Monitoring the Mean Vector and the Covariance Matrix of Bivariate Processes

This paper proposes the joint use of two charts based on the non-central chi-square statistic (NCS statistic) for monitoring the mean vector and the covariance matrix of bivariate processes, named as the joint NCS charts. The expression to compute the ARL, which is defined as the average number of samples the joint charts need to signal an out-of-control condition, is derived. The joint NCS charts might be more sensitive to changes in the mean vector or, alternatively, more sensitive to changes in the covariance matrix, accordingly to the values of their design parameters. In general, the joint NCS charts are faster than the combined T2 and |S| charts in signaling out-of-control conditions. Once the proposed scheme signals, the user can immediately identify the out-of-control variable. The risk of misidentifying the out-of-control variable is small (less than 5.0%)

Gráficos de controle para monitoramento de processos multivariados

Esta tese oferece algumas contribuições à área de monitoramento de processos multivariados. Com respeito ao monitoramento do vetor de médias, investigou-se o desempenho dos gráficos de 2 T baseados em componentes principais e também o desempenho dos gráficos de médias utilizados em conjunto, sendo que cada gráfico monitora a média de uma das características de qualidade. Com respeito ao monitoramento da matriz de covariâncias, foi proposta uma nova estatística baseada nas variâncias amostrais (estatística de VMAX). O gráfico de VMAX é mais eficiente do que o gráfico da variância amostral generalizada S , que é o gráfico usual para o monitoramento da matriz de covariâncias. Uma vantagem adicional dessa nova estatística é que o usuário já está bem familiarizado com o cálculo de variâncias amostrais; o mesmo não pode ser dito em relação à variância amostral generalizada S . O desempenho do gráfico de VMAX foi também avaliado quando se utiliza a amostragem dupla, quando se variam os parâmetros do gráfico de controle, quando se adota o esquema de EWMA e quando se aplicam regras especiais de decisão. Investigou-se também o desempenho dos gráficos de controle destinados ao monitoramento simultâneo do vetor de médias e da matriz de covariâncias.This thesis offers some contributions to the field of monitoring multivariate processes. Regarding to the monitoring of the mean vector, we investigated the performance of the 2 T charts based on principal components and also the performance of the mean charts used simultaneously, where each chart is assigned to control one quality characteristic. Regarding to the monitoring of the covariance matrix, we propose a new statistic based on the sample variances (the VMAX statistic). The VMAX chart is more efficient than the generalized variance S chart, which is the usual chart for monitoring the covariance matrix. An additional advantage of this new statistic is that the user is already well familiar with the calculation of sample variances; we can’t say the same regarding to the generalized variance S statistic. We also studied the performance of the VMAX chart with double sampling, with adaptive schemes, with the EWMA procedure and also with special run rules. We also investigated the performance of the control charts designed for monitoring the mean vector and the covariance matrix simultaneously.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

Estudo das propriedades dos gráficos de controle bivariados com amostragem dupla

Assim como o gráfico deXBARRA, o gráfico T2 de Hotelling é lento na detecção de pequenas a moderadas pertubações no processo. Estudos consagrados mostram que o desempenho do gráfico de XBARRA melhora em muito com o uso da amostragem dupla. Com base nestes resultados, este trabalho se dedica ao estudo das propriedades dos gráficos T2 com amostragem dupla para processos bivariados. Através de uma rotação dos eixos cartesianos é possível transformar as variáveis originais, que em geral são altamente correlacionadas, em variáveis independentes. Com as novas variáveis e trabalhando com coordenadas polares foi possível obter o número médio de amostras (NMA) que o gráfico proposto necessita para detectar uma alteração no processo. Por meio de comparações dos NMAs foi possível verificar que o gráfico de controle proposto é, na maioria das vezes, mais eficiente que os gráficos adaptativos em que o tamanho das amostras e/ou o intervalo entre retirada de amostras são variáveis.Similarly to the X chart, the T2 chart is slow to detect small or even moderate process disturbances. Earlier studies have shown that the use of the double sampling procedure improves substabtially the X chart performance. Based on that, we propose here to study the performance of the T2 chart with double sampling applied to control bivariate processes. An appropriate rotation transforms the original bivariate variables, in general presenting high correlation, in independent variables. With these equivalent variables and working with polar coordinates, it was possible to obtain the average run length (ARL) that measures the effectiveness of the proposed chart in detecting a process change. By comparisons of ARLs it was possible to verify that the proposed control chart is, frequently, more efficient than the adaptive charts with variable sample size or variable sampling interval.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES

