A clustering-based coordinate exchange algorithm for generating G-optimal experimental designs

In the optimal experimental design literature, the G-optimality is defined as minimizing the maximum prediction variance over the entire experimental design space. Although the G-optimality is a highly desirable property in many applications, there are few computer algorithms developed for constructing G-optimal designs. Some existing methods employ an exhaustive search over all candidate designs, which is time-consuming and inefficient. In this paper, a new algorithm for constructing G-optimal experimental designs is developed for both linear and generalized linear models. The new algorithm is made based on the clustering of candidate or evaluation points over the design space and it is a combination of point exchange algorithm and coordinate exchange algorithm. In addition, a robust design algorithm is proposed for generalized linear models with modification of an existing method. The proposed algorithm are compared with the methods proposed by Rodriguez et al. [Generating and assessing exact G-optimal designs. J. Qual. Technol. 2010;42(1):3–20] and Borkowski [Using a genetic algorithm to generate small exact response surface designs. J. Prob. Stat. Sci. 2003;1(1):65–88] for linear models and with the simulated annealing method and the genetic algorithm for generalized linear models through several examples in terms of the G-efficiency and computation time. The result shows that the proposed algorithm can obtain a design with higher G-efficiency in a much shorter time. Moreover, the computation time of the proposed algorithm only increases polynomially when the size of model increases.     

Construction and analysis of edge designs from skew-symmetric supplementary difference sets

The purpose of screening experiments is to identify the dominant variables from a set of many potentially active variables which may affect some characteristic y. Edge designs were recently introduced in the literature and are constructed by using conferences matrices and were proved to be robust. We introduce a new class of edge designs which are constructed from skew-symmetric supplementary difference sets. These designs are particularly useful since they can be applied for experiments with an even number of factors and they may exist for orders where conference matrices do not exist. Using this methodology, examples of new edge designs for 6, 14, 22, 26, 38, 42, 46, 58, and 62 factors are constructed. Of special interest are the new edge designs for studying 22 and 58 factors because edge designs with these parameters have not been constructed in the literature since conference matrices of the corresponding order do not exist. The suggested new edge designs achieve the same model-robustness as the traditional edge designs. We also suggest the use of a mirror edge method as a test for the linearity of the true underlying model. We give the details of the methodology and provide some illustrating examples for this new approach. We also show that the new designs have good D-efficiencies when applied to first order models.     

The performance of multivariate CUSUM control charts with estimated parameters

In this paper, we study the effect of estimating the vector of means and the variance–covariance matrix on the performance of two of the most widely used multivariate cumulative sum (CUSUM) control charts, the MCUSUM chart proposed by Crosier [Multivariate generalizations of cumulative sum quality-control schemes, Technometrics 30 (1988), pp. 291–303] and the MC1 chart proposed by Pignatiello and Runger [Comparisons of multivariate CUSUM charts, J. Qual. Technol. 22 (1990), pp. 173–186]. Using simulation, we investigate and compare the in-control and out-of-control performances of the competing charts in terms of the average run length measure. The in-control and out-of-control performances of the competing charts deteriorate significantly if the estimated parameters are used with control limits intended for known parameters, especially when only a few Phase I samples are used to estimate the parameters. We recommend the use of the MC1 chart over that of the MCUSUM chart if the parameters are estimated from a small number of Phase I samples.     

New classes of D-optimal edge designs

Edge designs are screening experimental designs that allow a model-independent estimate of the set of relevant variables, thus providing more robustness than traditional designs. In this paper we construct new classes of D-optimal edge designs. This construction uses weighing matrices of order n and weight k together with permutation matrices of order n to obtain D-optimal edge designs. One linear and one quadratic simulated screening scenarios are studied and compared using linear regression and edge designs analysis.     

A unification and some corrections to Markov chain approaches to develop variable ratio sampling scheme control charts

We provide some unifying definitions, make some corrections to articles by Faraz and Parsian (J Stat Pap 47:569–593, 2006) and Costa (J Qual Technol 26:155–163, 1994; 29:197–204, 1997), and using these provide more accurate tables of results and comparisons of control charts. We also investigate the impact of an incorrectly specified process shift on signal frequency.     

BALANCED ARRAYS AND WEIGHING DESIGNS

Dey (19711, Saha (1975), Kageyama & Saha (1983) and others have shown how optimum chemical balance weighing designs can be constructed from the incidence matrices of balanced incomplete block (BIB) designs. In this paper, it is shown that weighing designs can be constructed from some suitably chosen two-symbol balanced arrays of strength two, which need not always be incidence matrices of BIB designs. The findings lead us to construct new optimum chemical balance weighing designs from incidence matrices of BIB designs.     

Bayesian estimation of parameters of mixed geometric failure models from Type I group censored sample

The estimation problem of the parameters of a mixed geometric lifetime model, using Bayesian approach and Type I group censored sample, will be investigated in the case of two subpopulations. The Bayes estimates are derived for squared error, minimum expected, general entropy and linex loss functions under informative and diffuse priors. A practical example given by Nelson (W.B. Nelson, Hazard plotting methods for analysis of the life data with different failure models, J. Qual. Technol. 2 (1970), pp. 126–149) is considered. A simulation study is carried out along with risk.     

