# Analysis and Optimization of Profile and Shape Response Experiments

Open Access

- Graduate Program:
- Industrial Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 24, 2011
- Committee Members:
- Enrique Del Castillo, Dissertation Advisor
- Enrique Del Castillo, Committee Chair
- M Jeya Chandra, Committee Member
- Paul Griffin, Committee Member
- Murali Haran, Committee Member

- Keywords:
- Profile
- Shape
- Gaussian Process
- Geometry
- shape optimization
- Statistical shape analysis
- process optimization.

- Abstract:
- An engineering process that exhibits a response in the form of a univariate (or one-dimensional) curve whenever new experimental conditions are tried is said to have a profile, or functional response. Likewise, a manufacturing process or engineering system where the response of interest is the geometry of a product or part is said to have a shape response. A shape response can relate to a planar (two-dimensional) geometrical feature, or to a three-dimensional one. The overall theme of this dissertation is the modeling and optimization of engineering processes that have either a profile or a shape response. The models and methods described in this dissertation have application mainly in manufacturing, engineering design, and computer experiments. Statistical Shape Analysis (SSA) is a relatively new area within Statistics. Traditionally the realm of biological applications, it has been recently applied to manufacturing problems. D.G. Kendall, in pioneering work conducted in the 1980’s, defined the shape of an object as the geometrical information that remains once certain similarity transformations, namely, rotations excluding reflections, translations, and dilatations (or dilations) are filtered out. His work is based on a landmark representation of an object, where a landmark consists of the coordinates of a point measured on the object together with a label, with labels that correspond from object to object. This representation turns out to be relevant in manufacturing, since data obtained using a coordinate measuring machine will typically have this appearance. Over the last 20 years, several SSA tests have been proposed to detect differences in the mean shape between objects, but little work exists on the relative merits of these methods. The first part of this dissertation consists of a comprehensive performance analysis of landmark-based tests for mean shape differences. Since the performance of these tests depends on the types of shapes being tested, we consider both shapes that have been studied in the scarce extant literature on the subject, namely triangles and arbitrary polygons with few landmarks, and also consider shapes of specific interest in manufacturing applications, such as circular and cylindrical geometries with tens to hundreds of landmarks. An additional problem studied in this dissertation is that of shape optimization, that is, find the best operating conditions that lead to the most desirable shape of the product under fabrication. Previous tests for shape differences are based on Kendall’s definition of shape, which neglects differences in size between objects since it removes dilation (scale) effects, and make up for this deficiency by testing separately for differences in size. As an alternative, we present statistical tests for differences in form between the objects, where we define the form of an object as the geometrical information that remains once the effect of rotations and translations, but not dilations, is filtered out. We further develop a form optimization method when noise factors are present, proposing in effect a method for the Robust Parameter Design problem for shape (form) responses. Noise factors are factors that for the purpose of a carefully designed experiment are controllable, but that during normal operation of a production process or during use of a product vary randomly. The goal is to find the controllable factor conditions of the process that achieve a desired part form in the presence of noise factor variability. The second part of this dissertation deals with profile response processes, their modeling, and subsequent optimization. Methods for this type of processes are mainly based on frequentist model estimation techniques, where the uncertainty in the parameter estimates is not considered during the optimization phase. As shown by J. Peterson, neglecting the uncertainty in the parameter estimates may lead to solutions that will very unlikely achieve the desired process performance. While there exist recent work in profile response systems where a Bayesian point of view is taken for model fitting that does incorporate the uncertainty of the model parameters into the subsequent optimization phase, those models are not flexible enough as they depend on a parametric regression model that is required to fit the mean profile well. As a more flexible alternative to these prior approaches, we present new modeling and optimization methodology for profile response processes based on a spatio-temporal Gaussian Random Function (GRF) model. In this model, the space of the controllable factors corresponds to the ``space" dimension, and the space of the locations over which the profile responses are observed corresponds to the ``temporal" dimension. The temporal dimension may or may not be actually time in some applications since in general it is equivalent to the ``signal" in Taguchi's signal response models. Similarly as in the first part of this dissertation, the goal here is to find controllable factor operating conditions that will lead to a desirable profile response with highest probability in the presence of noise factor variability. The approach is fully Bayesian, incorporating the uncertainty of all process parameters present in the model which leads to more reliable predictive posterior probabilities to conformance to specifications for a given optimal solution. A discussion of robustness to the underlying assumptions and tools for checking model assumptions are provided. An adaptive Markov Chain Monte Carlo method for the fitting of the GRF model is presented that shows good convergence behavior.