Some notes on the construction and evaluation of supersaturated designs

Supersaturated designs offer a potentially useful way to investigate many factors with few experiments. A super‐saturated design evaluates factors with experiments, where . In some supersaturated designs the number of factors being investigated may exceed the number of experiments by a factor of three or more. The problem of developing a supersaturated design is described in computational complexity terms and methods of construction are discussed. Using standard combinatorial formulation may permit researchers to use additional methods to develop additional designs. Designs are generated with a modified method and compared with published designs using existing criteria and a new criterion. Copyright © 2002 John Wiley & Sons, Ltd.     

Examining robustness of model selection with half‐normal and LASSO plots for unreplicated factorial designs

When there are constraints on resources, an unreplicated factorial or fractional factorial design can allow efficient exploration of numerous factor and interaction effects. A half‐normal plot is a common graphical tool used to compare the relative magnitude of effects and to identify important effects from these experiments when no estimate of error from the experiment is available. An alternative is to use a least absolute shrinkage and selection operation plot to examine the pattern of model selection terms from an experiment. We examine how both the half‐normal and least absolute shrinkage and selection operation plots are impacted by the absence of individual observations or an outlier, and the robustness of conclusions obtained from these 2 techniques for identifying important effects from factorial experiments. The methods are illustrated with 2 examples from the literature.     

Influence Plots for LASSO

With many predictors in regression, fitting the full model can induce multicollinearity problems. Least Absolute Shrinkage and Selection Operation (LASSO) is useful when the effects of many explanatory variables are sparse in a high‐dimensional dataset. Influential points can have a disproportionate impact on the estimated values of model parameters. This paper describes a new influence plot that can be used to increase understanding of the contributions of individual observations and the robustness of results. This can serve as a complement to other regression diagnostics techniques in the LASSO regression setting. Using this influence plot, we can find influential points and their impact on shrinkage of model parameters and model selection. We illustrate the methods with two examples. Copyright © 2016 John Wiley & Sons, Ltd.     

An improved compromise ranking method for evaluation of environmentally conscious manufacturing programs

An environmentally responsible or environmentally conscious manufacturing (ECM) program addresses the environmental impact of the interrelated decisions that are made at various stages of product life, from conception to design, raw ECM programs consumption, processing, delivery, use, recycling and/or disposal. The evaluation of alternative ECM programs for producing a given product is similar to many strategic initiatives and their justification methodologies. This similarity arises from the fact that there are multiple attributes that need to be considered, many of which have long term and broad implications for an organisation. Only a few decision making methods were proposed in the past for ECM program evaluation for producing a given product. There is a need for a systematic and logical scientific method or mathematical tool to guide organisations in taking a proper decision about ECM program evaluation. This paper presents a multiple attribute decision making methodology for evaluation of environmentally conscious manufacturing programs for producing a given product. The methodology is based on an improved compromise ranking method considering the ECM program selection attributes and their relative importance. The proposed ‘ECM program selection index’ helps to evaluate and rank the ECM programs for producing a given product. An example is included to illustrate the approach.     

Pitfalls of using a single criterion for selecting experimental designs

For surrogate construction, a good experimental design (ED) is essential to simultaneously reduce the effect of noise and bias errors. However, most EDs cater to a single criterion and may lead to small gains in that criterion at the expense of large deteriorations in other criteria. We use multiple criteria to assess the performance of different popular EDs. We demonstrate that these EDs offer different trade‐offs, and that use of a single criterion is indeed risky. In addition, we show that popular EDs, such as Latin hypercube sampling (LHS) and D‐optimal designs, often leave large regions of the design space unsampled even for moderate dimensions. We discuss a few possible strategies to combine multiple criteria and illustrate them with examples. We show that complementary criteria (e.g. bias handling criterion for variance‐based designs and vice versa) can be combined to improve the performance of EDs. We demonstrate improvements in the trade‐offs between noise and bias error by combining a model‐based criterion, like the D‐optimality criterion, and a geometry‐based criterion, like LHS. Next, we demonstrate that selecting an ED from three candidate EDs using a suitable error‐based criterion helped eliminate potentially poor designs. Finally, we show benefits from combining the multiple criteria‐based strategies, that is, generation of multiple EDs using the D‐optimality and LHS criteria, and selecting one design using a pointwise bias error criterion. The encouraging results from the examples indicate that it may be worthwhile studying these strategies more rigorously and in more detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Robustness of response surface designs to missing data

In this paper, minimax loss response surface designs are constructed. These designs are more robust to one missing design point than the original designs. The proposed designs are compared with the designs in the literature, and they are better in terms of loss and number of runs. Moreover, the new suggestion for the value of α generates designs not only with less losses but also with higher D‐efficiency.     

