Spectral analysis of heart rate variability (HRV) has been applied in many medical research to study autonomic nervous system activity. In these studies, the researchers found that (i) ratio of low frequency (LF) to high frequency (HF) spectrum power is a useful measure of sympathetic/parasympathetic balance, and (ii) low heart rate variability is an important risk factors for patients. Therefore, continuous monitoring of the ratio and heart rate variability have the potential to early detect physiological deterioration of patients. This thesis consists of the following two parts. In the first part, we establish control charts monitoring heart rates and low HRV. Numerical method is applied to compute exact control limits of the EWRMS and EWMV charts. The distribution of the conventional LF/HF ratio statistic is difficult to derive, significant of alterations in HRV parameters can not be assessed efficiently. We resolve this problem in the second part, a new equivalently useful ratio statistic is proposed whose distribution can be derived more easily. Based on the derived distribution, the probability control limits of the proposed statistic are calculated. In application, we construct Shewhart charts of the newly proposed ratio statistic and EWRMS, EWMV charts of the heart rate variability to monitor the risk factors of patients in vascular intensive care unit. Furthermore, we define a risk score which combining the two risk factors together, heart rate variability and LF/HF spectrum power ratio. The results show that the higher risk scores corresponding to patients after operation in severer condition.

Multivariate control charts are valuable tools for multivariate statistical process control (MSPC) used to monitor industrial processes and to detect abnormal process behavior. It has been shown in the literature that Bayesian control charts are optimal tools to control the process compared with the non-Bayesian charts. To use any control chart, three control chart parameters must be specified, namely the sample size, the sampling interval and the control limit. Traditionally, control chart design is based on its statistical performance. Recently, industrial practitioners and academic researchers have increasingly recognized the cost benefits obtained by applying the economically designed control charts to quality control, equipment condition monitoring, and maintenance decision-making. The primary objective of this research is to design multivariate Bayesian control charts (MVBCH) both for quality control and conditional-based maintenance (CBM) applications. Although considerable research has been done to develop MSPC tools under the assumption that the observations are independent, little attention has been given to the development of MSPC tools for monitoring multivariate autocorrelated processes. In this research, we compare the performance of the squared predication error (SPE) chart using a vector autoregressive moving average with exogenous variables (VARMAX) model and a partial least squares (PLS) model for a multivariate autocorrelated process. The study shows that the use of SPE control charts based on the VARMAX model allows rapid detection of process disturbances while reducing false alarms. Next, the economic and economic-statistical design of a MVBCH for quality control considering the control limit policy proved to be optimal by Makis(2007) is developed. The computational results illustrate that the MVBCH performs considerably better than the MEWMA chart, especially for smaller mean shifts. Sensitivity analyses further explore the impact of the misspecified out-of-control mean on the actual average cost. Finally, design of a MVBCH for CBM applications is considered using the same control limit policy structure and including an observable failure state. Optimization models for the economic and economic statistical design of the MVBCH for a 3 state CBM model are developed and comparison results show that the MVBCH performs better than recently developed CBM Chi-square chart.

Multivariate control charts are valuable tools for multivariate statistical process control (MSPC) used to monitor industrial processes and to detect abnormal process behavior. It has been shown in the literature that Bayesian control charts are optimal tools to control the process compared with the non-Bayesian charts. To use any control chart, three control chart parameters must be specified, namely the sample size, the sampling interval and the control limit. Traditionally, control chart design is based on its statistical performance. Recently, industrial practitioners and academic researchers have increasingly recognized the cost benefits obtained by applying the economically designed control charts to quality control, equipment condition monitoring, and maintenance decision-making. The primary objective of this research is to design multivariate Bayesian control charts (MVBCH) both for quality control and conditional-based maintenance (CBM) applications. Although considerable research has been done to develop MSPC tools under the assumption that the observations are independent, little attention has been given to the development of MSPC tools for monitoring multivariate autocorrelated processes. In this research, we compare the performance of the squared predication error (SPE) chart using a vector autoregressive moving average with exogenous variables (VARMAX) model and a partial least squares (PLS) model for a multivariate autocorrelated process. The study shows that the use of SPE control charts based on the VARMAX model allows rapid detection of process disturbances while reducing false alarms. Next, the economic and economic-statistical design of a MVBCH for quality control considering the control limit policy proved to be optimal by Makis(2007) is developed. The computational results illustrate that the MVBCH performs considerably better than the MEWMA chart, especially for smaller mean shifts. Sensitivity analyses further explore the impact of the misspecified out-of-control mean on the actual average cost. Finally, design of a MVBCH for CBM applications is considered using the same control limit policy structure and including an observable failure state. Optimization models for the economic and economic statistical design of the MVBCH for a 3 state CBM model are developed and comparison results show that the MVBCH performs better than recently developed CBM Chi-square chart.

