V. Roshan Joseph

Blind Kriging: A New Method for Developing Metamodels

Kriging is a useful method for developing metamodels for product design optimization. The most popular kriging method, known as ordinary kriging, uses a constant mean in the model. In this article, a modified kriging method is proposed, which has an unknown mean model. Therefore, it is called blind kriging. The unknown mean model is identified from experimental data using a Bayesian variable selection technique. Many examples are presented, which show remarkable improvement in prediction using blind kriging over ordinary kriging. Moreover, a blind kriging predictor is easier to interpret and seems to be more robust against mis-specification in the correlation parameters.

Structured variable selection and estimation

In linear regression problems with related predictors, it is desirable to do variable selection and estimation by maintaining the hierarchical or structural relationships among predictors. In this paper we propose non-negative garrote methods that can naturally incorporate such relationships defined through effect heredity principles or marginality principles. We show that the methods are very easy to compute and enjoy nice theoretical properties. We also show that the methods can be easily extended to deal with more general regression problems such as generalized linear models. Simulations and real examples are used to illustrate the merits of the proposed methods.

An Efficient Variable Selection Approach for Analyzing Designed Experiments

The analysis of experiments in which numerous potential variables are examined is driven by the principles of effect sparsity, effect hierarchy, and effect heredity. We propose an efficient variable selection strategy to specifically address the unique challenges faced by such analysis. The proposed methods are natural extensions of the LARS general-purpose variable selection algorithm. They can be computed very rapidly and can find sparse models that better satisfy the goals of experiments. Simulations and real examples are used to illustrate the wide applicability of the proposed methods.

Statistical Adjustments to Engineering Models

Statistical models are commonly used in quality-improvement studies. However, such models tend to perform poorly when predictions are made away from the observed data points. On the other hand, engineering models derived using the underlying physics of the process do not always match satisfactorily with reality. This article proposes engineering—statistical models that overcome the disadvantages of engineering models and statistical models. The engineering—statistical model is obtained through some adjustments to the engineering model using experimental data. The adjustments are done in a sequential way and are based on empirical Bayes methods. We also develop approximate frequentist procedures for adjustments that are computationally much easier to implement. The usefulness of the methodology is illustrated using a problem of predicting surface roughness in a microcutting process and the optimization of a spot-welding process.

Composite Gaussian process models for emulating expensive functions

A new type of nonstationary Gaussian process model is developed for approximating computationally expensive functions. The new model is a composite of two Gaussian processes, where the first one captures the smooth global trend and the second one models local details. The new predictor also incorporates a flexible variance model, which makes it more capable of approximating surfaces with varying volatility. Compared to the commonly used stationary Gaussian process model, the new predictor is numerically more stable and can more accurately approximate complex surfaces when the experimental design is sparse. In addition, the new model can also improve the prediction intervals by quantifying the change of local variability associated with the response. Advantages of the new predictor are demonstrated using several examples.

A Bayesian Approach to the Design and Analysis of Fractionated Experiments

Specifying a prior distribution for the large number of parameters in the statistical model is a critical step in a Bayesian approach to the design and analysis of experiments. This article shows that the prior distribution can be induced from a functional prior on the underlying transfer function. The functional prior requires specification of only a few hyperparameters and thus can be easily implemented in practice. The usefulness of the approach is demonstrated through the analysis of some experiments. The article also proposes a new class of design criteria and establishes their connections with the minimum aberration criterion.

Quality Loss Functions for Nonnegative Variables and Their Applications

Loss functions play a fundamental role in every quality engineering method. A new set of loss functions is proposed based on Taguchis societal loss concept. Its applications to robust parameter design are discussed in detail. The loss functions are shown to posses some interesting properties and lead to theoretical results that cannot be handled with other loss functions.

Functionally Induced Priors for the Analysis of Experiments

This work develops the idea of using functional priors for the design and analysis of three-level and higher-level experiments. Developing a prior distribution for model parameters is challenging, because a factor can be qualitative or quantitative. We propose appropriate correlation functions and coding schemes so that the prior distribution is simple and the results are interpretable. The prior incorporates well-known principles, such as effect hierarchy and effect heredity, which helps resolve the aliasing problems in fractional designs almost automatically. The usefulness of the new approach is illustrated through the analysis of some real experiments.

Robust Parameter Design With Feed-Forward Control

When there exists strong noise factors in the process, robust parameter design alone may not be effective, and a control strategy can be used to compensate for the effect of noise. In this article a parameter design methodology in the presence of a feed-forward control is developed. In particular, performance measures for evaluating control factor settings in measurement systems, simple response systems, and multiple target systems are developed. Strategies for the design and analysis of experiments are discussed. The approach is illustrated using an example on gold plating.

Statistical Modeling and Analysis for Robust Synthesis of Nanostructures

We systematically investigate the best process conditions that ensure synthesis of different types of one-dimensional cadmium selenide nanostructures with high yield and reproducibility. Through a designed experiment and rigorous statistical analysis of experimental data, models linking the probabilities of obtaining specific morphologies to the process variables are developed. A new iterative algorithm for fitting a multinomial generalized linear model is proposed and used. The optimum process conditions, which maximize the preceding probabilities and make the synthesis process robust (i.e., less sensitive) to variations in process variables around set values, are derived from the fitted models using Monte Carlo simulations. Cadmium selenide has been found to exhibit one-dimensional morphologies of nanowires, nanobelts, and nanosaws, often with the three morphologies being intimately intermingled within the as-deposited material. A slight change in growth condition can result in a totally different morphology. To identify the optimal process conditions that maximize the yield of each type of nanostructure and, at the same time, make the synthesis process robust (i.e., less sensitive) to variations of process variables around set values, a large number of trials were conducted with varying process conditions. Here, the response is a vector whose elements correspond to the number of appearances of different types of nanostructures. The fitted statistical models would enable nanomanufacturers to identify the probability of transition from one nanostructure to another when changes, even tiny ones, are made in one or more process variables. Inferential methods associated with the modeling procedure help in judging the relative impact of the process variables and their interactions on the growth of different nanostructures. Owing to the presence of internal noise, that is, variation around the set value, each predictor variable is a random variable. Using Monte Carlo simulations, the mean and variance of transformed probabilities are expressed as functions of the set points of the predictor variables. The mean is then maximized to find the optimum nominal values of the process variables, with the constraint that the variance is under control.