Nonparametric and Semiparametric Nonlinear Profile Monitoring with Multiple Predictors

Monitoring data arising from a process a practitioner desires to be in-control is a typical task in Statistical Process Control (SPC). These data are generated sequentially over time, and the goal of a SPC tool called a control chart is to detect an out-of-control process with as little delay as possible while also minimizing false alarms. The assumptions made on the data generating process guides control chart design. We consider a general task called profile monitoring wherein the data being monitored are a set of noisy responses and predictors, and we wish to detect changes in the functional relationship $f$ between these two. Ideally, the profile monitoring method should minimize the number and strength of the assumptions about the data as much as possible while being computationally fast and having both desirable sensitivity and specificity. As such we consider estimating the functional relationship $f$ in either a semiparametric or nonparametric model. The dissertation involves two completed projects and preliminary results in a promising direction. The first project proposes a semiparametric approach using a single-index model (SIM), where we monitor an $l_2$ based statistic on the parametric component called the index parameter. The SIM approach outcompetes its competitors in detection delay and is the first profile monitoring method to use SIMs to model $f$. The second project provides a nonparametric control chart that is fast, has small detection delay, and is able to avoid false alarms magnitudes better than what is found in the control chart literature. Typically, control charts are designed to achieve an in-control average run length (ARL) of 200 or 370. The proposed eigenvector perturbation control chart achieves an in-control ARL that is greater that $10^6$ while needing only a small (often mere a single) number of out-of-control observations to correctly flag an alarm. The dissertation concludes with a promising direction of research. Although the estimate of a SIM in the first project is computationally fast enough for many applications, it is insufficient for streaming data. The final project aims to fill this gap and we provide preliminary results demonstrating a faster approach that empirically gives the same predictive power as its competitor.