Profile control charts based on nonparametric L-1 regression methods

Abstract

Classical statistical process control often relies on univariate characteristics. In many contemporary applications, however, the quality of products must be characterized by some functional relation between a response variable and its explanatory variables. Monitoring such functional profiles has been a rapidly growing field due to increasing demands. This paper develops a novel nonparametric L-1 location-scale model to screen the shapes of profiles. The model is built on three basic elements: location shifts, local shape distortions, and overall shape deviations, which are quantified by three individual metrics. The proposed approach is applied to the previously analyzed vertical density profile data, leading to some interesting insights.

Phase I monitoring with nonparametric mixed‐effect models

“…Recently, in phase I and II profiles monitoring, several hypothesis‐testing procedures have been studied. See Jensen et al, Jensen and Birch, Zhang and Albin, Wei et al, and Abdel‐Salam et al The present article falls in this category. It is benefitted from the methodologies in Qiu et al and Yu et al and uses a multiple testing approach.…”

“…Zhang and Albin treated profiles as vectors in a high‐dimensional space and applied a χ 2 ‐type control chart to identify outliers by comparing each profile to a central vector. Also, Wei et al developed a nonparametric L1 location‐scale model to screen shapes of profiles in phase II analysis.…”

Asymptotics of nonparametric L-1 regression models with dependent data

“…Using a coupling argument to approximate the dependent process by an m -dependent process with a diverging m , we study the modulus of continuity of weighted empirical processes, and the latter result serves as a key tool in establishing our uniform Bahadur representations. These Bahadur representations provide deep insights into the asymptotic behavior of the estimates, and in particular they provide theoretical justification for the profile control chart methodologies in Wei, Zhao and Lin (2012). These technical treatments are also of interest in other nonparametric problems involving dependent data.…”

“…In this example, measurements depend on both the left and right neighboring measurements, and it is reasonable to impose the dependence structure (2.1). See Wei, Zhao and Lin (2012) for a detailed analysis. Also, as will be discussed in Section 6, the two-sided framework (2.1) can be extended to spatial lattice settings.…”

“…The basic idea is to use all but one subject to do model fitting, validate the fitted model using the left-out subject, and finally choose the optimal bandwidth by minimizing the overall prediction error: $$b_{\text{LS}}^{\ast }=\underset{b}{\text{argmin}}\sum _{i=1}^n\sum _{j=1}^{m_i}{\{Y_{i,j}-{\stackrel{\sim }{\mu }}^{(-i)}(x_{i,j})\}}^2,$$ where μ̃ ( −i ) ( x ) represents the estimator of μ ( x ) based on data from all but i th subject. As in Wei, Zhao and Lin (2012), we replace the square loss by absolute deviation: $$b_{\text{LAD}}^{\ast }=\underset{b}{\text{argmin}}\sum _{i=1}^n\sum _{j=1}^{m_i}\mid Y_{i,j}-{\stackrel{\sim }{\mu }}^{false(-ifalse)}false(x_{i,j}false)\mid .$$…”