Robust parameter design: Optimization of combined array approach with orthogonal arrays

Robust parameter design, originally proposed by Taguchi (1987. System of Experimental Design, vols. I and II. UNIPUB, New York), is an off-line production technique for reducing variation and improving a product's quality by using product arrays. However, the use of product arrays results in an exorbitant number of runs. To overcome the drawbacks of the product array several scientists proposed the use of combined arrays, where the control and noise factors are combined in a single array. In this paper, we use certain orthogonal arrays that are embedded into Hadamard matrices as combined arrays, in order to identify a model that contains all the main effects (control and noise) and their control-by-noise interactions with high efficiency. Aliasing of effects in each case is also discussed.     

Combined array approach for optimal designs

Robust parameter design, originally proposed by Taguchi ( 1987 ) is an offline production technique for reducing variation and improving product's quality To achieve this objective Taguchi proposed the use of product arrays. However. the product array approach, results in an exorbitant number of runs To overcome the drawbacks of the product array Welch, Wu, Kang and Sacks ( 1990 ), Shoemaker, Tsui and Wu ( 1991 ) and Montgomery ( 1991a ) proposed the use of combined arrays, where the control factors and noise factors are combined in a single array. In this paper we study the concept of combined array for an intermediate class of designs where n = 1 (mod4), n = 2 (mod4) and n = 3 (mod4). The designs presented in this paper, though not orthogonal, offer a great reduction in the run-size.     

On the pade tablé and its relationship to the r and sarrays and arm a modeling

The Box-Jenkins method is a popular and important technique for modeling and forecasting of time series. Unfortunately the problem of determining the appropriate ARMA forecasting model (or indeed if an ARMA model holds) is a major drawback to the use of the Box-Jenkins methodology. Gray et al. (1978) and Woodward and Gray (1979) have proposed methods of estimating p and qin ARMA modeling based on the R and Sarrays that circumvent some of these modeling difficulties.

In this paper we generalize the R and S arrays by showing a relationship to Padé approximunts and then show that these arrays have a much wider application than in just determining model order. Particular non-ARMA models can be identified as well. This includes certain processes that consist of deterministic functions plus ARMA noise, indeed we believe that the combined R and S arrays are the best overall tool so fur developed for the identification of general 2nd order (not just stationary) time scries models.     

Mean squared error of prediction approach to the analysis of a combined array

SUMMARY The combined array provides a powerful, more statistically rigorous alternative to Taguchi's crossed-array approach to robust parameter design. The combined array assumes a single linear model in the control and the noise factors. One may then find conditions for the control factors which will minimize an appropriate loss function that involves the noise factors. The most appropriate loss function is often simply the resulting process variance, recognizing that the noise factors are actually random effects in the process. Because the major focus of such an experiment is to optimize the estimated process variance, it is vital to understand the resulting prediction properties. This paper develops the mean squared error for the estimated process variance for the combined array approach, under the assumption that the model is correctly specified. Specific combined arrays are compared for robustness. A practical example outlines how this approach may be used to select appropriate combined arrays within a particular experimental situation.     

Split-plot designs for robust product experimentation

Genichi Taguchi has emphasized the use of designed experiments in several novel and important applications. In this paper we focus on the use of statistical experimental designs in designingproducts to be robust to environmental conditions. The engineering concept of robust product design is very important because it is frequently impossible or prohibitively expensive to control or eliminate variation resulting from environmental conditions. Robust product design enablesthe experimenter to discover how to modify the design of the product to minimize the effect dueto variation from environmental sources. In experiments of this kind, Taguchi's total experimental arrangement consists of a cross-product of two experimental designs:an inner array containing the design factors and an outer array containing the environmental factors. Except in situations where both these arrays are small, this arrangement may involve a prohibitively large amount of experimental work. One of the objectives of this paper is to show how this amount of work can be reduced. In this paper we investigate the applicability of split-plot designs for thisparticular experimental situation. Consideration of the efficiency of split-plot designs and anexamination of several variants of split-plot designs indicates that experiments conductedin a split-plot mode can be of tremendous value in robust product design since they not only enable the contrasts of interest to be estimated efficiently but also the experiments can be considerably easier to conduct than the designs proposed by Taguchi.     

