Giovanna Capizzi

An Adaptive Exponentially Weighted Moving Average Control Chart

Lucas and Saccucci showed that exponentially weighted moving average (EWMA) control charts can be designed to quickly detect either small or large shifts in the mean of a sequence of independent observations. But a single EWMA chart cannot perform well for small and large shifts simultaneously. Furthermore, in the worst-case situation, this scheme requires a few observations to overcome its initial inertia. The main goal of this article is to suggest an adaptive EWMA (AEWMA) chart that weights the past observations of the monitored process using a suitable function of the current “error.” The resulting scheme can be viewed as a smooth combination of a Shewhart chart and an EWMA chart. A design procedure for the new control schemes is suggested. Comparisons of the standard and worst-case average run length profiles of the new scheme with those of different control charts show that AEWMA schemes offer a more balanced protection against shifts of different sizes.

A Least Angle Regression Control Chart for Multidimensional Data

In multidimensional applications, it is very rare that all variables shift at the same time. A statistical process control procedure would have superior efficiency when limited to the subset of variables likely responsible for the out-of-control conditions. The key idea of this article consists of combining a variable selection method with a multivariate control chart to detect changes in both the mean and variability of a multidimensional process with Gaussian errors. In particular, we develop a control chart for Phase II monitoring which integrates the least angle regression algorithm with a multivariate exponentially weighted moving average. Comparisons with related multivariate control schemes demonstrate the efficiency of the proposed control chart in a wide range of practical applications, including profile and multistage process monitoring. Further, the proposed scheme may also provide valuable diagnostic information for fault isolation. Supplemental materials, including an R package, are available online.

Combined Shewhart–EWMA control charts with estimated parameters

Shewhart and EWMA control charts can be suitably combined to obtain a simple monitoring scheme sensitive to both large and small shifts in the process mean. So far, the performance of the combined Shewhart–EWMA (CSEWMA) has been investigated under the assumption that the process parameters are known. However, parameters are often estimated from reference Phase I samples. Since chart performances may be even largely affected by estimation errors, we study the behaviour of the CSEWMA with estimated parameters in both in- and out-of-control situations. Comparisons with standard Shewhart and EWMA charts are presented. Recommendations are given for Phase I sample size requirements necessary to achieve desired in-control performance.

Practical Design of Generalized Likelihood Ratio Control Charts for Autocorrelated Data

Control charts based on generalized likelihood ratio (GLR) tests are attractive from both theoretical and practical points of view. In particular, in the case of an autocorrelated process, the GLR test uses the information contained in the time-varying response after a change and, as shown by Apley and Shi, is able to outperfom traditional control charts applied to residuals. In addition, a GLR chart provides estimates of the magnitude and the time of occurrence of the change. In this article we present a practical approach to implementating GLR charts for monitoring an autoregressive moving average process assuming that only a phase I sample is available. The proposed approach, based on automatic time series identification, estimates the GLR control limits through stochastic approximation using bootstrap resampling and thus is able to take into account the uncertainty about the underlying model. A Monte Carlo study shows that our methodology can be used to design, in a semiautomatic fashion, a GLR chart with a prescribed rate of false alarms when as few as 50 phase I observations are available. A real example is used to illustrate the designing procedure.

Recent Advances in Process Monitoring: Nonparametric and Variable-Selection Methods for Phase I and Phase II

ABSTRACT The main aim of this article is to review and discuss two particular topics of statistical process monitoring: the need for a nonparametric approach to Phase I analysis and the use of variable selection–based control charts in multivariate Phase II monitoring. After discussing a number of critical issues related to these topics, I present recently proposed solutions and summarize several research problems that require further investigation.

Self-Starting CUSCORE Control Charts for Individual Multivariate Observations

In some manufacturing settings, such as during process start-up and in the case of short production runs, process parameters are unknown, and Phase I samples cannot be gathered to accurately estimate control limits for prospective monitoring. Self-starting charts can be applied to these low-volume applications. In this article, two new self-starting multivariate control charts, both based on a CUSCORE-type procedure, are proposed for monitoring the unknown mean of a multivariate normal distribution. These charting procedures, which weight current observations according to the information contained in the fault signature, are able to outperform the previously suggested self-starting charts, which neglect the dynamic pattern of the mean change.