D-optimal designs for beta regression models with single predictor

The problem of construction of D-optimal designs for beta regression models involving one predictor is considered for the mean-precision parameterization suggested by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions. J Appl Stat. 2004;31:799–815]. Here we use the logit link function for the mean sub-model. These designs are presented and discussed for unrestricted as well as restricted design regions by considering the precision parameter as (1) a known constant and (2) an unknown constant. Efficiency comparison of obtained designs with commonly used equi-weighted, equi-spaced designs is made to recommend designs for practical use. Real-life applications are given to show the usefulness of these designs.     

OPTIMUM SINGULAR SPRING BALANCE WEIGHING DESIGNS WITH NON-HOMOGENEITY OF THE VARIANCES OF ERRORS FOR ESTIMATING THE TOTAL WEIGHT

The problem of estimation of the total weight of objects using a singular spring balance weighing design with non-homogeneity of the variances of errors has been dealt with in this paper. Based on a theorem by Katulska (1984) giving a lower bound for the variance of the estimated total weight, a necessary and sufficient condition for this lower bound to be attained is obtained. It is shown that weighing designs for which the the lower bound is attainable, can be constructed from the incidence matrices of (α₁,.,α_t)-resolvable block designs, α-resolvable block designs, singular group divisible designs, and semi-regular group divisible designs.     

Extensions of D-optimal minimal designs for symmetric mixture models

The purpose of mixture experiments is to explore the optimum blends of mixture components, which will provide the desirable response characteristics in finished products. D-optimal minimal designs have been considered for a variety of mixture models, including Scheffé's linear, quadratic, and cubic models. Usually, these D-optimal designs are minimally supported since they have just as many design points as the number of parameters. Thus, they lack the degrees of freedom to perform the lack-of-fit (LOF) tests. Also, the majority of the design points in D-optimal minimal designs are on the boundary: vertices, edges, or faces of the design simplex. In this article, extensions of the D-optimal minimal designs are developed for a general mixture model to allow additional interior points in the design space to enable prediction of the entire response surface. Also a new strategy for adding multiple interior points for symmetric mixture models is proposed. We compare the proposed designs with Cornell (1986 Cornell, J.A. (1986). A comparison between two ten-point designs for studying three-component mixture systems. J. Qual. Technol. 18(1):1–15.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) two 10-point designs for the LOF test by simulations.     

A-optimal designs for optimum mixture in an additive quadratic mixture model

In the analysis of experiments with mixtures, quadratic models have been widely used. The optimum designs for the estimation of optimum mixing proportions in a quadratic mixture model have been studied by Pal and Mandal [Optimum designs for optimum mixtures. Statist Probab Lett. 2006;76:1369–1379] and Mandal et al. [Optimum mixture designs: a pseudo-Bayesian approach. J Ind Soc Agric Stat. 2008;62(2):174–182; Optimum mixture designs under constraints on mixing components. Statist Appl. 2008;6(1&2) (New Series): 189–205], using a pseudo-Bayesian approach. In this paper, a similar approach has been employed to obtain the A-optimal designs for the estimation of optimum proportions in an additive quadratic mixture model, proposed by Darroch and Waller [Additivity and interaction in three-component experiments with mixture. Biometrika. 1985;72:153–163], when the number of components is 3, 4 and 5. It has been shown that the vertices of the simplex are necessarily the support points of the optimum design, and the other support points include barycentres of depth at most 2.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

Multivariate CUSUM chart: properties and enhancements

The multivariate CUSUM#1 control chart of Pignatiello and Runger (J. Qual. Technol. 22:173–186, 1991) is widely used in practical applications due to its good ability to detect shifts of small and medium size in a process of interest. This paper investigates properties and suggests several refinements of this chart. The performance of the competing procedures is evaluated within a Monte Carlo simulation study. The suggested log MCUSUM chart proves to be the best among the investigated alternatives for the considered performance criteria.     

Catalogue of small runs nonregular designs from hadamard matrices with generalized minimum aberration

Fries and Hunter ( 1980 ) proposed the Minimum Aberration criterion (MA) for selecting regular designs. The regular designs with MA are msot commonly used because they are considered as the best designs. How ever, as pointed out by Chen, Sun and Wu ( 1993 ), there are situations that other designs may better meet the design need. Therefore, they catalogued some two-level and three-level fractional factorial regular designs with small (16,27,32,64) runs. For nonregular designs, such as the ones taken from Hadamard matrices, the MA criterion is not appUcable. Deng and Tang ( 1999 ) introduced Generalized Minimum Aberration Criterion (GMA) as a natural extension to the MA criterion. Similar to the case in the regular designs, other designs may better meet practical need, In this paper, we use the GMA criterion to give a catalogue of nonregular designs with smaU (16,20,24) runs.     

