Essays on Monitoring Profile Data

Author:

Zhu, Junjia

Graduate Program:

Statistics

Degree:

Doctor of Philosophy

Document Type:

Dissertation

Date of Defense:

August 18, 2008

Committee Members:

Dennis Kon Jin Lin, Committee Chair/Co-Chair
Donald Richards, Committee Member
Damla Senturk, Committee Member
Enrique Del Castillo, Committee Member

Keywords:

Profile Monitoring
Linear Profiles
Likelihood Ratio Test
Generalized Likelihood Ratio Test
Average Run Length
Statistical Process Control

Abstract:

Profile data emerges when the quality of a product or process is characterized by a relationship between two or more variables. Each complete sampling process yields a collection of data points that can be represented by a functional curve (or profile). The purpose of this work is to study novel models and procedures regarding monitoring profile data. Motivated by a vertical density profile problem, we first focus on monitoring the slopes of linear profiles. A Shewhart-type control chart for monitoring slopes of linear profiles is proposed. Both Phase I and Phase II applications are discussed. The performance of the proposed chart in Phase I applications is demonstrated using both real-life data in an illustrative example and simulated data in a probability of signal study. It is shown that the average run length (ARL) of the proposed control chart depends only on the shifts of slopes; whereas the ARL of the multivariate $T^2$ chart depends on both the shifts of slopes and the correlation between the estimated slope and the intercept. When such a correlation is low, the proposed control chart has a better ARL performance than the $T^2$ chart. We next focus on Phase II monitoring of profile data. The goal is to monitor the functional relationship between the response variable ($Y$) with one or more explanatory variables ($x's$), which is assumed to be adequately modeled by the Generalized Linear Model (GLM). Shewhart-type and EWMA-type control charts based on the likelihood ratio test (LRT) statistic are proposed to detect the shift of on-line profiles from the baseline profile. The proposed control chart can be applied on both linear and nonlinear profiles as long as the likelihood function of the profile model can be written. Linear profiles, multiple linear regression profiles and logistic regression profiles are used to demonstrate the use of proposed control charts. It is shown that the LRT-based control charts can efficiently detect shift of parameters. In particular, the EWMA-type LRT chart has better ARL performance than the Shewhart-type LRT chart within the scope of our simulation. Finally we propose a general method based on the Generalized Likelihood Ratio Test (GLRT) to perform Phase II monitoring of profile data. Unlike most existing methods in profile monitoring area, the proposed method uses nonparametric regression to estimate the on-line profiles and thus does not have any requirement of the functional form of the profiles. Both Shewhart-type and EWMA-type control charts are considered. The ARL performance of the proposed method is studied by using a real-life nonlinear profile dataset. It is shown that the proposed GLRT-based control chart can efficiently detect both location and dispersion shifts of the on-line profiles from the baseline profile. An upper control limit (UCL) corresponding to a desired in-control ARL value is constructed.