Likelihood and Bayesian Methods in SPC
Session Chair: Will Guthrie, NISTThe Missing-Index Problem in Process Control
Joseph G. Voelkel
Rochester Institute of Technology
Abstract: Many processes in which routine data are collected are indexed on two variables, the first of which is time. The second index variable could be spindle, station, or cavity, for example. When this second index is known, process analysis can be performed in several ways - these methods allow for estimates of mean differences due to time and to the second variable. We briefly review these methods. However, it is not unusual for information on the index of the second variable to be missing: in an eight-spindle machine, eight parts may be sampled and measured consecutively, but the first part sampled may have come from any of the eight spindles; in a six-cavity mold, the six parts may be ejected into a bin and the cavity-index information is lost; in a sixty-four-cavity mold, eight parts may be sampled at random from one shot each hour, again without identifying information.
For the first two of these three situations, we investigate what information may be extracted about mean differences due to the missing-index variable. We formulate several models consistent with missing-index scenarios and frame these in the context of finite-mixture models; consider a likelihood approach using the EM algorithm; and examine these methods on both real and simulated data sets.
Pareto Charting Using Unsupervised, Freestyle Text Data and Two-Stage "Human-Assisted Modeling"
Theodore T. Allen and Hui P. Xiong
The Ohio State University
Abstract: This article proposes a method for Pareto charting that is based on unsupervised, freestyle text such as rework or scrap cause descriptions. The proposed procedure is based on Bayesian clustering techniques and a novel two-stage human assisted modeling (HAM) framework that permits subject matter experts (SMEs) or other users to introduce "high-level" data into the process after an initial analysis phase. All of the proposed methods are intended to foster interpretable causal information supporting easily identified actions. The methods are illustrated using a real-world case study which involves scrap data from a stone manufacturer and an image database.
Controlling Attribute Type Data From a Bayesian Perspective
Panagiotis Tsiamyrtzis
Athens University of Economics & Business.
Abstract: We consider a process for which one or more quality characteristics are used to classify the outcome as acceptable or not. As a result we obtain attribute data of Binomial type distribution with unknown "success probability" that expresses the percent of nonconforming. In this work we focus in modeling dynamically this probability and detect (in an online fashion) of whether it shifts to higher values (i.e. we have process degradation) or not. The modeling will be based on a sequentially updated mixture of Beta distributions that will allow the process readings to contribute as they become available, making the approach particularly attractive in case of Phase I, or in general short run, attribute data. Decision making and prediction issues will be provided.
A Multivariate CUSUM Chart Accommodating Prior Information About Potential Shifts
Wei Liu and Peihua Qiu
University of Minnesota
We consider statistical process control (SPC) for Phase II monitoring of the process mean when process measurements are multivariate. In cases when the direction of a potential mean shift is known, Healy (1987) generalized the univariate cumulative sum (CUSUM) chart to multivariate cases, using the likelihood ratio inferences, and the generalized CUSUM chart was shown to be efficient. Other existing multivariate control charts usually do not use any prior information about the potential mean shift, making them less efficient in cases when such prior information is available. In this paper, we suggest a multivariate CUSUM chart for applications in which the mean shift direction is not completely known but it follows a prior distribution. This chart can be regarded as a compromise of Healy’s CUSUM chart and other existing multivariate control charts, in terms of proper accommodation of prior information about the potential shift. Numerical studies show that it performs well in cases when such prior information is available.