Projection Properties of Certain Three-Level Main Effect Plans with Quantitative Factors

ABSTRACT

Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a “projection design”. In this article, we have identified all the geometric non isomorphic projection designs of an OA(27,13,3,2), an OA(18,7,3,2) and an OA(36,13,3,2) into k = 3,4, and 5 factors when they are used for screening out active quantitative experimental factors, with regard to the prior selection of the middle level of factors. We use the popular D-efficiency criterion to evaluate the ability of each design found in estimating the parameters of a second order model.     

An orthogonal arrays approach to robust parameter designs methodology

Robust parameter design methodology was originally introduced by Taguchi [14 Taguchi, G. 1986. Introduction to Quality Engineering: Designing Quality Into Products and Process, Tokyo: Asian Productivity Organization.  [Google Scholar]] as an engineering methodology for quality improvement of products and processes. A robust design of a system is one in which two different types of factors are varied; control factors and noise factors. Control factors are variables with levels that are adjustable, whereas noise factors are variables with levels that are hard or impossible to control during normal conditions, such as environmental conditions and raw-material properties. Robust parameter design aims at the reduction of process variation by properly selecting the levels of control factors so that the process becomes insensitive to changes in noise factors. Taguchi [14 Taguchi, G. 1986. Introduction to Quality Engineering: Designing Quality Into Products and Process, Tokyo: Asian Productivity Organization.  [Google Scholar] 15 Taguchi, G. 1987. System of Experimental Design, Vol. I and II, New York: UNIPUB.  [Google Scholar]] proposed the use of crossed arrays (inner–outer arrays) for robust parameter design. A crossed array is the cross-product of an orthogonal array (OA) involving control factors (inner array) and an OA involving noise factors (outer array). Objecting to the run size and the flexibility of crossed arrays, several authors combined control and noise factors in a single design matrix, which is called a combined array, instead of crossed arrays. In this framework, we present the use of OAs in Taguchi's methodology as a useful tool for designing robust parameter designs with economical run size.     

Statistical methods for continuous outcomes in partially clustered designs

We address statistical issues involved in the partially clustered design where clusters are only employed in the intervention arm, but not in the control arm. We develop a cluster adjusted t-test to compare group treatment effects with individual treatment effects for continuous outcomes in which the individual level data are used as the unit of the analysis in both arms, we develop an approach for determining sample sizes using this cluster adjusted t-test, and use simulation to demonstrate the consistent accuracy of the proposed cluster adjusted t-test and power estimation procedures. Two real examples illustrate how to use the proposed methods.     

Bayesian nonparametric smoothers for regular processes

This report is about the analysis of stochastic processes of the form R = S + N, where S is a “smooth” functional and N is noise. The proposed methods derive from the assumption that the observed R-values and unobserved values of R, the assumed inferential objectives of the analysis, are linearly related through Taylor series expansions of observed about unobserved values. The expansion errors and all other priori unspecified quantities have a joint multivariate normal distribution which expresses the prior uncertainty about their values. The results include interpolators, predictors, and derivative estimates, with credibility-interval estimates automatically generated in each case. An analysis of an acid-rain wet-deposition time series is included to indicate the efficacy of the proposed method. It was this problem which led to the methodological developments reported in this paper.     

A critical look at Taguchi's modelling approach for robust design

The objective of Taguchi's robust design method is to reduce the output variation from the target (the desired output) by making the performance insensitive to noise, such as manufacturing imperfections, environmental variations and deterioration. This objective has been recognized to be very effective in improving product and manufacturing process design. In application, however, Taguchi's analysis approach of modelling the average loss (or signal-to-noise ratios) may lead to non-optimal solutions, efficiency loss and information loss. In addition, since his modelling loss approach requires a special experimental format that contains a cross-product of two separate arrays for control and noise factors, this leads to less flexible and unnecessarily expensive experiments. The response model approach, an alternative approach proposed by Welch et al. , Box and Jones, Lucas and Shoemaker et al. , does not have these problems. However, this alternative approach also has its own problems. This paper reviews and discusses the potential problems of Taguchi's modelling approach. We illustrate these problems with examples and numerical studies. We also compare the advantages and disadvantages of Taguchi's approach and the alternative approach.     

