Variance Dispersion Graphs Research Articles

Prediction variance capability of orthogonal uniform composite designs and orthogonal array composite designs in the spherical region

In this paper, the prediction variance capabilities of two new second-order designs orthogonal uniform composite designs and orthogonal array composite designs are examined using the variance dispersion graph and the fraction of design space plot in the spherical region. Also, the D- and G-optimality criteria are evaluated for these two response surface designs. The orthogonal array composite designs were shown to have a better prediction variance in terms of the fraction of design space plot and the variance dispersion graphs over the orthogonal uniform composite designs.

Prediction Variance Properties of Third-Order Response Surface Designs in the Hypersphere

The variance dispersion graphs (VDGs) and the fraction of design space (FDS) graphs are two graphical methods that effectively describe and evaluate the points of best and worst prediction capability of a design using the scaled prediction variance properties. These graphs are often utilized as an alternative to the single-value criteria such as D- and E- when they fail to describe the true nature of designs. In this paper, the VDGs and FDS graphs of third-order orthogonal uniform composite designs (OUCD₄) and orthogonal array composite designs (OACD₄) using the scaled-prediction variance properties in the spherical region for 2 to 7 factors are studied throughout the design region and over a fraction of space. Single-valued criteria such as D-, A- and G-optimality are also studied. The results obtained show that the OUCD₄ is more optimal than the OACD₄ in terms of D-, A- and G-optimality. The OUCD₄ was shown to possess a more stable and uniform scaled-prediction variance throughout the design region and over a fraction of design space than the OACD₄ although the stability of both designs slightly deteriorated towards the extremes.

Prediction Variance Capabilities of Third-Order Response Surface Designs for Cuboidal Regions

Experimenters often evaluate the steadiness and consistency of designs over the region of interest by means of the prediction variance capabilities using the variance dispersion graph and the fraction of design space graph. The variance dispersion graph and the fraction of design space graph effectively describe the prediction variance capabilities of a design in the region of interest. However, the prediction variance capabilities of third-order response surface designs have not been studied in the literature. In this paper, the prediction variance capabilities of two third-order response surface designs term augmented orthogonal uniform composite designs and orthogonal array composite designs in the cuboidal region for 3≤k≤7 with <img src=image/13424429_01.gif> center points are examined. The prediction variance capabilities are evaluated using the variance dispersion graph and the fraction of design space graph. Also, D-, E-, G-and T-optimality criteria are used in evaluating these designs in terms of single-value criterion. The results obtained show that the augmented orthogonal uniform composite designs have better prediction variance capabilities in the cuboidal region in the terms of the variance dispersion graphs for factors 3 and 4. The augmented orthogonal uniform composite designs also have better prediction variance capabilities for 3≤k≤7 compare to the orthogonal array composite designs in terms of the fraction of design space graph. The augmented orthogonal uniform composite designs are shown to be superior over the orthogonal array composite designs in terms of D-, E-, G-and T-optimality criteria for single-value criterion. This shows that the performances of the prediction variance capabilities of third-order response surface designs can be clearly visualized by means of the variance dispersion graph and fraction of design space and should be consider over the single-value criteria even though the single value-criteria show some degree of design performance. The augmented orthogonal uniform composite design is should often be considered in experimentation over the orthogonal array composite design since the augmented orthogonal uniform composite design performance better.

Factorwise variance dispersion graphs

In this paper, new plots, called factorwise variance dispersion graphs (FVDGs) with accompanying coordinate trace plots (CTPs), are introduced. FVDGs display prediction variances throughout the design space across the levels of each design factor, while CTPs plot the coordinates associated with minimum and maximum prediction variances, and hence G-efficiencies. FVDGs and CTPs can supplement the use of other graphical tools (e.g., variance dispersion graphs and fraction of design space plots) when assessing the prediction variance properties of a response surface design. They also can be used to assess the robustness of designs with respect to model reduction, design augmentation, and with an irregular design space as in a mixture experiment. Examples and the benefits of FVDGs and CTPs will be presented.

