Confidence Regions for Response Surface Optima based on Parametric and Nonparametric Models
Open Access
- Author:
- Chen, Peng
- Graduate Program:
- Industrial Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- August 17, 2020
- Committee Members:
- Enrique Del Castillo, Dissertation Advisor/Co-Advisor
Enrique Del Castillo, Committee Chair/Co-Chair
Necdet S Aybat, Committee Member
Eunhye Song, Committee Member
Bharath Kumar Sriperumbudur, Outside Member
Wen Shen, Outside Member
Steven James Landry, Program Head/Chair - Keywords:
- Response Surface Methodology
Regression
Random Processes
Optimization
Statistical Inference - Abstract:
- A fundamental problem in response surface analyses based on designed experiments is to locate the optimal operating conditions of an industrial process or the best design of a product. It is well-known that the global optimum of a fitted response surface model is only a point estimate for the global optimum of the true underlying function. The goal of this research is to compute an inference region, in the controllable factor space, that can be said to contain the true optimum with certain guarantees. Inference regions on response surface optima have important applications in many areas of engineering, such as industrial and chemical engineering, pharmaceutical research and development, semiconductor manufacturing, and have found recent applications in experiments conducted by Evolutionary Biologists. However, previous methods to compute the regions have their limitations. They are often based on polynomial models, which are not flexible enough to capture complex response surfaces, depend on strong distributional assumptions that are not always valid, and are mostly all frequentist methods which can only provide a confidence region but not a probability guarantee. In this research, we propose a set of algorithms to compute both Bayesian credible regions and frequentist confidence regions on the locations of the optima of response surfaces. The credible regions are based on Polynomial and Gaussian Process models. The confidence regions are based on Polynomial, Gaussian Process, and Ordinary Kriging models, some of which are distribution-free. The Bootstrap and Markov Chain Monte Carlo methods are used to develop the confidence regions and credible regions, respectively. The proposed regions are valid, unbiased, and accurate, as shown by numerical evidence. The proposed inference regions can be used as basic building blocks for more advanced experiments. For instance, in experiments with high-dimensional responses, the regions can be used to locate the optima of the first principal components of the responses. In sequential experiments, the regions can be generated iteratively to discover the optimum of a latent response function as more and more experimental runs are added. Confidence regions for some other popular response surface models, such as Bayesian LASSO Regression, Projection Pursuit Regression, and Generalized Additive Models, are also studied. The proposed algorithms are implemented as R functions, which are included in the form of an R package.