Contributions to improve the power, efficiency and scope of control-chart methods : a thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Albany, New Zealand

Listed in 2019 Dean's List of Exceptional ThesesDetection of outliers and other anomalies in multivariate datasets is a particularly difficult problem which spans across a range of systems, such as quality control in factories, microarrays or proteomic analyses, identification of features in image analysis, identifying unauthorized access in network traffic patterns, and detection of changes in ecosystems. Multivariate control charts (MCC) are popular and sophisticated statistical process control (SPC) methods for monitoring characteristics of interest and detecting changes in a multivariate process. These methods are divided into memory-less and memory-type charts which are used to monitor large and small-to-moderate shifts in the process, respectively. For example, the multivariate χ2 is a memory-less control chart that uses only the most current process information and disregards any previous observations; it is typically used where any shifts in the process mean are expected to be relatively large. To increase the sensitivity of the multivariate process control tool for the detection of small-to-moderate shifts in the process mean vector, different multivariate memory-type tools that use information from both the current and previous process observations have been proposed. These tools have proven very useful for multivariate independent normal or "nearly" normal distributed processes. Like most univariate control-chart methods, when the process parameters (i.e., the process mean vector or covariance parameters, or both) are unknown, then MCC methods are based on estimated parameters, and their implementation occurs in two phases. In Phase I (retrospective phase), a historical reference sample is studied to establish the characteristics of the in-control state and evaluate the stability of the process. Once the in-control reference sample has been deemed to be stable, the process parameters are estimated from Phase I, and control chart limits are obtained for use in Phase II. The Phase II aspect initiates ongoing regular monitoring of the process. If successive observed values obtained at the beginning of Phase II fall within specified desired in-control limits, the process is considered to be in control. In contrast, any observed values during Phase II which fall outside the specified control limits indicate that the process may be out of control, and remedial responses are then required. Although conventional MCC are well developed from a statistical point of view, they can be difficult to apply in modern, data-rich contexts. This serious drawback comes from the fact that classical MCC plotting statistics requires the inversion of the covariance matrix, which is typically assumed to be known. In practice, the covariance matrix is seldom known and often empirically estimated, using a sample covariance matrix from historical data. While the empirical estimate of the covariance matrix may be an unbiased and consistent estimator for a low-dimensional data matrix with an adequate prior sample size, it performs inconsistently in high-dimensional settings. In particular, the empirical estimate of the covariance matrix can lead to in ated false-alarm rates and decreased sensitivity of the chart to detect changes in the process. Also, the statistical properties of traditional MCC tools are accurate only if the assumption of multivariate normality is satisfied. However, in many cases, the underlying system is not multivariate normal, and as a result, the traditional charts can be adversely affected. The necessity of this assumption generally restricts the application of traditional control charts to monitoring industrial processes. Most MCC applications also typically focus on monitoring either the process mean vector or the process variability, and they require that the process mean vector be stable, and that the process variability be independent of the process mean. However, in many real-life processes, the process variability is dependent on the mean, and the mean is not necessarily constant. In such cases, it is more appropriate to monitor the coefficient of variation (CV). The univariate CV is the ratio of the standard deviation to the mean of a random variable. As a relative dispersion measure to the mean, it is useful for comparing the variability of populations having very different process means. More recently, MCC methods have been adapted for monitoring the multivariate coefficient of variation (CV). However, to date, studies of multivariate CV control charts have focused on power - the detection of out-of-control parameters in Phase II, while no study has investigated their in-control performance in Phase I. The Phase I data set can contain unusual observations, which are problematic as they can in uence the parameter estimates, resulting in Phase II control charts with reduced power. Relevant Phase I analysis will guide practitioners with the choice of appropriate multivariate CV estimation procedures when the Phase I data contain contaminated samples. In this thesis, we investigated the performance of the most widely adopted memory-type MCC methods: the multivariate cumulative sum (MCUSUM) and the multivariate exponentially weighted moving average (MEWMA) charts, for monitoring shifts in a process mean vector when the process parameters are unknown and estimated from Phase I (chapters 2 and 3). We demonstrate that using a shrinkage estimate of the covariance matrix improves the run-length performance of these methods, particularly when only a small Phase I sample size is available. In chapter 4, we investigate the Phase I performance of a variety of multivariate CV charts, considering both diffuse symmetric and localized CV disturbance scenarios, and using probability to signal (PTS) as a performance measure. We present a new memory-type control chart for monitoring the mean vector of a multivariate normally distributed process, namely, the multivariate homogeneously weighted moving average (MHWMA) control chart (chapter 5). We present the design procedure and compare the run length performance of the proposed MHWMA chart for the detection of small shifts in the process mean vector with a variety of other existing MCC methods. We also present a dissimilarity-based distribution-free control chart for monitoring changes in the centroid of a multivariate ecological community (chapter 6). The proposed chart may be used, for example, to discover when an impact may have occurred in a monitored ecosystem, and is based on a change-point method that does not require prior knowledge of the ecosystem's behaviour before the monitoring begins. A novel permutation procedure is employed to obtain the control-chart limits of the proposed charting test-statistic to obtain a suitable distance-based model of the target ecological community through time. Finally, we propose enhancements to some classical univariate control chart tools for monitoring small shifts in the process mean, for those scenarios where the process variable is observed along with a correlated auxiliary variable (chapters 7 through 9). We provide the design structure of the charts and examine their performance in terms of their run length properties. We compare the run length performance of the proposed charts with several existing charts for detecting a small shift in the process mean. We offer suggestions on the applications of the proposed charts (in chapters 7 and 8), for cases where the exact measurement of the process variable of interest or the auxiliary variable is diffcult or expensive to obtain, but where the rank ordering of its units can be obtained at a negligible cost. Thus, this thesis, in general, will aid practitioners in applying a wider variety of enhanced and novel control chart tools for more powerful and effcient monitoring of multivariate process. In particular, we develop and test alternative methods for estimating covariance matrices of some useful control-charts' tools (chapters 2 and 3), give recommendations on the choice of an appropriate multivariate CV chart in Phase I (chapter 4), present an efficient method for monitoring small shifts in the process mean vector (chapter 5), expand MCC analyses to cope with non-normally distributed datasets (chapter 6) and contribute to methods that allow efficient use of an auxiliary variable that is observed and correlated with the process variable of interest (chapters 7 through 9)