Quando há interesse em monitorar a variância de uma característica da qualidade de interesse através de gráfico de controle por variáveis, o gráfico S2 é a alternativa mais usual. Entretanto, há situações onde mensurar a característica da qualidade é caro, consome mais tempo por unidade de inspeção, requer maior esforço dos operadores quanto à obtenção dos dados ou envolve ensaios destrutivos. Nestes casos, a classificação da variável contínua em categorias através de um dispositivo torna-se uma alternativa interessante. A avaliação pode ser mais rápida, a análise e o equipamento utilizado podem ser mais simples, de modo que o custo final da inspeção seja menor. O objetivo do trabalho é propor um gráfico de controle por atributos para monitoramento da variabilidade. Para tanto a estatística GS2 é calculada e gráfico sinaliza se GS2 > LC, LC limite de controle determinado de modo que minimize o ARL1, fixado um valor de ARL0. Como resultado a performance do gráfico GS2 é comparada ao gráfico S2 em termos de ARL1. / In cases aiming at monitoring the variance of a products quality characteristics using a variable control chart, chart S2 is the most used alternative. However, in some situations, this solution can be expensive, demand more time per individual inspected unit, demand greater efforts from operators to acquire data or involve destructive tests. In such cases, the use of a gauge measurement tool to classify the continuous variable into categories, becomes an interesting alternative. The assessment can be faster, the analysis and the tool used can be simple, resulting in less costly final inspections. This work proposes the use of an attribute control chart to monitor variability. Statistics GS2 is calculated and control chart signalize if GS2 > CL, whereas CL is the determined control limit, minimizing ARL1 for a fixed value of ARL0. GS2 control chart performance is compared to S2 chart based on ARL1.

Quando há interesse em monitorar a variância de uma característica da qualidade de interesse através de gráfico de controle por variáveis, o gráfico S2 é a alternativa mais usual. Entretanto, há situações onde mensurar a característica da qualidade é caro, consome mais tempo por unidade de inspeção, requer maior esforço dos operadores quanto à obtenção dos dados ou envolve ensaios destrutivos. Nestes casos, a classificação da variável contínua em categorias através de um dispositivo torna-se uma alternativa interessante. A avaliação pode ser mais rápida, a análise e o equipamento utilizado podem ser mais simples, de modo que o custo final da inspeção seja menor. O objetivo do trabalho é propor um gráfico de controle por atributos para monitoramento da variabilidade. Para tanto a estatística GS2 é calculada e gráfico sinaliza se GS2 > LC, LC limite de controle determinado de modo que minimize o ARL1, fixado um valor de ARL0. Como resultado a performance do gráfico GS2 é comparada ao gráfico S2 em termos de ARL1. / In cases aiming at monitoring the variance of a products quality characteristics using a variable control chart, chart S2 is the most used alternative. However, in some situations, this solution can be expensive, demand more time per individual inspected unit, demand greater efforts from operators to acquire data or involve destructive tests. In such cases, the use of a gauge measurement tool to classify the continuous variable into categories, becomes an interesting alternative. The assessment can be faster, the analysis and the tool used can be simple, resulting in less costly final inspections. This work proposes the use of an attribute control chart to monitor variability. Statistics GS2 is calculated and control chart signalize if GS2 > CL, whereas CL is the determined control limit, minimizing ARL1 for a fixed value of ARL0. GS2 control chart performance is compared to S2 chart based on ARL1.

The determination of economic control charts and the determination of specification limits with minimum cost are two different research topics. In this study, we first combine the design of economic control charts and the determination of specification limits to maximize the expected profit per unit time for the smaller the better quality variable following the gamma distribution. Because of the asymmetric distribution, we design the EWMA control chart with asymmetric control limits. We simultaneously determine the economic EWMA control chart and upper specification limit with maximum expected profit per unit time. Then, extend the approach to determine the economic variable sampling interval EWMA control chart and upper specification limit with maximum expected profit per unit time. In all our numerical examples of the two profit models, the optimum expected profit per unit time under inspection is higher than that of no inspection. The detection ability of the EWMA chart with an appropriate weight is always better than the X-bar probability chart. The detection ability of the VSI EWMA chart is also superior to that of the fixed sampling interval EWMA chart. Sensitivity analyses are provided to determine the significant parameters for the optimal design parameters and the optimal expected profit per unit time.