Estimation of missing observations in two‐level split‐plot designs

Inserting estimates for the missing observations from split‐plot designs restores their balanced or orthogonal structure and alleviates the difficulties in the statistical analysis. In this article, we extend a method due to Draper and Stoneman to estimate the missing observations from unreplicated two‐level factorial and fractional factorial split‐plot (FSP and FFSP) designs. The missing observations, which can either be from the same whole plot, from different whole plots, or comprise entire whole plots, are estimated by equating to zero a number of specific contrast columns equal to the number of the missing observations. These estimates are inserted into the design table and the estimates for the remaining effects (or alias chains of effects as the case with FFSP designs) are plotted on two half‐normal plots: one for the whole‐plot effects and the other for the subplot effects. If the smaller effects do not point at the origin, then different contrast columns to some or all of the initial ones should be discarded and the plots re‐examined for bias. Using examples, we show how the method provides estimates for the missing observations that are very close to their actual values. Copyright © 2007 John Wiley & Sons, Ltd.     

Selecting a near‐optimal design for multiple criteria with improved robustness to different user priorities

In a decision‐making process, relying on only one objective can often lead to oversimplified decisions that ignore important considerations. Incorporating multiple, and likely competing, objectives is critical for balancing trade‐offs on different aspects of performance. When multiple objectives are considered, it is often hard to make a precise decision on how to weight the different objectives when combining their performance for ranking and selecting designs. We show that there are situations when selecting a design with near‐optimality for a broad range of weight combinations of the criteria is a better test selection strategy compared with choosing a design that is strictly optimal under very restricted conditions. We propose a new design selection strategy that identifies several top‐ranked solutions across broad weight combinations using layered Pareto fronts and then selects the final design that offers the best robustness to different user priorities. This method involves identifying multiple leading solutions based on the primary objectives and comparing the alternatives using secondary objectives to make the final decision. We focus on the selection of screening designs because they are widely used both in industrial research, development, and operational testing. The method is illustrated with an example of selecting a single design from a catalog of designs of a fixed size. However, the method can be adapted to more general designed experiment selection problems that involve searching through a large design space.     

Using a genetic algorithm to generate D‐optimal designs for mixture‐process variable experiments

This article presents and develops a genetic algorithm (GA) to generate D‐efficient designs for mixture‐process variable experiments. It is assumed the levels of a process variable are controlled during the process. The GA approach searches design points from a set of possible points over a continuous region and works without having a finite user‐defined candidate set. We compare the performance of designs generated by the GA with designs generated by two exchange algorithms (DETMAX and k‐exchange) in terms of D‐efficiencies and fraction of design space (FDS) plots which are used to evaluate a design's prediction variance properties. To illustrate the methodology, examples involving three and four mixture components and one process variable are proposed for creating the optimal designs. The results show that GA designs have superior prediction variance properties in comparison with the DETMAX and k‐exchange algorithm designs when the design space is the simplex or is a highly‐constrained subspace of the simplex.     

Three‐level and mixed‐level orthogonal arrays for lean designs

Orthogonal arrays (OA's) are widely used in design of experiments. Each OA has a specific number of rows that is fixed by the number of factors in the OA and the number of levels in each factor. In a practical application of an industrial experiment, however, because of various operational constraints it could happen that the number of runs of the experiment cannot be set exactly equal to the number of rows of an OA. In this case, a lean design can be used. A lean design is obtained by removing some specific rows and columns from the extended design matrix formed from an OA, so that the resulting sub‐matrix still allows efficient estimation of the effects of some of the factors. Tables for 2‐level lean designs are already available in the literature. In this paper, the authors will investigate 3‐level lean designs and mixed‐level lean designs, and construct tables for such designs for convenient use. Copyright © 2009 John Wiley & Sons, Ltd.     

Graphical Methods for Influential Data Points in Cluster Analysis

In cluster analysis, many numerical measures to detect which data points are influential have been proposed in the past literature. These numerical measures provide only limited information about which data points are influential but fail to reveal deeper relationships between the observations. They describe an overall pattern but fail to provide details about the mechanism that exists among the influential data points. In this paper, several graphical methods are described for detecting this mechanism. In the process, each data point is decomposed to show the pattern, how it influences other observations and the partitioning in cluster analysis. The approach also allows comparison of different clustering methods and how these options impact the relationship between observations. Copyright © 2014 John Wiley & Sons, Ltd.     

