Peihua Qiu

The changepoint model for statistical process control

Statistical process control (SPC) requires statistical methodologies that detect changes in the pattern of data over time. The common methodologies, such as Shewhart, cumulative sum (cusum), and exponentially weighted moving average (EWMA) charting, require the in-control values of the process parameters, but these are rarely known accurately. Using estimated parameters, the run length behavior changes randomly from one realization to another, making it impossible to control the run length behavior of any particular chart. A suitable methodology for detecting and diagnosing step changes based on imperfect process knowledge is the unknown-parameter changepoint formulation. Long recognized as a Phase I analysis tool, we argue that it is also highly effective in allowing the user to progress seamlessly from the start of Phase I data gathering through Phase II SPC monitoring. Despite not requiring specification of the post-change process parameter values, its performance is never far short of that of the optimal cusum chart which requires this knowledge, and it is far superior for shifts away from the cusum shift for which the cusum chart is optimal. As another benefit, while changepoint methods are designed for step changes that persist, they are also competitive with the Shewhart chart, the chart of choice for isolated non-sustained special causes.

A Rank-Based Multivariate CUSUM Procedure

We consider statistical process control when measurements are multivariate. A cumulative sum (CUSUM) procedure is suggested in detecting a shift in the mean vector of the measurements, which is based on the cross-sectional antiranks of the measurements. At each time point, the measurements are ordered and their antiranks, which are the indices of the order statistics, are recorded. When the process is in control and the joint distribution of the multivariate measurements satisfies some regularity conditions, the antirank vector at each time point has a given distribution. This distribution changes to some other distribution when the process is out of control and the components of the shift in the mean vector of the process are not all the same. This CUSUM can therefore detect shifts in all directions except the one in which the components of the shift in the mean vector are all the same but not 0. The shift with equal components, however, can be easily detected by another univariate CUSUM. The former CUSUM procedure is distribution free in the sense that all its properties depend on the distribution of the antirank vector only.

NONPARAMETRIC PROFILE MONITORING BY MIXED EFFECTS MODELING

In some applications, the quality of a process is characterized by the functional relationship between a response variable and one or more explanatory variables. Profile monitoring is for checking the stability of this relationship over time. Control charts for monitoring nonparametric profiles are useful when the relationship is too complicated to be described parametrically. Most existing control charts in the literature are for monitoring parametric profiles. They require the assumption that within-profile measurements are independent of each other, which is often invalid in practice. This article focuses on nonparametric profile monitoring when within-profile data are correlated. A novel control chart is suggested, which incorporates local linear kernel smoothing into the exponentially weighted moving average (EWMA) control scheme. In this method, within-profile correlation is described by a nonparametric mixed-effects model. Our proposed control chart is fast to compute and convenient to use. Numerical examples show that it works well in various cases. Some technical details are provided in an Appendix available online as supplemental materials.

Fuzzy Modeling and Fuzzy Control

preparation of statements that are fair, clear, and helpful to courts; and responding to questions by judges and juries.” The author does a good job of meeting these goals through discussions on how to quantify DNA evidence for presentation in court or preparing legal statements. This book will be helpful to statisticians and others with technical backgrounds, who might be called on as expert witnesses in deciding what kind of information is considered a valid evidence and what should be presented. For example, in Chapter 8, the author mentions that an expert witness needs sufficient information to answer two questions for a jury: (1) How likely is the evidence if the defendant s is guilty? and (2) how likely is the evidence if s is innocent and i is the true culprit? The author discusses different ways to answer these questions. Although the writing in this book is fairly nontechnical, some mathematical theory behind the results is presented, but not for courtroom statements. It is given for forensic scientists to provide insight into the reasoning behind results. The concept of p-value is very difficult to understand for those unfamiliar with statistical terminology. The author presents this concept in plain English in very easy-to-understand language without actually mentioning the term “pvalue.” Although the concepts of hypotheses testing are used in discussions from the beginning, it is introduced formally only at the end of Chapter 8, where the standard definition of p-value also is given. Similarly, the discussions include descriptions of the concepts of conditional probabilities and two types of error rates, which also are not easy to understand for members of a jury. The errors of logic, referred to as the prosecutor’s fallacy and the defendant’s fallacy, are described using simple examples. An interesting discussion also shows how a lower level of language comprehension by jurors can lead to confusion about differences in P(A|B) and P(B|A). There is more emphasis on evaluating evidence using likelihood ratios and the Bayes theorem. In today’s world of expanding use of scientific methods to resolve conflicts by the judicial system and recent increases in use of DNA profiling as evidence, there is a need for more literature that describes these concepts in simple language. This book is a good example of how statistics can be explained in plain English to a nontechnical audience, a skill that every statistician needs to master for improved communication.

