focuses on expensive simulations, which have small sample sizes.The paper estimates the gradients (in the KKT conditions) through low-order polynomials, fitted locally.These polynomials are estimated using Ordinary Least Squares (OLS), which also enables estimation of the variability of the estimated gradients.Using these OLS results, the paper applies the bootstrap (resampling) method to test the KKT conditions.Furthermore, it applies the classic Student t test to check whether the simulation outputs are feasible, and whether any constraints are binding.The paper applies the new procedure to both a synthetic example and an inventory simulation; the empirical results are encouraging.stopping rule;metaheuristics;RSM;design of experiments