Comparison of Simplex Designs for Quadratic Mixture Models

Designs for quadratic regression are considered when the possible values of the controlable variable are mixtures x = (x ₁, x ₂, …, x _{q + 1}) of nonnegative components x _i with Σ ^{q + 1} ₁ x _i = 1. The designs that are optimum with respect to the D-, A-, and E-optimality criteria are compared in their performance relative to these and other criteria. Computational routines for obtaining these designs are developed, and the geometry of optimum structures is discussed. Except when q = 2, the A-optimum design is supported by the vertices and midpoints of edges of the simplex, as is the case for the previously known D-optimum design. Although the E-optimum design requires more observation points, it is more robust in its efficiency, under variation of criterion: but all three designs perform reasonably well in this sense.     

Cost‐penalized estimation and prediction evaluation for split‐plot designs

Comparisons between different designs have traditionally focused on balancing the quality of estimation or prediction relative to the overall size of the design. For split‐plot designs with two levels of randomization, the total number of observations may not accurately summarize the true cost of the experiment, because different costs are likely associated with setting up the whole and subplot levels. In this paper, we present several flexible measures for design assessment based on D‐, G‐ and V‐optimality criteria that take into account potentially different cost structures for the split‐plot designs. The new measures are illustrated with two examples: a 2³ factorial experiment for first‐order models, where all possible designs are considered, and selective designs for a three‐factor second‐order model. Copyright © 2006 John Wiley & Sons, Ltd.     

Robust designs with high projection efficiency

Alphabetic optimality criteria, such as the D, A, and I criteria, require specifying a model to select optimal designs. They are not model‐free, and the designs obtained by them may not be robust. Recently, many extensions of the D and A criteria have been proposed for selecting robust designs with high estimation efficiency. However, approaches for finding robust designs with high prediction efficiency are rarely studied in the literature. In this paper, we propose a compound criterion and apply the coordinate‐exchange 2‐phase local search algorithm to generate robust designs with high estimation, high prediction, or balanced estimation and prediction efficiency for projective submodels. Examples demonstrate that the designs obtained by our method have better projection efficiency than many existing designs.     

Constructing Tables of Minimum Aberration Pn–m Designs

Fries and Hunter (1980) presented a practical algorithm for selecting standard 2 ^n–m fractional factorial designs based on a criterion they called “minimum aberration.” In this article some simple results are presented that enable the Fries–Hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p ≥ 2 is prime. Examples of minimum aberration 2 ^n–m designs with resolution R ≥ 4 are given for m, n – m < 9. A matrix is given for generating 3 ^n–m designs with m, n – m ≤ 6, which have, or nearly have, minimum aberration.     

Elementary Statistics

Fractional two-level factorial designs are often used in the early stages of an investigation to screen for important factors. Traditionally, 2 ^n-k fractional factorial designs of resolution III, IV, or V have been used for this purpose. When the investigator is able to specify the set of nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design with fewer runs than a standard textbook design by searching within a wider class of designs called parallel-flats designs. The run sizes in this class of designs do not necessarily need to be powers of 2. We discuss an algorithm for constructing orthogonal parallel-flats designs to meet user specifications. Several examples illustrate the use of the algorithm.     

Optimal Experimental Designs for Estimating the Independent Variable in Regression

Estimation of the independent variable in a regression situation for a measured value of the dependent variable (inverse estimation) is discussed along with the corresponding design problems under correct classification and misclassification of the model.

Optimal designs are derived using a linear approximation for the particular cases when the true model is linear or quadratic. Figures are provided to enable the experimenter to locate optimal designs for various sets of the parameters.

The design criterion considered in this paper is minimization of the integral over the range of x of E( – x)² with respect to design.     

Comparison of designs in presence of a possible correlation in observations

We often assume the standard linear model with uncorrelated observations for comparison of designs without realizing a possible presence of correlation in observations. In this paper we present several change of variance functions including the one given in Zhou (2001) for comparing designs in presence of possible correlation in observations. We find a design by minimizing one of our proposed change of variance functions in a simple response surface setup. We then compare its performance with all variance design, all bias design, and the design making the average variance equal to the average squared bias. We also compare a second order rotatable design with a non-rotatable design. The rotatable design is better than the non-rotatable design with respect to A-, D-, and E- optimality criterion functions under the standard linear model with uncorrelated observations. We observe that the rotatable design may not perform better than the non-rotatable design with respect to the change of variance functions. We present some important properties of the change of variance functions. We find that the A-optimum designs may perform poorly with respect to a change of variance function.     