Statistical process control (SPC) is a collection of scientific tools developed and engineered to diagnose unnecessary variation in the output of a production process and eliminate it or perhaps accommodate it by adjusting process settings. The task of quality control (QC) is of fundamental importance in manufacturing processes when a change in the process causes misleading results, this alteration should be detected and corrected as soon as possible. Statistical QC charts originated in the late 1920s by Dr. W. A. Shewhart provide a powerful tool for monitoring production lines in manufacturing industries. They are also have been implemented in various disciplines, such as sequential monitoring of internet traffic flows, health care systems, and more. Shewhart-type charts are effective in detecting large shifts in the process but ineffective in detecting small to moderate shifts. This blind spot allows small shifts (smaller than one standard deviation) to continue undetected in the process, thereby incurring larger total costs for manufacturers. This thesis addresses this issue by augmenting current time-weighted charts (charts that use all the information from the start of a process until the most recent sample/observation) with a Double Generally Weighted Moving Average (DGWMA) chart, leading to more effective process monitoring. The objective of this thesis is to provide the fundamentals and introduce the researcher/practitioner to the essentials of the univariate DGWMA chart from both parametric and nonparametric perspectives. Numerous concepts and characteristics of proposed DGWMA charts are discussed comprehensively. Theoretical expressions and detailed calculations have been provided to aid the interested reader to familiarize and study the topic more thoroughly. This thesis paints a bigger picture of the DGWMA chart in a sense that other time-weighted charts such as the Generally Weighted Moving Average (GWMA), Exponentially Weighted Moving Average (EWMA), Double Exponentially Weighted Moving Average (DEWMA) and Cumulative Sum (CUSUM) fall under this umbrella. Both real-life data and simulated examples have been embedded throughout the thesis. We make use of R and Mathematica software packages to calculate numerical results related to the run length distribution and its associated characteristics in this thesis. We only consider control charts for monitoring the process location parameter. However, our conclusions and recommendations are extendable for the process dispersion parameter. In this thesis, we consider the DGWMA chart as the main chart and the EWMA, DEWMA, GWMA, and CUSUM charts as special cases. The thesis consists of the following chapters with a short description for each chapter as follows: Chapter 1 provides a brief introduction to SPC concepts and gives a literature review in terms of background information for the research conducted in this thesis. The scope and objectives of the present research are highlighted in detail. Chapter 2 provides an overview and a theoretical background on the design and implementation of the DGWMA chart derived from the SPC literature review. The properties of the DGWMA chart, including the plotting statistic, the structure for the weights, the control limits (exact/steady-state), etc. are considered in detail. The weighting structure of the DGWMA chart and its special case are discussed and pictured to emphasize the impact of weights in increasing the detection capability of time-weighted charts. Three approaches are described and investigated for calculating the run length distribution and its associated characteristics for the DGWMA chart and its special case the DEWMA chart; this includes: (i) exact approach; (ii) Markov chain approach; (iii) Monte Carlo simulation. In Chapter 3 we develop a one-sided generalized parametric chart (denoted by DGWMA-TBE) for monitoring the time between events (TBE) of nonconformities items originating from the high-yield processes when the underlying process distribution is gamma and the parameter of interest is known (Case K) and unknown (Case U). A Markov chain approach is implemented to derive the run length distribution and its associated characteristics for the DGWMA and DEWMA charts. An exact approach is also used to derive closed-form expressions for the run length distribution of the proposed chart. Performance analysis has been undertaken to execute a comparative study with several existing time-weighted charts. The proposed chart encompasses one-sided GWMA-TBE, EWMA-TBE, DEWMA-TBE, and Shewhart-type charts as limiting or special cases. The CUSUM-TBE chart is also included in the performance comparison. The necessary design parameters are provided to aid the implementation of the proposed chart and finding the optimal design and near optima design that is useful for practitioners. Alternative discrete distributions are considered for the weights of the GWMA-TBE chart and a discussion is provided to address the connection between new weights originating from the suggested distributions and the chart’s capability in detecting shifts. As a result, one can design an optimal GWMA-TBE chart by replacing weights from the discrete Weibull distribution without the implementation of the double exponential smoothing technique. Chapter 4 focuses on developing a two-sided nonparametric (distribution-free) DGWMA control chart based on the exceedance (EX) statistic, denoted as DGWMA-EX when the parameter of interest is unknown (Case U) and the underlying process distribution is continuous and symmetric. An exact approach and a Markov chain approach are considered to calculate the run length distribution and its associated characteristics for the proposed chart. A performance comparison has been undertaken to execute analysis with other nonparametric time-weighted charts available in the SPC literature. The proposed chart en-compasses two-sided GWMA-EX, EWMA-EX, DEWMA-EX, and Shewhart-type charts as limiting or special cases. The CUSUM-EX chart is also included in the performance comparison Also, the performance of the proposed DGWMA-EX chart has been evaluated under different symmetric and skewed distributions in comparison with its main counterparts, and the necessary results and recommendations are provided for practitioners to design an optimal chart. Chapter 5 encloses this thesis with a summary of the research conducted and provides concluding remarks concerning future research opportunities. / Thesis (PhD (Mathematical Statistics))--University of Pretoria, 2020. / This research was supported in part by the National Research Foundation (NRF) under Grant Number 71199 and the postgraduate research bursary supported by the University of Pretoria. Any findings, opinions, and conclusions expressed in this thesis are those of the author and do not necessarily reflect the views of the parties. / Statistics / PhD (Mathematical Statistics) / Unrestricted