Multivariate control charts based on Bayesian state space models

This paper develops a new multivariate control charting method for vector autocorrelated and serially correlated processes. The main idea is to propose a Bayesian multivariate local level model, which is a generalization of the Shewhart-Deming model for autocorrelated processes, in order to provide the predictive error distribution of the process and then to apply a univariate modified EWMA control chart to the logarithm of the Bayes' factors of the predictive error density versus the target error density. The resulting chart is proposed as capable to deal with both the non-normality and the autocorrelation structure of the log Bayes' factors. The new control charting scheme is general in application and it has the advantage to control simultaneously not only the process mean vector and the dispersion covariance matrix, but also the entire target distribution of the process. Two examples of London metal exchange data and of production time series data illustrate the capabilities of the new control chart.Comment: 19 pages, 6 figure

Multivariate Statistical Process Control Charts: An Overview

In this paper we discuss the basic procedures for the implementation of multivariate statistical process control via control charting. Furthermore, we review multivariate extensions for all kinds of univariate control charts, such as multivariate Shewhart-type control charts, multivariate CUSUM control charts and multivariate EWMA control charts. In addition, we review unique procedures for the construction of multivariate control charts, based on multivariate statistical techniques such as principal components analysis (PCA) and partial lest squares (PLS). Finally, we describe the most significant methods for the interpretation of an out-of-control signal.quality control, process control, multivariate statistical process control, Hotelling's T-square, CUSUM, EWMA, PCA, PLS

Learning Curves and p-charts for a preliminary estimation of asymptotic performances of a manufacturing process

This paper presents a method for a preliminary estimation of asymptotic performances of a manufacturing process based on the knowledge of its learning curve estimated during the setting up of p-chart. The main novelties of the method are the possibility of estimating the asymptotic variability of a process and providing a simple approach for evaluating the period of revision of process control limits. An application of the method to a real example taken from the literature is also provided

Monitoring of the BTA Deep Hole Drilling Process Using Residual Control Charts

Deep hole drilling methods are used for producing holes with a high lengthto- diameter ratio, good surface finish and straightness. The process is subject to dynamic disturbances usually classified as either chatter vibration or spiralling. In this work, we propose to monitor the BTA drilling process using control charts to detect chatter as early as possible and to secure production with high quality. These control charts use the residuals obtained from a model which describes the variation in the amplitude of the relevant frequencies of the process. The results showed that chatter is detected and some alarm signals are related to changing physical conditions of the process. --