Firework Plots for Evaluating the Impact of Outliers and Influential Observations in Ridge Regression

With many predictors in regression, fitting the full model can induce multicollinearity problems. Thus, ridge regression provides a beneficial means of stabilizing the coefficient estimates in the fitted model. Outliers can distort many measures in data analysis and statistical modeling, while influential points can have disproportionate impact on the estimated values of model parameters. Graphical summaries, called firework plots, are simple tools for evaluating the impact of outliers and influential points in regression. Variations of the plots focus on allowing visualization of the impact on the estimated parameters and variability. This paper describes how three‐dimensional and pairwise firework plots as well as scalable waterfall–firework plots can be used to increase understanding of contributions of individual observations and as a complement to other regression diagnostic techniques in the ridge regression setting. Using these firework plots, we can find outliers and influential points and their impact on model parameters and show how in some applications, the type of analysis used changes the impact of various observations. We illustrate the methods with two examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Firework Plot as a Graphical Exploratory Data Analysis Tool for Evaluating the Impact of Outliers in Data Exploration and Regression

Outliers can distort many measures for data analysis. We propose a new set of graphical summaries, called firework plots, as simple tools for evaluating the impact of outliers in data exploration and regression assessment. One variation of the plot focuses on the impact of extreme observations on the mean and standard deviation by using curves that trace the relative contribution to the overall summary as weights for individual observations are changed from 1 to 0 in a univariate data set. Similarly, other variations for bivariate data allow examination of the impact of changing weights on combinations of the correlation coefficient and mean with two‐ or three‐dimensional firework plots. One variation of the plot focuses on the impact on the estimated intercept, the estimated slope, and the estimated standard deviation by using curves based on the relative contribution to the overall summary as weights for individual observations are changed from 1 to 0 in a simple linear regression analysis. Similarly, other variations for a multiple regression allow the practitioner to examine the impact of changing weights on combinations of the estimated regression coefficients and the standard error with the pairwise firework plot matrix. Copyright © 2013 John Wiley & Sons, Ltd.     

Collaborative Design of Hollow Nanocubes,In Situ Cross‐Linked Binder,and Amorphous Void@SiOx@C as a Three‐Pronged Strategy for Ultrastable Lithium Storage

Although silicon‐based materials are ideal candidate anodes for high energy density lithium‐ion batteries, the large volumetric expansion seriously damages the integrity of the electrodes and impedes commercial processes. Reasonable electrode design based on adjustable structures of silicon and strong binders prepared by a facile method is still a great challenge. Herein, a three‐pronged collaborative strategy via hollow nanocubes, amorphous Void@SiO_x@C, and in situ cross‐linked polyacrylic acid and d ‐sorbitol 3D network binder (c‐PAA‐DS) is adopted to maintain structural/electrode integrality and stability. The all‐integrated c‐PAA‐DS/Void@SiO_x@C electrode delivers excellent mechanical property, which is attributed to ductility of the c‐PAA‐DS binder and high adhesion energy between Void@SiO_x@C and c‐PAA‐DS calculated by density functional theory. Benefiting from the synergistic effect of accommodation of the hollow structure, protection of outer carbon shell, amorphous Void@SiO_x@C, and strong adhesive c‐PAA‐DS binder, c‐PAA‐DS/Void@SiO_x@C shows excellent electrochemical performance. Long cycling stability with a reversible capacity of 696 mAh g^?1 is obtained, as well as tiny capacity decay after 500 cycles at 0.5 A g^?1 and high‐rate performance. The prelithiated Void@SiO_x@C||LiNi_0.5Co_0.2Mn_0.3O₂ (NCM523) full cell is also assembled and shows a reversible capacity of 157 mAh g^?1 at 0.5 C, delivering an excellent capacity retention of 94% after 160 cycles.     

Estimation of three‐dimensional stress intensity factor for structural ‘T’ details

Three‐dimensional (3D) opening mode stress intensity factors (SIFs) for structural steel‐welded ‘T’ details were investigated by the finite element method. A 3D shape‐dependent correction factor is proposed for semi‐elliptical surface cracks. The aspect ratio (a/c) of a semi‐elliptical crack plays a key role in the approximation of 3D‐SIF values, and in the present study, it was estimated for a 3D crack analysis. The estimated 3D‐SIF was determined through a correlation between the a/c ratio and the two‐dimensional SIF for semi‐elliptical cracks in the thickness direction adjacent to the web‐flange junction of a welded ‘T’. The resulting equation can be used to estimate the 3D‐SIF values from the two‐dimensional SIF without much ambiguity.     

A single camera three‐dimensional digital image correlation system for the study of adiabatic shear bands

We describe the capability of a high‐resolution three‐dimensional digital image correlation (DIC) system specifically designed for high strain‐rate experiments. Utilising open‐source camera calibration and two‐dimensional DIC tools within the MATLAB framework, a single camera three‐dimensional DIC system with submicron displacement resolution is demonstrated. The system has a displacement accuracy of up to 200 times the optical spatial resolution, matching that achievable with commercial systems. The surface strain calculations are benchmarked against commercially available software before being deployed on quasi‐static tests showcasing the ability to detect both in‐ and out‐of‐plane motion. Finally, a high strain‐rate (1.2×10³ s^?1) test was performed on a top‐hat sample compressed in a split‐Hopkinson pressure bar in order to highlight the inherent camera synchronisation and ability to resolve the adiabatic shear band phenomenon.