Monitoring the mean vector and the covariance matrix of multivariate processes with sample means and sample ranges
Monitoramento do vetor de médias e da matriz de covariâncias de processos multivariados baseado nas médias e nas amplitudes amostrais
Costa, Antonio Fernando B.; Machado, Marcela Aparecida G.
http://dx.doi.org/10.1590/S0103-65132011005000029
Prod, vol.21, n2, p.197-208, 2011
Abstract
The joint X and R charts and the joint X 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 X and R charts are slightly inferior to the joint X 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.
Keywords
Control charts. Mean vector. Covariance matrix. Multivariate processes.
Resumo
Os gráficos conjuntos de X e R e X 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 X e R em uso conjunto são ligeiramente inferior aos gráficos de X 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.
Palavras-chave
Gráficos de controle. Vetor de médias. Matriz de covariância. Processos multivariados.
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