Optimal prediction variance properties of some central composite designs in the hypercube

Measures of the prediction variance performances of three variations of central composite designs when the region is hypercube were examined and compared using the G-efficiency and I-optimality criterion so as to determine the economical design(s) that perform(s) better than the other. The hypercube is the multidimensional cuboidal region with the axial distance, . The designs are the standard central composite designs (CCD), the small composite design (SCD) and minimum run resolution V (minResV) design. Two prediction variance based optimality criteria, I-optimality and G-efficiency were determined, a plot of variance dispersion graph and fraction of design space are used to give a comprehensive picture of the behavior of the prediction variance throughout the region of interest. Comparing the three designs mentioned above for 3, 4, and 5 factors on cuboidal region, the result showed that CCD performed better than SCD and MinResVdesign considering 2, 3, and 5 center points.

Graphical evaluation of robust parameter designs based on extended scaled prediction variance and extended spherical average prediction variance

ABSTRACTFor any response surface design, there are locations in the design region where responses are estimated well and locations where estimation is relatively poor. Consequently, graphical evaluation—such as variance dispersion graphs and the fraction of design space—is used as an alternative to a single-valued criterion. Such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher. In this article, we propose the extended scaled prediction variance and extended spherical average prediction variance as prediction methods. We also illustrate how graphical methods can be employed to evaluate robust parameter designs.

Flexible Graphical Assessment of Experimental Designs in R: The vdg Package

The R package vdg provides a flexible interface for producing various graphical summaries of the prediction variance associated with specific linear model specifications and experimental designs. These methods include variance dispersion graphs, fraction of design space plots and quantile plots which can assist in choosing between a catalogue of candidate experimental designs. Instead of restrictive optimization methods used in traditional software to explore design regions, vdg utilizes sampling methods to introduce more flexibility. The package takes advantage of R's modern graphical abilities via ggplot2 (Wickham 2009), adds facilities for using a variety of distance methods, allows for more flexible model specifications and incorporates quantile regressions to help with model comparison.

Comparison of Prediction Capabilities of Partially Replicated Central Composite Designs in Cuboidal Region

Three variations of partially replicated central composite designs have been compared. Their prediction capabilities with respect to their prediction variances are assessed using variance dispersion graphs in cuboidal region when the axial distance is α = 1. The results show that for n0 = 1, 2, and 3 center points, the replicated star designs have better prediction potentials than the replicated cube designs when interest is in predicting responses in the face-centered cube. The face-centered cube is the second-order central composite design with axial distance, α = 1.

Graphical Evaluation of the Prediction Capabilities of Partially Replicated Orthogonal Central Composite Designs

In this work, the performances of the partially replicated‐cube and ‐star (axial) portions of orthogonal central composite designs (CCD) in spherical regions are studied. The replication of the cube and star portions of the CCD without replicating the centre point for k = 3, 4, 5 and 6 number of factors is the main focus. The interest is to evaluate the performance of the CCD with respect to the stability, small prediction variance and prediction capability, when replication is at other points away from the centre of the design. The variance dispersion graphs and fraction of design space plots are the two graphical techniques used while D‐ and G‐efficiencies are the single‐value optimality criteria utilized in the assessment. The results show that replicating the star portions of the CCD considerably reduces the prediction variance, thereby improving the G‐efficiency in the spherical region and, therefore, recommended for prediction with precision and with smaller number of runs than replicating the cube. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Vdgraph: A Package for Creating Variance Dispersion Graphs

Vdgraph: A Package for Creating Variance Dispersion Graphs

Three‐Dimensional Quantile Plots and Dynamic Quantile Plots of the Prediction Variance for Response Surface Designs

A historically common choice for evaluating response surface designs is to use alphabetic optimality criteria. Single‐number criteria such as D, A, G, and V optimality do not completely reflect the estimation or prediction variance characteristics of the designs in question. For prediction‐based assessment, alternatives to single‐number summaries include the graphical displays of the prediction variance across the design regions. Variance dispersion graphs, fraction of design space plots, and quantile plots have been suggested to evaluate the overall prediction capability of response surface designs. The quantile plots use the percentiles. These quantile plots use the percentiles of the distribution at a given radius instead of just the mean, maximum, and minimum prediction variance values on concentric spheres inside the region of the interest. Previously, the user had to select several values of radius and draw corresponding quantile plots to evaluate the overall prediction capability of response surface designs. The user‐specified choice of radii to examine makes the plot somewhat subjective. Alternately, we propose to remove this subjectivity by using a three‐dimensional quantile plot. As another extension of the quantile plots, we suggest dynamic quantile plots to animate the quantile plots and use them for comparing and evaluating response surface designs. Copyright © 2011 John Wiley &amp; Sons, Ltd.

