Robust Parameter Design: A Penalized Likelihood Approach

Open Access
Author:
Kankam, Kwame Adu
Graduate Program:
Statistics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
September 29, 2014
Committee Members:
  • James Landis Rosenberger, Dissertation Advisor
  • James Landis Rosenberger, Committee Chair
  • Dennis Kon Jin Lin, Committee Member
  • Matthew Logan Reimherr, Committee Member
  • Lingzhou Xue, Committee Member
  • Enrique Del Castillo, Committee Member
Keywords:
  • Penalized Likelihood
  • Design of Experiments
  • Coordinate Descent
Abstract:
Robust Parameter Design (RPD) is an engineering methodology aimed at designing quality into industrial products and processes by optimizing a quality characteristic with respect to the controllable input variables. It involves designing a system to withstand unavoidable variation while meeting its intended goal. Proposed and popularized by Japanese engineer and quality expert, Genichi Taguchi, it has entered the statistical mainstream and several approaches have been proposed. In the dual response RPD, response surfaces for the mean and the variance are obtained and depending on the goal of the experiment an objective function is determined. This is then optimized with respect to the control factors in order to find the best levels at which industrial processes should be carried out in order to obtain the best products. In this dissertation, we propose a penalized likelihood approach to the dual response RPD which we call Adaptive Penalized Likelihood Effects Selection (APLES). We begin with a heteroscedastic linear model and specify a parametric variance function (specifically the log-linear variance function). We maximize the loglikelihood subject to constraints on the $L_1$ norms of the mean and variance parameter vectors in order to obtain simultaneous variable selection and estimation of the mean and variance parameters. For fixed values of the variance parameters, the problem reduces to a weighted least squares regression with an adaptive Lasso penalty. For fixed values of the mean parameters, the problem is equivalent to a gamma-error generalized linear model (GLM) with log link and an adaptive Lasso penalty. By iterating between these two minimization problems, we obtain the final set of APLES estimates for the non-zero mean and variance parameters. We describe and utilize the cyclic coordinate descent (CCD) algorithm which is very fast and has good convergence properties for the data sets that typically arise in applications. We apply APLES to RPD and show using simulations that it has good performance in a wide variety of settings for the mean parameters, variance parameters, noise variables, and different choices of `quality' objective functions. We also illustrate the use of APLES by analyzing some well-known data sets from the literature. Another advantage of our approach is that it can be used for identification of location and dispersion effects in screening experiments. Screening experiments are the initial experiments carried out on a new process in which a potentially large number of factors are studied. Unlike RPD, the aim is not to optimize the subsequent response surfaces, but to select important variables for subsequent experimentation. We show that APLES performs equally well in these situations compared to other methods.