Regular E-Optimal Spring Balance Weighing Designs with Correlated Errors

The problems linked with an E-optimal spring balance weighing design with correlated errors are discussed. The topic is focus on the determining the maximal eigenvalue of the inverse of the information matrix of estimators. The constructing method of the E-optimal design, based on the incidence matrices of balanced incomplete block designs, is presented.     

Crisis cycles in the slightly unstable block random model

ABSTRACT

The paper proposes a new approach for studying the time to time appearing breakdowns in economy. Block random model can describe stability of large complicated systems with variable number of participants. Theoretical background of the model is given by a theorem about the eigenvalues of block random matrices [Juhász F. On the characteristic values of non-symmetric block random matrices. J Theoret Probab. 1990;67:199–205; On the structural eigenvalues of block random matrices. Linear Algebra Appl. 1996;246:225–231]. The model takes into account not only effects of participants but of groups formed from them as well. Slight instability means group level stability and participant level instability [Juhász F. On the turbulence of slightly unstable block random systems. In: Taylor C, et al., editors. Numerical methods for laminar and turbulent flow. Atlanta; 1995. p. 113–121]. Lability index of block random systems is introduced for measuring instability. It is showed that lability index of a slightly unstable block random model is growing while number of participants increases. Alteration in the number of participants makes it possible to describe crisis cycles.     

A review of uniform cross-over designs

Uniform cross-over designs form an important family of experimental designs. They have been applied in many scientific disciplines including clinical trials, agricultural studies and psychological experiments. In this paper we consider the four types of uniform cross-over design, as given by Williams [1949. Experimental designs balanced for the estimation of residual effects of treatments. Aust. J. Sci. Res. 2, 149–168], Cheng and Wu [1980. Balanced repeated measurements designs. Ann. Statist. 8, 1272–1283. Corrigendum 11 (1983) 349], Bate and Jones [2006. The construction of nearly balanced and nearly strongly balanced uniform cross-over designs. J. Statist. Plann. Inference 136, 3248–3267] and Kunert [1983. Optimal design and refinement of the linear model with applications to repeated measurements designs. Ann. Statist. 11, 247–257]. The efficiency of these designs, existence criteria and methods of construction are described.     

Foldover Conference Designs for Screening Experiments

A conference matrix is a square matrix C with zeros on the diagonal and ±1s off the diagonal, such that C ^T C = CC ^T  = (n ? 1)I, where I is the identity matrix. Conference matrices are an important class of combinatorial designs due to their many applications in several fields of science, including statistical-experimental designs, telecommunications, elliptic geometry, and more. In this article, conference matrices and their full foldover design are combined together to obtain an alternative method for screening active factors in complicated problems. This method provides a model-independent estimate of the set of active factors and also gives a linearity test for the underlying model.     

A new approach for monitoring process variance

ABSTRACT

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having non-normal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used in such circumstances. In this paper, we propose a new variance chart based on a simple statistic to monitor process variance shifts. We explore the sampling properties of the new monitoring statistic and calculate the average run lengths (ARLs) of the proposed variance chart. Furthermore, an arcsine transformed exponentially weighted moving average (EWMA) chart is proposed because the ARLs of this modified chart are more intuitive and reasonable than those of the variance chart. We compare the out-of-control variance detection performance of the proposed variance chart with that of the non-parametric Mood variance (NP-M) chart with runs rules, developed by Zombade and Ghute [Nonparametric control chart for variability using runs rules. Experiment. 2014;24(4):1683–1691], and the nonparametric likelihood ratio-based distribution-free exponential weighted moving average (NLE) chart and the combination of traditional exponential weighted moving average (EWMA) mean and EWMA variance (CEW) control chart proposed by Zou and Tsung [Likelihood ratio-based distribution-free EWMA control charts. J Qual Technol. 2010;42(2):174–196] by considering cases in which the critical quality characteristic has a normal, a double exponential or a uniform distribution. Comparison results showed that the proposed chart performs better than the NP-M with runs rules, and the NLE and CEW control charts. A numerical example of service times with a right-skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the application of the proposed variance chart and of the arcsine transformed EWMA chart and to compare them with three existing variance (or standard deviation) charts. The proposed charts show better detection performance than those three existing variance charts in monitoring and detecting shifts in the process variance.     

Improved maximum-likelihood estimation in a regression model with general parametrization

We analyse the finite-sample behaviour of two second-order bias-corrected alternatives to the maximum-likelihood estimator of the parameters in a multivariate normal regression model with general parametrization proposed by Patriota and Lemonte [A.G. Patriota and A.J. Lemonte, Bias correction in a multivariate regression model with genereal parameterization, Stat. Prob. Lett. 79 (2009), pp. 1655–1662]. The two finite-sample corrections we consider are the conventional second-order bias-corrected estimator and the bootstrap bias correction. We present the numerical results comparing the performance of these estimators. Our results reveal that analytical bias correction outperforms numerical bias corrections obtained from bootstrapping schemes.