The use of probabilistic models to produce optimal graphical displays of high-dimensional data sets

Summary Several techniques for exploring ann×p data set are considered in the light of the statistical framework: data-structure+noise. The first application is to Principal Component Analysis (PCA), in fact generalized PCA with any metric M on the unit space ℝ _p . A natural model for supporting this analysis is the fixed-effect model where the expectation of each unit is assumed to belong to some q-dimensional linear manyfold defining the structure, while the variance describes the noise. The best estimation of the structure is obtained for a proper choice of metric M and dimensionality q: guidelines are provided for both choices in section 2. The second application is to Projection Pursuit which aims to reveal structure in the original data by means of suitable low-dimensional projections of them. We suggest the use of generalized PCA with suitable metric M as a Projection Pursuit technique. According to the kind of structure which is looked for, two such metrics are proposed in section 3. Finally, the analysis ofn×p contingency tables is considered in section 4. Since the data are frequencies, we assume a multinomial or Poisson model for the noise. Several models may be considered for the structural part; we can say that Correspondence Analysis rests on one of them, spherical factor analysis on another one; Goodman association models also provide an alternative modelling. These different approaches are discussed and compared from several points of view.     

A TWO-STAGE UNRELATED RANDOMISED RESPONSE PROCEDURE

In a sample survey, questions requiring personal or controversial assertions often give rise to resistance. A randomised response procedure can be used to help the researcher gather accurate data in this case. This paper describes a new two-stage unrelated randomised response procedure that combines the use of two randomisation devices (Mangat & Singh, 1990) and an unrelated question (Horwitz et al. 1967). It examines the situation where the respondents are not completely truthful in their answers. The efficiency of this new method is compared with the original one-stage procedure proposed by Horwitz et al. (1967), and guidelines for choosing the values of different parameters for the procedures are provided. Results from an empirical study which examines the efficiency and feasibility of the proposed method are given.     

On designing new optimal synthetic Tukey’s control charts

Tukey’s control chart is generally used for monitoring the processes where the measurement process physically damages the product. It is based on single observation and robust to outliers. In this paper, two optimal synthetic Tukey’s control charts are proposed by integrating the conforming run length chart with the Tukey’s control chart and its modification. The performance comparison of the proposed charts with the existing Tukey’s control charts is made by using out-of-control average run length and extra quadratic loss as performance metrics. The proposed charts offer better protection against the process shifts as compare to the existing Tukey’s control charts when the underlying process distribution is symmetric or asymmetric. Simulation studies also establish the supremacy of the proposed control charts over the existing Tukey’s control charts. In the end, an illustrative example based on a real data set of the combined cycle power plant is provided for practical implementation.     

Estimation of patterned covariance in the multivariate linear models: an outer product least-squares approach

In this article, we present a framework of estimating patterned covariance of interest in the multivariate linear models. The main idea in it is to estimate a patterned covariance by minimizing a trace distance function between outer product of residuals and its expected value. The proposed framework can provide us explicit estimators, called outer product least-squares estimators, for parameters in the patterned covariance of the multivariate linear model without or with restrictions on regression coefficients. The outer product least-squares estimators enjoy the desired properties in finite and large samples, including unbiasedness, invariance, consistency and asymptotic normality. We still apply the framework to three special situations where their patterned covariances are the uniform correlation, a generalized uniform correlation and a general q-dependence structure, respectively. Simulation studies for three special cases illustrate that the proposed method is a competent alternative of the maximum likelihood method in finite size samples.     

Constructing non-regular robust parameter designs

In recent years there has been considerable attention paid to robust parameter design as a strategy for variance reduction. Of particular concern is the selection of a good experimental plan in light of the two different types of factors in the experiment (control and noise) and the asymmetric manner in which effects of the same order are treated. Recent work has focussed on the selection of regular fractional factorial designs in this setting. In this article, we consider the construction and selection of optimal non-regular experiment plans for robust parameter design. Our approach defines the word-length pattern for non-regular fractional factorial designs with two different types of factors which allows for the choice of optimal design to emphasize the estimation of the effects of interest. We use this new word-length pattern to rank non-regular robust parameter designs. We show that one can easily find minimum aberration robust parameter designs from existing orthogonal arrays. The methodology is demonstrated by finding optimal assignments for control and noise factors for 12, 16 and 20-run orthogonal arrays.     