Sample Size For Tolerance Limits on a Normal Distribution

Sample size tables are given for tolerance limits on a normal distribution. Wald-Wolfowitz two-sided limits and one-sided limits are considered. The criterion used for determining sample size is as follows: For a tolerance limit such that Pr (coverage ≥ P) = γ, choose P′ > P and δ (small) and require Pr (coverage ≥ P′) ≤ δ. Five levels of P, three levels of γ, three levels of P′, and three levels of δ are used in the tables. The tables are given for the common case where the degrees of freedom for the x² is one less than the sample size, but it is shown how to use the tables for other cases which occur in simple linear regression and some experimental designs. Examples are given to illustrate the use of the tables.     

Two-Level Multifactor Designs for Detecting the Presence of Interactions

A design optimality criterion, tr (L)-optimality, is applied to the problem of designing two-level multifactor experiments to detect the presence of interactions among the controlled variables. We give rules for constructing tr (L)-optimal foldover designs and tr (L)-optimal fractional factorial designs. Some results are given on the power of these designs for testing the hypothesis that there are no two-factor interactions. Augmentation of the tr (L)-optimal designs produces designs that achieve a compromise between the criteria of D-optimality (for parameter estimation in a first-order model) and tr (L)-optimality (for detecting lack of fit). We give an example to demonstrate an application to the sensitivity analysis of a computer model.     

On robust fixed‐order filter design

Abstract

The paper investigates the filter design problem based on linear matrix inequality (LMI) formulation. Filter design approaches to achieve bounded robust H ₈ norm, robust H ₂ norm, and robust l ₂ – l ₈ gain are developed. It is shown that these designs can all be cast in terms of LMIs. Moreover, the LMIs for robust performance assessment and filter design bear similar form with respect to these different measures. The fixed‐order filter design problem is also tackled in terms of LMIs. An example is provided to illustrate the filter design procedure.     

Near Minimal Resolution IV Designs for the sn Factorial Experiment

A fractional factorial design is of resolution IV if all main effects are estimable in the presence of two-factor interactions. For the sⁿ factorial experiment such a design requires at least N = s[(s – I)n – (s – 2)] runs. In this paper a series of resolution IV designs are given for the s” factorial, s = p ^α where p is prime, in N = s(s – I)n runs. A special case of the construction method produces a series of generalized foldover designs for the sⁿ experiment, s ≥ 3 and n ≥ 3, in N = s(s – I)n + s runs. These foldover designs permit estimation of the general mean in addition to all main effects and provide s degrees of freedom for estimating error. A section on analysis is included.     

Factorial Designs,the |X′X| Criterion,and Some Related Matters

Two of the basic approaches to choosing an n-point experimental design in many industrial situations are (i) to set down a simple factorial or fractional factorial design in the factors being studied, or (ii) to choose a design based on the well-known |X′X| criterion. Experimenters often prefer (i) due to its simplicity; our viewpoint here is that (ii) is much better. We first indicate some situations for which (when all the factors are restricted to a cuboidal region) the factorial approach is optimal, as judged by the |X′X| criterion, but the assumed models are often not sensible ones in practical work. We then examine what (similarly restricted) designs are optimal under the |X′X| criterion for the standard linear models of first and second order; because of the very rapid increase in computational difficulties, we consider only “cube plus star” type designs for k ≥ 3 (except for k = 3, n = 10). In spite of computational requirements, we recommend use of the |X′X| criterion in general rather than the indiscriminate use of factorials and we briefly discuss the reasons why, both for linear and nonlinear model situations.     

Saturated Fractions of 2 n and 3 n Factorial Designs

Saturated fractions of 2 ⁿ and 3 ⁿ factorial designs which permit the estimation of both main effects and first order interactions are described. A simple method of generating these particular designs is given. In addition to presenting the specific designs for n = 3, …, 10, tables of variances and relative efficiencies are included to assist the potential user in assessing the suitability of the described designs.     