On comparing the prediction variances of some central composite designs in spherical regions: a review

Three second-order response surface designs, namely: central composite design, small composite design and minimum-run resolution V design are compared for factors 3 to 7 in spherical regions using the G - and I - optimality criteria. The structures of the response variances of the designs are displayed graphically using the variance dispersion graphs. The maximum and average response variances are determined analytical as functions of the radii of the design region. Results for the G - and I - optimality criteria are obtained for a spherical region. The results suggest that none of the designs is uniformly better than the others with respect to the optimality criteria and the variance dispersion graphs for the factors considered.

Discussion of “Response surface design evaluation and comparison”

Selecting the best possible experimental design for a given situation is not a simple matter. This is because there are a lot of criteria that ought to be taken into account when choosing one out of many alternative design options. In their article, the authors focus on the use of graphical methods for comparing experimental designs. In particular, the article reviews most of the literature on variance dispersion graphs and fraction of design space plots. It is explained that sophisticated variance dispersion graphs have been proposed in the literature for assessing model robustness, for evaluating split-plot designs, for evaluating the impact of measurement error on mixture designs, and for mixture experiments involving process variables. There is no doubt that variance dispersion graphs and fraction of design space plots are very useful tools for comparing alternative design options. Oftentimes, experimental design options are selected using some design optimality criterion, such as the estimation-based -optimality criterion and the prediction-based - and -optimality criteria (see, e.g., [Atkinson and Donev, 1992] and [Myers and Montgomery, 2002]). Rightly so, such an approach is often criticized because the design selection is then based on one-number summaries of the properties of the design options. The -optimality criterion, for example, favours experimental designs that have the smallest maximum prediction variance over the region of interest without considering the distribution of the magnitude of the prediction variance throughout that region. This shortcoming of the -optimality criterion is overcome by variance dispersion plots and fraction of design space plots, which provide a detailed picture of the predictive quality of experimental designs throughout the entire region of interest. That software packages such as Design Expert and JMP have implemented similar graphical methods for evaluating the predictive performance of experimental designs should therefore be lauded. One regrettable aspect of the vast literature on the construction of variance dispersion graphs and fraction of design space plots is the emphasis on the use of scaled prediction variances.

A New Design Criterion for Robust Parameter Experiments

Varying noise factors in an experiment allows us to find robust conditions against such variation. The levels of the noise factors need to be selected carefully. Levels that are too wide may be infeasible or too costly, and if too narrow they may not provide a good model for prediction and control purposes. We propose a noise-factor separation (NFS) criterion for designs used in robust parameter design, in which a design is preferred to another if it provides the same expected mean square error for the noise part of the fitted model but for a smaller range of the noise factors in uncoded units. We evaluate several experimental designs that can be used to fit a response model that is quadratic in the controllable factors and also contains noise main effects and noise × control interactions. It is shown how the new criterion is related to variance dispersion graphs for the slope and may be used in conjunction with these graphs to assess a given experimental design. The new criterion is incorporated into an optimization formulation used to find new three-level designs that also includes traditional design criteria, such a D-efficiency. A Genetic algorithm was developed to solve such formulation. It is shown how the new designs are competitive in terms of design size, noise-factor separation, and variance dispersion for the mean and slope with respect to composite mixed resolution designs.