Skewness Reduction Approach in Multi-Attribute Process Monitoring

Since the product quality of many industrial processes depends upon more than one dependent variable or attribute, they are either multivariate or multi-attribute in nature. Although multivariate statistical process control is receiving increased attention in the literature, little work has been done to deal with multi-attribute processes. In this article, we develop a new methodology to monitor multi-attribute processes. To do this, first we transform multi-attribute data in a way that their marginal probability distributions have almost zero skewness. Then, we estimate the transformed covariance matrix and apply the well-known T ² control chart. In order to illustrate the proposed method and evaluate its performance, we use two simulation experiments and compare the results with the ones from both MNP chart and the χ² control chart.     

Rate efficient estimation of realized Laplace transform of volatility with microstructure noise

In this paper, we consider the problem of estimating the Laplace transform of volatility within a fixed time interval [0,T] using high‐frequency sampling, where we assume that the discretized observations of the latent process are contaminated by microstructure noise. We use the pre‐averaging approach to deal with the effect of microstructure noise. Under the high‐frequency scenario, we obtain a consistent estimator whose convergence rate is , which is known as the optimal convergence rate of the estimation of integrated volatility functionals under the presence of microstructure noise. The related central limit theorem is established. The simulation studies justify the finite‐sample performance of the proposed estimator.     

Construction of more Flexible and Efficient P–rep Designs

Early generation variety trials are very important in plant and tree breeding programs. Typically many entries are tested, often with very few resources available. Unreplicated trials using control plots are popular and it is common to repeat the trials at a number of locations. An alternative is to use partially replicated (p–rep) designs, where a proportion of the test entries are replicated at each location. We extend a method for the generation of p–rep designs based on α–arrays to allow for a much broader class of designs to be constructed. Updating procedures for the average efficiency factor and its upper bound are developed for application to the computer generation of efficient p–rep designs.     

Testing for additivity and joint effects in multivariate nonparametric regression using Fourier and wavelet methods

We consider the problem of testing for additivity and joint effects in multivariate nonparametric regression when the data are modelled as observations of an unknown response function observed on a d-dimensional (d 2) lattice and contaminated with additive Gaussian noise. We propose tests for additivity and joint effects, appropriate for both homogeneous and inhomogeneous response functions, using the particular structure of the data expanded in tensor product Fourier or wavelet bases studied recently by Amato and Antoniadis (2001) and Amato, Antoniadis and De Feis (2002). The corresponding tests are constructed by applying the adaptive Neyman truncation and wavelet thresholding procedures of Fan (1996), for testing a high-dimensional Gaussian mean, to the resulting empirical Fourier and wavelet coefficients. As a consequence, asymptotic normality of the proposed test statistics under the null hypothesis and lower bounds of the corresponding powers under a specific alternative are derived. We use several simulated examples to illustrate the performance of the proposed tests, and we make comparisons with other tests available in the literature.     

Monitoring Variation in a Multivariate Process When the Dimension is Large Relative to the Sample Size

A control procedure is presented for monitoring changes in variation for a multivariate normal process in a Phase II operation where the subgroup size, m, is less than p, the number of variates. The methodology is based on a form of Wilk' statistic, which can be expressed as a function of the ratio of the determinants of two separate estimates of the covariance matrix. One estimate is based on the historical data set from Phase I and the other is based on an augmented data set including new data obtained in Phase II. The proposed statistic is shown to be distributed as the product of independent beta distributions that can be approximated using either a chi-square or F-distribution. An ARL study of the statistic is presented for a range of conditions for the population covariance matrix. Cases are considered where a p-variate process is being monitored using a sample of m observations per subgroup and m < p. Data from an industrial multivariate process is used to illustrate the proposed technique.