Root cause analysis strategy for robust design domain recognition

Design domain identification with desirable attributes (e.g. feasibility, robustness and reliability) provides advantages when tackling large-scale engineering optimization problems. For the purpose of dealing with feasibility robustness design problems, this article proposes a root cause analysis (RCA) strategy to identify desirable design domains by investigating the root causes of performance indicator variation for the starting sampling initiation of evolutionary algorithms. The iterative dichotomizer 3 method using a decision tree technique is applied to identify reduced feasible design domain sets. The robustness of candidate domains is then evaluated through a probabilistic principal component analysis-based criterion. The identified robust design domains enable optimal designs to be obtained that are relatively insensitive to input variations. An analytical example and an automotive structural optimization problem are demonstrated to show the validity of the proposed RCA strategy.     

Orthogonal Main-Effect 2 n 3 m Designs and Two-Factor Interaction Aliasing

Methods are presented for the determination of the alias matrix of two-factor interactions for the orthogonal main-effect 2 ⁿ 3 ^m plans catalogued by Addelman and Kempthorne. This catalogue includes Placket-Burman designs and designs obtained by replacement in 2 ^n–p plans or collapsing in 3 ^m–m plans. Systematic methods are included to facilitate the data computations. For standard r ^n–p factorial designs, techniques are given to determine a set of live factors, a generating set of linear sum congruences and the alias matrix. Additional orthogonal main-effect 2 ⁿ 3 ^m designs are constructed to supplement the Addelman-Kempthorne catalogue of designs.     

Screening Designs for Poly-Factor Experimentation

A survey is given of the following types of screening designs: Incomplete 2 ^k designs, srlpersaturated and grollp-screening designs. These designs are compared with each other. Some new results for group-screening are derived.     

Designing Experiments for Causal Networks

Causal networks are directed graphs representing cause-effect relationships and are multiple-response generalizations of Ishikawa's cause–effect diagrams. Emphasizing tolerance design applications, this article describes an algorithm for designing suitable experiments when the factors and responses are organized as a causal network. The causal network is transformed into a so-called causal map, which represents all factors and responses as points in a common D-dimensional metric space. The design approach is algorithmic, optimizing the entropy criterion due to Wynn. This criterion is applied to maximize dispersion among the multiple responses, using a distance-in-space coefficients model. A key constraint is for the blocks to be self-contained; this implies that each block can be analyzed without reference to other blocks. This is to be complemented by a unified, all-block analysis. The resulting designs are evaluated for efficiency, response dispersion, and resolution V column rank. Particular attention is given to skewing each block by shifting one or a few factors off-center.     

Robust design of slow-light tapers in periodic waveguides

This article concerns the design of tapers for coupling power between uniform and slow-light periodic waveguides. New optimization methods are described for designing robust tapers, which not only perform well under nominal conditions, but also over a given set of parameter variations. When the set of parameter variations models the inevitable variations typical in the manufacture or operation of the coupler, a robust design is one that will have a high yield, despite these parameter variations. The ideas of successive refinement, and robust optimization based on multi-scenario optimization with iterative sampling of uncertain parameters, using a fast method for approximately evaluating the reflection coefficient, are introduced. Robust design results are compared to a linear taper, and to optimized tapers that do not take parameter variation into account. Finally, robust performance of the resulting designs is verified using an accurate, but much more expensive, method for evaluating the reflection coefficient.     

Quality Management: Tools and Methods for Improvement

Results from the comparison of D, G, and A efficiencies and the scaled average prediction variance IV criterion are presented for the central composite, small composite, Notz, Hoke, Box–Draper, and computer-generated designs. These design optimality criteria are evaluated over the cuboidal design region for three, four, and five design variables.     

New sampling strategies to reduce the effect of autocorrelation on the synthetic T2 chart to monitor bivariate process

In this paper, synthetic T² chart is developed to monitor bivariate process with correlated variables and autocorrelated observations. The proposed chart is a combination of the Hotelling's T² chart and the conforming run length chart. The operation and design of the chart are described when observations are autocorrelated and cross correlated. The first‐order vector autoregressive process VAR (1) is used to model the bivariate data from an autocorrelated process of interest. Using an average run length as performance measure criterion in the VAR (1) model, it is observed that autocorrelation seriously impact the performance of the synthetic T² chart. To reduce the effect of autocorrelation on the performance of the synthetic T² chart, the skip and mixed sampling strategies are implemented to form rational subgroups in the construction of synthetic T² chart. The average run length performance of the synthetic T² chart implementing these strategies is compared with that of the standard strategy of formation of rational subgroups. It is observed that implementing skip and mixed sampling strategies within rational subgroup improves the performance of the synthetic T² chart.