Multivariate Statistical Process Control Using LASSO

This article develops a new multivariate statistical process control (SPC) methodology based on adapting the LASSO variable selection method to the SPC problem. The LASSO method has the sparsity property of being able to select exactly the set of nonzero regression coefficients in multivariate regression modeling, which is especially useful in cases where the number of nonzero coefficients is small. In multivariate SPC applications, process mean vectors often shift in a small number of components. Our primary goals are to detect such a shift as soon as it occurs and to identify the shifted mean components. Using this connection between the two problems, we propose a LASSO-based multivariate test statistic, and then integrate this statistic into the multivariate EWMA charting scheme for Phase II multivariate process monitoring. We show that this approach balances protection against various shift levels and shift directions, and thus provides an effective tool for multivariate SPC applications. This article has supplementary material online.

Distribution-free multivariate process control based on log-linear modeling

This paper considers Statistical Process Control (SPC) when the process measurement is multivariate. In the literature, most existing multivariate SPC procedures assume that the in-control distribution of the multivariate process measurement is known and it is a Gaussian distribution. In applications, however, the measurement distribution is usually unknown and it needs to be estimated from data. Furthermore, multivariate measurements often do not follow a Gaussian distribution (e.g., cases when some measurement components are discrete). We demonstrate that results from conventional multivariate SPC procedures are usually unreliable when the data are non-Gaussian. Existing statistical tools for describing multivariate non-Gaussian data, or transforming the multivariate non-Gaussian data to multivariate Gaussian data, are limited, making appropriate multivariate SPC difficult in such cases. In this paper, we suggest a methodology for estimating the in-control multivariate measurement distribution when a set of in-control data is available, which is based on log-linear modeling and which takes into account the association structure among the measurement components. Based on this estimated in-control distribution, a multivariate CUSUM procedure for detecting shifts in the location parameter vector of the measurement distribution is also suggested for Phase II SPC. This procedure does not depend on the Gaussian distribution assumption; thus, it is appropriate to use for most multivariate SPC problems.

A nonparametric multivariate cumulative sum procedure for detecting shifts in all directions

Summary. The fairly limited range of tools for multivariate statistical process control generally rests on the assumption that the data vectors follow a multivariate normal distribution-an assumption that is rarely satisfied. We discuss detecting possible shifts in the mean vector of a multivariate measurement of a statistical process when the multivariate distribution of the measurement is non-Gaussian. A nonparametric cumulative sum procedure is suggested which is based both on the order information among the measurement components and on the order information between the measurement components and their in-control means. It is shown that this procedure is effective in detecting a wide range of possible shifts. Several numerical examples are presented to evaluate its performance. This procedure is also applied to a data set from an aluminium smelter.

Edge-preserving image denoising and estimation of discontinuous surfaces

In this paper, we are interested in the problem of estimating a discontinuous surface from noisy data. A novel procedure for this problem is proposed based on local linear kernel smoothing, in which local neighborhoods are adapted to the local smoothness of the surface measured by the observed data. The procedure can therefore remove noise correctly in continuity regions of the surface and preserve discontinuities at the same time. Since an image can be regarded as a surface of the image intensity function and such a surface has discontinuities at the outlines of objects, this procedure can be applied directly to image denoising. Numerical studies show that it works well in applications, compared to some existing procedures

On Nonparametric Statistical Process Control of Univariate Processes

This article considers statistical process control (SPC) of univariate processes when the parametric form of the process distribution is unavailable. Most existing SPC procedures are based on the assumption that a parametric form (e.g., normal) of the process distribution can be specified beforehand. In the literature, it has been demonstrated that their performance is unreliable in cases when the prespecified process distribution is invalid. To overcome this limitation, some nonparametric (or distribution-free) SPC charts have been proposed, most of which are based on the ordering information of the observed data. This article tries to make two contributions to the nonparametric SPC literature. First, we propose an alternative framework for constructing nonparametric control charts, by first categorizing observed data and then applying categorical data analysis methods to SPC. Under this framework, some new nonparametric control charts are proposed. Second, we compare our proposed control charts with several representative existing control charts in various cases. Some empirical guidelines are provided for users to choose a proper nonparametric control chart for a specific application. This article has supplementary materials online.

Distribution-free cumulative sum control charts using bootstrap-based control limits

This paper deals with phase II, univariate, statistical process control when a set of in-control data is available, and when both the in-control and out-of-control distributions of the process are unknown. Existing process control techniques typically require substantial knowledge about the in-control and out-of-control distributions of the process, which is often difficult to obtain in practice. We propose (a) using a sequence of control limits for the cumulative sum (CUSUM) control charts, where the control limits are determined by the conditional distribution of the CUSUM statistic given the last time it was zero, and (b) estimating the control limits by bootstrap. Traditionally, the CUSUM control chart uses a single control limit, which is obtained under the assumption that the in-control and out-of-control distributions of the process are Normal. When the normality assumption is not valid, which is often true in applications, the actual in-control average run length, defined to be the expected time duration before the control chart signals a process change, is quite different from the nominal in-control average run length. This limitation is mostly eliminated in the proposed procedure, which is distribution-free and robust against different choices of the in-control and out-of-control distributions.