Phase I Distribution-Free Analysis of Multivariate Data

ABSTRACT In this study, a new distribution-free Phase I control chart for retrospectively monitoring multivariate data is developed. The suggested approach, based on the multivariate signed ranks, can be applied to individual or subgrouped data for detection of location shifts with an arbitrary pattern (e.g., isolated, transitory, sustained, progressive, etc.). The procedure is complemented with a LASSO-based post-signal diagnostic method for identification of the shifted variables. A simulation study shows that the method compares favorably with parametric control charts when the process is normally distributed, and largely outperforms other multivariate nonparametric control charts when the process distribution is skewed or heavy-tailed. An R package can be found in the supplementary material.

Adaptive Generalized Likelihood Ratio Control Charts for Detecting Unknown Patterned Mean Shifts

Traditional statistical control schemes have been mainly focused on detecting constant mean shifts. In many practical applications, however, the mean of the observed sequence can exhibit a time-varying behavior after the fault occurrence. Shewhart control charts supplemented with sensitizing run rules have been suggested for detecting nonrandom dynamic mean patterns. However, the choice of a particular set of run rules requires some prior knowledge of the possible expected patterns. In addition, the resulting schemes can have poor performance against small shifts. When the mean shift pattern is known in advance, generalized likelihood ratio (GLR), cumulative score (CUSCORE), and optimal general linear filter (OGLF) control charts have also been proposed for change-point detection. Further, some adaptive CUSCORE schemes have been recently developed for detecting an unknown patterned mean shift. However, these adaptive CUSCOREs assume the occurrence of one-sided mean shifts. Hence, their performance may be poor in the case of an oscillatory behavior of the process mean. To overcome these limitations, we propose estimating a possible pattern in the mean using an exponentially weighted moving average (EWMA), which is especially effective with one-sided shifts in the mean, combined with a wavelet smoother, which is effective in estimating oscillatory mean patterns. The two estimates then drive two separate conventional GLR tests for significance. A control chart is proposed for observations from normal distributions and then extended to a completely distribution-free setting. Extensive simulation results demonstrate the efficiency of the proposed charts with a wide variety of patterned mean shifts in both the independent and autocorrelated scenarios and with a wide variety of distributions. In addition, the proposed scheme provides post-signal diagnostic information that estimates the change-point location and the shift pattern of the mean. An R package is available online as supplementary material.

Evaluation of the run-length distribution for a combined Shewhart-EWMA control chart

A simple algorithm is introduced for computing the run length distribution of a monitoring scheme combining a Shewhart chart with an Exponentially Weighted Moving Average control chart. The algorithm is based on the numerical approximation of the integral equations and integral recurrence relations related to the run-length distribution. In particular, a Clenshaw-Curtis product-integration rule is applied for handling discontinuities in the integrand function due to the simultaneous use of the two control schemes. The proposed algorithm, implemented in R and publicy available, compares favourably with the Markov chain approach originally used to approximate the run length properties of the combined Shewhart-EWMA.

An Enhanced Control Chart for Start-Up Processes and Short Runs

Abstract Classic charting procedures are usually designed assuming that process parameters are known or may be estimated using large Phase I samples gathered before a production run. However, in some manufacturing settings, such as during the process start-up, historical data cannot be collected to accurately estimate the in-control process parameters. In this article, we suggest a new self-starting control chart which uses consecutive observations to jointly update the parameter estimates and check for out-of-control conditions. In particular, we introduce a charting procedure, ACUSCORE, that updates the reference pattern of a type-CUSCORE chart using an adaptive EWMA. The proposed control chart seems to outperform traditional self-starting control charts which neglect the dynamic pattern of the mean change.