Three-Dimensional Variance Dispersion Graphs for Split-Plot Designs

Complete randomization for many industrial and agricultural experiments is frequently impractical due to constraints in time, cost, or existence of one or more hard-to-change factors. In these situations, restrictions on randomization may lead to split-plot designs, allowing certain factor levels to be randomly applied to the whole plot units and remaining factor levels randomly applied to the subplot units. Separate random errors in whole and subplot units from the two randomizations results in a compound symmetric error structure, which affects estimation, inference, and choice of design. In this article, we consider the prediction properties of split-plot designs, expanding the comparison between designs beyond parameter estimation properties and present three-dimensional variance dispersion graphs. These graphs can be used for evaluating the prediction capability of split-plot designs as well as for developing design strategies. We demonstrate, through several examples, that three-dimensional variance dispersion plots offer a more comprehensive study of competing designs than what is offered when comparing designs with single number optimality criteria (such as D-, G-, and V-criteria).

Prediction Variance Properties of Second-Order Designs for Cuboidal Regions

Much information is available about the construction and evaluation of the prediction variance properties of response surface designs for fitting a second-order model in a spherical region of interest. There is less information available about the prediction variance properties of second-order designs for a cuboidal region. In this paper, we construct and evaluate designs that are appropriate for the second-order model in cuboidal regions of interest. Several standard experimental designs are investigated and some new designs created using the G and I optimality criteria. We evaluate the designs using the variance of the predicted response over the design region. Variance dispersion graphs and fraction of design space plots are utilized to characterize the scaled prediction variance properties of the most promising designs.

Evaluating Mixture-Process Designs with Control and Noise Variables

Mixture-process experiments involve two distinct types of variables. The mixture variables are varied through their relative proportions and cannot be adjusted independently. The process variables can be varied independently of one another and of the mixture components. When some of the process variables are noise variables, variables that cannot be controlled in normal process operation, we are in a robust design setting. In these situations, it is customary to fit a response model combining mixture, process, and noise variables and to derive a model for the mean response and a model for the variance of the response for this response model. The variance model is directly related to the directional derivative (slope) of the response surface in the directions of the noise variables. We develop expressions for the scaled and unsealed prediction variances of both the mean and slope models. We then demonstrate the use of variance dispersion graphs (VDGs) and fraction of design space (FDS) plots of the prediction variance values to evaluate a variety of competing designs for such settings.

Fraction of Design Space Plots for Assessing Mixture and Mixture-Process Designs

Variance dispersion graphs are useful tools for evaluating various types of designs, including mixture and mixture-process designs. They allow an experimenter to see patterns of scaled prediction variances (SPV) throughout a design space. We introduce a complementary fraction of design space (FDS) plot that provides additional information on the distribution of the SPV throughout a design space. These plots display the fraction of design space where the SPV is less than or equal to specific values. The FDS plots for combined mixture-process experiments also show which of the two types of variables has more influence on the SPV. The FDS plots for mixture and mixture-process experiments are developed and then demonstrated with several examples.

Three-Dimensional Variance Dispersion Graphs for Mixture-Process Experiments

In a mixture experiment, the design factors are the proportions of the components of a mixture, and the response variables depend only on these component proportions. In addition to the mixture components, the experimenter may be interested in other variables that can be varied independently of one another and of the mixture components. Such mixture-process experiments are common in industry. There are many strategies based on different design criteria that are used to create designs involving both types of variables. We develop variance dispersion graphs (VDGs) to evaluate mixture-process designs and illustrate how the graphs are used with two examples.

Fraction of Design Space to Assess Prediction Capability of Response Surface Designs

Variance Dispersion Graphs (VDGs) are useful summaries for comparing competing designs on a fixed design space. However, they do not give all useful information about the prediction capability of a design. We propose the Fraction of Design Space (FDS) technique, which addresses some of the shortcomings of VDGs. The new technique gives the researcher more detailed information by quantifying the fraction of design space where the scaled prediction variance (SPV) is less than or equal to any pre-specified value. The FDS graph gives the researcher information about the distribution of the SPV in the region based on the ranges and proportions of possible SPV values. Several variations on the graph, including plotting the variance of the estimated mean response, are also presented to allow for specialized consideration of different designs. The FDS technique complements the use of VDGs to give the researcher more insight into the prediction capability of a design. Several standard designs with different numbers of factors are studied with both methods.