Variable parameter and double sampling (X)over-bar charts in the presence of correlation: The Markov chain approach

The general assumption under which the (X) over bar chart is designed is that the process mean has a constant in-control value. However, there are situations in which the process mean wanders. When it wanders according to a first-order autoregressive (AR (1)) model, a complex approach involving Markov chains and integral equation methods is used to evaluate the properties of the (X) over bar chart. In this paper, we propose the use of a pure Markov chain approach to study the performance of the (X) over bar chart. The performance of the chat (X) over bar with variable parameters and the (X) over bar with double sampling are compared. (C) 2011 Elsevier B.V. All rights reserved

Variable parameter and double sampling charts in the presence of correlation: The Markov chain approach

The general assumption under which the chart is designed is that the process mean has a constant in-control value. However, there are situations in which the process mean wanders. When it wanders according to a first-order autoregressive (AR (1)) model, a complex approach involving Markov chains and integral equation methods is used to evaluate the properties of the chart. In this paper, we propose the use of a pure Markov chain approach to study the performance of the chart. The performance of the chat with variable parameters and the with double sampling are compared.Markov chain chart Correlation Variable parameter Double sampling

The steady-state behavior of the synthetic and side-sensitive synthetic double sampling (X)over-bar charts

The steady-state average run length is used to measure the performance of the recently proposed synthetic double sampling (X) over bar chart (synthetic DS chart). The overall performance of the DS X chart in signaling process mean shifts of different magnitudes does not improve when it is integrated with the conforming run length chart, except when the integrated charts are designed to offer very high protection against false alarms, and the use of large samples is prohibitive. The synthetic chart signals when a second point falls beyond the control limits, no matter whether one of them falls above the centerline and the other falls below it; with the side-sensitive feature, the synthetic chart does not signal when they fall on opposite sides of the centerline. We also investigated the steady-state average run length of the side-sensitive synthetic DS X chart. With the side-sensitive feature, the overall performance of the synthetic DS X chart improves, but not enough to outperform the non-synthetic DS X chart. Copyright (C) 2014 John Wiley &Sons, Ltd

Monitoring the mean vector and the covariance ­matrix of multivariate processes with sample means and sample ranges

Os gráficos conjuntos de e R e S² são os mais utilizados para o monitoramento da média e da dispersão do processo. Com os tamanhos de amostra usuais de 4 e 5, os gráficos de R em uso conjunto são ligeiramente inferior aos gráficos de e S² em uso conjunto em termos da eficiência em detectar alterações no processo. Neste artigo, mostra-se que para o caso multivariado, os gráficos baseados nas médias amostrais padronizadas e amplitudes amostrais (gráfico MRMAX) ou nas médias amostrais padronizadas e variâncias amostrais (gráfico MVMAX) são similares em termos da eficiência em detectar alterações no vetor de médias e/ou na matriz de covariâncias. A familiaridade do usuário com o cálculo de amplitudes amostrais é um aspecto favorável do gráfico MRMAX. Um exemplo é apresentado para ilustrar a aplicação do gráfico proposto.The joint and S² charts are the most common charts used for monitoring the process mean and dispersion. With the usual sample sizes of 4 and 5, the joint and R charts are slightly inferior to the joint and S² charts in terms of efficiency in detecting process shifts. In this article, we show that for the multivariate case, the charts based on the standardized sample means and sample ranges (MRMAX chart) or on the standardized sample means and sample variances (MVMAX chart) are similar in terms of efficiency in detecting shifts in the mean vector and/or in the covariance matrix. User's familiarity with the computation of sample ranges is a point in favor of the MRMAX chart. An example is presented to illustrate the application of the proposed chart.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq