Leslie M. Moore

Minimax and maximin distance designs

Abstract Beginning with an arbitrary set and a distance defined on it, we develop the notions of minimax and maximin distance sets (designs). These are intended for use in the selection-of-sites problem when the underlying surface is modeled by a prior distribution and observations are made without error. It is shown that such designs have quite general asymptotically optimum (and dual) characteristics under what are termed the G- and D-criteria. There are many examples given, dealing espeacially with the unit square and with k factors at two levels.

Using Orthogonal Arrays in the Sensitivity Analysis of Computer Models

We consider a class of input sampling plans, called permuted column sampling plans, that are popular in sensitivity analysis of computer models. Permuted column plans, including replicated Latin hypercube sampling, support estimation of first-order sensitivity coefficients, but these estimates are biased when the usual practice of random column permutation is used to construct the sampling arrays. Deterministic column permutations may be used to eliminate this estimation bias. We prove that any permuted column sampling plan that eliminates estimation bias, using the smallest possible number of runs in each array and containing the largest possible number of arrays, can be characterized by an orthogonal array of strength 2. We derive approximate standard errors of the first-order sensitivityindices for this sampling plan. We give two examples demonstrating the sampling plan, behavior of the estimates, and standard errors, along with comparative results based on other approaches.

Batch sequential design to achieve predictive maturity with calibrated computer models

Sequential experiment design strategies have been proposed for efficiently augmenting initial designs to solve many problems of interest to computer experimenters, including optimization, contour and threshold estimation, and global prediction. We focus on batch sequential design strategies for achieving maturity in global prediction of discrepancy inferred from computer model calibration. Predictive maturity focuses on adding field experiments to efficiently improve discrepancy inference. Several design criteria are extended to allow batch augmentation, including integrated and maximum mean square error, maximum entropy, and two expected improvement criteria. In addition, batch versions of maximin distance and weighted distance criteria are developed. Two batch optimization algorithms are considered: modified Fedorov exchange and a binning methodology motivated by optimizing augmented fractional factorial skeleton designs.

Using Genetic Algorithms to Design Experiments: A Review

Genetic algorithms (GAs) have been used in many disciplines to optimize solutions for a broad range of problems. In the last 20 years, the statistical literature has seen an increase in the use and study of this optimization algorithm for generating optimal designs in a diverse set of experimental settings. These efforts are due in part to an interest in implementing a novel methodology as well as the hope that careful application of elements of the GA framework to the unique aspects of a designed experiment problem might lead to an efficient means of finding improved or optimal designs. In this paper, we explore the merits of using this approach, some of the aspects of design that make it a unique application relative to other optimization scenarios, and discuss elements which should be considered for an effective implementation. We conclude that the current GA implementations can, but do not always, provide a competitive methodology to produce substantial gains over standard optimal design strategies. We consider both the probability of finding a globally optimal design as well as the computational efficiency of this approach. Copyright

Bayesian Prediction Intervals and Their Relationship to Tolerance Intervals

We consider Bayesian prediction intervals that contain a proportion of a finite number of observations with a specified probability. Such intervals arise in numerous applied contexts and are closely related to tolerance intervals. Several examples are provided to illustrate this methodology, and simulation studies are used to demonstrate potential pitfalls of using tolerance intervals when prediction intervals are required.

Minimax distance designs in two-level factorial experiments

Abstract A minimax distance criterion was set forth in Johnson et al. (1990) for the purpose of selection among experimental designs. Unlike the usual design criteria such as D-, E- or G-optimality, minimax distance presumes no underlying model and, in turn, is not concerned with the rank of an associated design matrix. In situations where either the model is unknown or it is not possible to run enough experiments to estimate all parameters of an assumed model, this criterion is considered as a viable tool in the task of design selection. This paper deals with the design space associated with n factors, each of which can take two levels. We exhibit minimax distance designs that compare favorably with designs chosen to do well on classical grounds.

Approximate one-sided tolerance bounds on the number of failures using Poisson regression

Safety studies of a nuclear reactor often center their interest on the probable number, of component failures in a time span of duration To . Using the asymptotic normality of the estimator from Poisson regression, we develop approximate upper tolerance bounds for the distribution of the number of failures. Tables are given for easy computation of such bounds when the bound itself is less than 50. An example consisting of 90 failure records for nuclearreactor valve types provides illustration of the tolerance bound computations and dataanalytic techniques for validating a Poisson regression model.

Combined array experiment design

Abstract Experiment plans formed by combining two or more designs, such as orthogonal arrays primarily with 2- and 3-level factors, creating multi-level arrays with subsets of different strength are proposed for computer experiments to conduct sensitivity analysis. Specific illustrations are designs for 5-level factors with fewer runs than generally required for 5-level orthogonal arrays of strength 2 or more. At least 5 levels for each input are desired to allow for runs at a nominal value, 2-values either side of nominal but within a normal, anticipated range, and two, more extreme values either side of nominal. This number of levels allows for a broader range of input combinations to test the input combinations where a simulation code operates. Five-level factors also allow the possibility of up to fourth-order polynomial models for fitting simulation results, at least in one dimension. By having subsets of runs with more than strength 2, interaction effects may also be considered. The resulting designs have a “checker-board” pattern in lower-dimensional projections, in contrast to grid projection that occurs with orthogonal arrays. Space-filling properties are also considered as a basis for experiment design assessment.

Statistical methods for sensitivity and performance analysis in computer experiments

We describe statistical methods for sensitivity and performance analysis of complex computer simulation experiments. Graphical methods, such as trellis plots, are suggested for exploratory analysis of individual or aggregate performance metrics, conditional on different experiment inputs. More formal statistical methods, such as analysis of variance based methods and regression tree analysis, are used to determine variables having substantive influence on the experimental results and to investigate the structure of the underlying relationship between inputs and outputs. The methods are discussed in relation to a supply chain model of the textile manufacturing process having many possible input and output variables of interest and for a computer model used to describe the flow of material in an ecosystem.

Constructing rigorous and broad biosurveillance networks for detecting emerging zoonotic outbreaks.

Determining optimal surveillance networks for an emerging pathogen is difficult since it is not known beforehand what the characteristics of a pathogen will be or where it will emerge. The resources for surveillance of infectious diseases in animals and wildlife are often limited and mathematical modeling can play a supporting role in examining a wide range of scenarios of pathogen spread. We demonstrate how a hierarchy of mathematical and statistical tools can be used in surveillance planning help guide successful surveillance and mitigation policies for a wide range of zoonotic pathogens. The model forecasts can help clarify the complexities of potential scenarios, and optimize biosurveillance programs for rapidly detecting infectious diseases. Using the highly pathogenic zoonotic H5N1 avian influenza 2006-2007 epidemic in Nigeria as an example, we determined the risk for infection for localized areas in an outbreak and designed biosurveillance stations that are effective for different pathogen strains and a range of possible outbreak locations. We created a general multi-scale, multi-host stochastic SEIR epidemiological network model, with both short and long-range movement, to simulate the spread of an infectious disease through Nigerian human, poultry, backyard duck, and wild bird populations. We chose parameter ranges specific to avian influenza (but not to a particular strain) and used a Latin hypercube sample experimental design to investigate epidemic predictions in a thousand simulations. We ranked the risk of local regions by the number of times they became infected in the ensemble of simulations. These spatial statistics were then complied into a potential risk map of infection. Finally, we validated the results with a known outbreak, using spatial analysis of all the simulation runs to show the progression matched closely with the observed location of the farms infected in the 2006-2007 epidemic.