Lulu Kang

Regression-Based Inverse Distance Weighting With Applications to Computer Experiments

Inverse distance weighting (IDW) is a simple method for multivariate interpolation but has poor prediction accuracy. In this article we show that the prediction accuracy of IDW can be substantially improved by integrating it with a linear regression model. This new predictor is quite flexible, computationally efficient, and works well in problems having high dimensions and/or large datasets. We also develop a heuristic method for constructing confidence intervals for prediction. This article has supplementary material online.

Bayesian Optimal Single Arrays for Robust Parameter Design

It is critical to estimate control-by-noise interactions in robust parameter design. This can be achieved by using a cross array, which is a cross product of a design for control factors and another design for noise factors. However, the total run size of such arrays can be prohibitively large. To reduce the run size, single arrays are proposed in the literature, where a modified effect hierarchy principle is used for the optimal selection of the arrays. In this article, we argue that effect hierarchy principle should not be altered for achieving the robustness objective of the experiment. We propose a Bayesian approach to develop single arrays which incorporate the importance of control-by-noise interactions without altering the effect hierarchy. The approach is very general and places no restrictions on the number of runs or levels or type of factors or type of designs. A modified exchange algorithm is proposed for finding the optimal single arrays. MATLAB code for implementing the algorithm is available as supplemental material in the online version of this article on the Technometrics web site. We also explain how to design experiments with internal noise factors, a topic that has received scant attention in the literature. The advantages of the proposed approach are illustrated using several examples.

Design and Modeling Strategies for Mixture-of-Mixtures Experiments

In mixture-of-mixtures experiments major components are defined as the components which themselves are mixtures of some other components, called minor components. Sometimes components are divided into different categories where each category is called a major component and the components within a major component become minor components. The special structure of the mixture-of-mixtures experiment makes the design and modeling approaches different from a typical mixture experiment. In this article we propose a new model called the major–minor model to overcome some of the limitations of the commonly used multiple-Scheffé model. We also provide a strategy for designing experiments that are much smaller in size than those based on the existing methods. We then apply the proposed design and modeling approach to a mixture-of-mixtures experiment conducted to formulate a new potato crisp. This article has supplementary material online.

Kernel Approximation: From Regression to Interpolation

In this paper we introduce a new interpolation method, known as kernel interpolation (KI), for modeling the output from expensive deterministic computer experiments. We construct it by repeating a generalized version of the classic Nadaraya--Watson kernel regression an infinite number of times. Although this development is numerical, we are able to provide a statistical framework for KI using a nonstationary Gaussian process. This enables us to quantify the uncertainty in the predictions as well as estimate the unknown parameters in the model using the empirical Bayes method. Through some theoretical arguments and numerical examples, we show that KI has better prediction performance than the popular kriging method in certain situations.

Product defect rate confidence bound with attribute and variable data

The upper confidence bound for a product defect rate is a very important index for evaluating the production process in industry. In this paper, we provide a bootstrap methodology to construct a (1−α)100% upper confidence bound for the overall defect rate of a product whose quality assessment involves multiple pass/fail binary data and multiple continuous data. When only the pass/fail data are included we propose using a bootstrap method which is consistent with the Clopper–Pearson one-sided confidence interval. When only the continuous data are included the BCa bootstrap method is recommended. These two methods are combined to provide an upper confidence bound for the overall defect rate of the product when multiple pass/fail binary data and multiple continuous data are present. All methods are clearly stated in algorithmic form, investigated through simulation and demonstrated using example data sets. In the simulation studies and examples the proposed algorithms show great advantages in both coverage probability and computational efficiency. Copyright

Comparing the Slack-Variable Mixture Model With Other Alternatives

There have been many linear regression models proposed to analyze mixture experiments including the Scheffé model, the slack-variable model, and the Kronecker model. The use of the slack-variable model is somewhat controversial within the mixture experiment research community. However, in situations that the slack-variable ingredient is used to fill in the formulation and the remaining ingredients have constraints such that they can be chosen independently of one another, the slack-variable model is extremely popular by practitioners mainly due to the ease of interpretation. In this article, we advocate that for some mixture experiments the slack-variable model has appealing properties including numerical stability and better prediction accuracy when model-term selection is performed. We also explain how the effects of the slack-variable model components should be interpreted and how easy it is for practitioners to understand the components effects. We also investigate how to choose the slack-variable component, what transformation should be used to reduce collinearity, and under what circumstances the slack-variable model should be preferred. Both simulation and practical examples are provided to support the conclusions.

A Bayesian hierarchical model for quantitative and qualitative responses

ABSTRACT In many science and engineering systems both quantitative and qualitative output observations are collected. If modeled separately the important relationship between the two types of responses is ignored. In this article, we propose a Bayesian hierarchical modeling framework to jointly model a continuous and a binary response. Compared with the existing methods, the Bayesian method overcomes two restrictions. First, it solves the problem in which the model size (specifically, the number of parameters to be estimated) exceeds the number of observations for the continuous response. We use one example to show how such a problem can easily occur if the design of the experiment is not proper; all the frequentist approaches would fail in this case. Second, the Bayesian model can provide statistical inference on the estimated parameters and predictions, whereas it is not clear how to obtain inference using the latest method proposed by Deng and Jin (2015), which jointly models the two responses via constrained likelihood. We also develop a Gibbs sampling scheme to generate accurate estimation and prediction for the Bayesian hierarchical model. Both the simulation and the real case study are shown to illustrate the proposed method.

Strategies for Formulations Development: A Step-by-Step Guide Using JMP®

Strategies for Formulations Development: A Step-by-Step Guide Using JMP® by Ronald D. Snee and Roger W. Hoerl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lulu Kang Basic Experimental Strategies and Data Analysis for Science and Engineering by John Lawson and John Erjavec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theresa L. Utlaut