A new chart for monitoring the covariance matrix of bivariate processes

In this article we consider a control chart based on the sample variances of two quality characteristics. The points plotted on the chart correspond to the maximum value of these two statistics. The main reason to consider the proposed chart instead of the generalized variance |S| chart is its better diagnostic feature, that is, with the new chart it is easier to relate an out-of-control signal to the variables whose parameters have moved away from their in-control values. We study the control chart efficiency considering different shifts in the covariance matrix. In this way, we obtain the average run length (ARL) that measures the effectiveness of a control chart in detecting process shifts. The proposed chart always detects process disturbances faster than the generalized variance |S| chart. The same is observed when the size of the samples is variable, except in a few cases in which the size of the samples switches between small size and very large size.

Control charts for monitoring the mean vector and the covariance matrix of bivariate processes

In this article, we propose new control charts for monitoring the mean vector and the covariance matrix of bivariate processes. The traditional tools used for this purpose are the T (2) and the |S| charts. However, these charts have two drawbacks: (1) the T (2) and the |S| statistics are not easy to compute, and (2) after a signal, they do not distinguish the variable affected by the assignable cause. As an alternative to (1), we propose the MVMAX chart, which only requires the computation of sample means and sample variances. As an alternative to (2), we propose the joint use of two charts based on the non-central chi-square statistic (NCS statistic), named as the NCS charts. Once the NCS charts signal, the user can immediately identify the out-of-control variable. In general, the synthetic MVMAX chart is faster than the NCS charts and the joint T (2) and |S| charts in signaling processes disturbances.

A single statistic for monitoring the covariance matrix of bivariate processes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Synthetic control charts with two-stage sampling for monitoring bivariate processes

In this article, we consider the synthetic control chart with two-stage sampling (SyTS chart) to control bivariate processes. During the first stage, one item of the sample is inspected and two correlated quality characteristics (x;y) are measured. If the Hotelling statistic T1 2 for these individual observations of (x;y) is lower than a specified value UCL 1 the sampling is interrupted. Otherwise, the sampling goes on to the second stage, where the remaining items are inspected and the Hotelling statistic T2 2 for the sample means of (x;y) is computed. When the statistic T2 2 is larger than a specified value UCL2, the sample is classified as nonconforming. According to the synthetic control chart procedure, the signal is based on the number of conforming samples between two neighbor nonconforming samples. The proposed chart detects process disturbances faster than the bivariate charts with variable sample size and it is from the practical viewpoint more convenient to administer.

Monitoring bivariate processes

The T2 chart and the generalized variance |S| chart are the usual tools for monitoring the mean vector and the covariance matrix of multivariate processes. The main drawback of these charts is the difficulty to obtain and to interpret the values of their monitoring statistics. In this paper, we study control charts for monitoring bivariate processes that only requires the computation of sample means (the ZMAX chart) for monitoring the mean vector, sample variances (the VMAX chart) for monitoring the covariance matrix, or both sample means and sample variances (the MCMAX chart) in the case of the joint control of the mean vector and the covariance matrix.

Attribute charts for monitoring the mean vector of bivariate processes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A Single Statistic for Monitoring the Covariance Matrix of Bivariate Processes

In this article, we present a new control chart for monitoring the covariance matrix in a bivariate process. In this method, n observations of the two variables were considered as if they came from a single variable (as a sample of 2n observations), and a sample variance was calculated. This statistic was used to build a new control chart specifically as a VMIX chart. The performance of the new control chart was compared with its main competitors: the generalized sampled variance chart, the likelihood ratio test, Nagao's test, probability integral transformation (v(t)), and the recently proposed VMAX chart. Among these statistics, only the VMAX chart was competitive with the VMIX chart. For shifts in both variances, the VMIX chart outperformed VMAX; however, VMAX showed better performance for large shifts (higher than 10%) in one variance.

Bivariate control charts with double sampling

In this article, we consider the T(2) chart with double sampling to control bivariate processes (BDS chart). During the first stage of the sampling, n(1) items of the sample are inspected and two quality characteristics (x; y) are measured. If the Hotelling statistic T(1)(2) for the mean vector of (x; y) is less than w, the sampling is interrupted. If the Hotelling statistic T(1)(2) is greater than CL(1), where CL(1) > w, the control chart signals an out-of-control condition. If w < T(1)(2) <= CL(1), the sampling goes on to the second stage, where the remaining n(2) items of the sample are inspected and T(2)(2) for the mean vector of the whole sample is computed. During the second stage of the sampling, the control chart signals an out-of-control condition when the statistic T(2)(2) is larger than CL(2). A comparative study shows that the BDS chart detects process disturbances faster than the standard bivariate T(2) chart and the adaptive bivariate T(2) charts with variable sample size and/or variable sampling interval.

The Synthetic Control Chart Based on Two Sample Variances for Monitoring the Covariance Matrix

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

The double sampling and the EWMA charts based on the sample variances

We propose a new statistic to control the covariance matrix of bivariate processes. This new statistic is based on the sample variances of the two quality characteristics, in short VMAX statistic. The points plotted on the chart correspond to the maximum of the values of these two variances. The reasons to consider the VMAX statistic instead of the generalized variance vertical bar S vertical bar is its faster detection of process changes and its better diagnostic feature; that is, with the VMAX statistic it is easier to identify the out-of-control variable. We study the double sampling (DS) and the exponentially weighted moving average (EWMA) charts based on the VMAX statistic. (C) 2008 Elsevier B.V. All rights reserved.

Gráfico de controle de VMAX para o monitoramento da matriz de covariâncias

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Gráfico de controle MCMAX para o monitoramento simultâneo do vetor de médias e da matriz de covariâncias

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Joint control charts for monitoring location and dispersion

The T-2 and the generalized variance vertical bar S vertical bar charts are used for monitoring the mean vector and the covariance matrix of multivariate processes. In this article, we propose for bivariate processes the use of the T-2 and the VMAX charts. The points plotted on the VMAX chart correspond to the maximum of the sample variances of the two quality characteristics. The reason to consider the VMAX statistic instead of the generalized variance vertical bar S vertical bar is the user's familiarity with the computation of simple sample variances; we can't say the same with regard to the computation of the generalized variance vertical bar S vertical bar.

Monitoring the covariance matrix with the VMAX chart

In this article, we propose a new statistic to control the covariance matrix of bivariate processes. This new statistic is based on the sample vat-lances of the two quality characteristics, shortly VMAX statistic. The points plotted on the chart correspond to the maximum of the values of these two variances. The reasons to consider the VMAX statistic instead of the generalized variance vertical bar S vertical bar are faster detection of process changes and better diagnostic feature, that is, with the VMAX statistic It is easier to identify the out-of-control variable.

Estudo das propriedades dos gráficos de controle bivariados com amostragem dupla

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)