無 / In this article, a two-step process is considered to investigate the effects of measurement errors on EWMA and cause-selecting EWMA control charts. At the end of current process, a pair of imprecise measurements of in-coming quality and out-going quality is randomly taken with individual units. The linear relationship between in-coming quality and out-going quality is assumed and four possible states of the process are defined with respective distributions of in-coming and out-going qualities derived. The EWMA control chart with measurement error is then constructed to monitor small-scale shift in mean for the previous process while the cause-selecting control chart, or EWMA control chart based on residuals, including measurement error, is proposed to diagnose the state of current process. Based on sensitivity analysis, the presence of imprecise measurement diminishes the power of both the EWMA and the proposed control charts and affects the detectability of process disturbances. Further, applications of proposed control charts are demonstrated through a numerical example to show some possible misuses of control charts. If the process mean shifts in a small scale when a single assignable cause occurs on each step, the proposed cause-selecting control chart is more sensitive than other control charts. The Hotelling T^2 control chart is also compared to illustrate the diagnostic advantage outweighed by proposed cause-selecting control chart.

The main focus of many agricultural, ecological and environmental studies is to develop well designed, cost-effective and efficient sampling designs. Ranked set sampling (RSS) is one of those sampling methods that can help accomplish such objectives by incorporating prior information and expert knowledge to the design. In this thesis, new RSS schemes are suggested for efficiently estimating the population mean. These sampling schemes can be used as cost-effective alternatives to the traditional simple random sampling (SRS) and RSS schemes. It is shown that the mean estimators under the proposed sampling schemes are at least as efficient as the mean estimator with SRS. We consider the best linear unbiased estimators (BLUEs) and the best linear invariant estimators (BLIEs) for the unknown parameters (location and scale) of a location-scale family of distributions under double RSS (DRSS) scheme. The BLUEs and BLIEs with DRSS are more precise than their counterparts based on SRS and RSS schemes. We also consider the BLUEs based on DRSS and ordered DRSS (ODRSS) schemes for the unknown parameters of a simple linear regression model using replicated observations. It turns out that, in terms of relative efficiencies, the BLUEs under ODRSS are better than the BLUEs with SRS, RSS, ordered RSS (ORSS) and DRSS schemes. Quality control charts are widely recognized for their potential to be a powerful process monitoring tool of the statistical process control. These control charts are frequently used in many industrial and service organizations to monitor in-control and out-of-control performances of a production or manufacturing process. The RSS schemes have had considerable attention in the construction of quality control charts. We propose new exponentially weighted moving average (EWMA) control charts for monitoring the process mean and the process dispersion based on the BLUEs obtained under ORSS and ODRSS schemes. We also suggest an improved maximum EWMA control chart for simultaneously monitoring the process mean and dispersion based on the BLUEs with ORSS scheme. The proposed EWMA control charts perform substantially better than their counterparts based on SRS and RSS schemes. Finally, some new EWMA charts are also suggested for monitoring the process dispersion using the best linear unbiased absolute estimators of the scale parameter under SRS and RSS schemes.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / John English / This research builds upon previous efforts to explore the use of Statistical Process Control (SPC) in lieu of cycle counting. Specifically a three pronged effort is developed. First, in the work of Huschka (2009) and Miller (2008), a mixture distribution is proposed to model the complexities of multiple Stock Keeping Units (SKU) within an operating department. We have gained access to data set from a large retailer and have analyzed the data in an effort to validate the core models. Secondly, we develop a recursive relationship that enables large samples of SKUs to be evaluated with appropriately with the SPC approach. Finally, we present a comprehensive set of type I and type II error rates for the SPC approach to inventory accuracy monitoring.