Simultaneous Univariate X_bar Charts to Control Bivariate Processes with Autocorrelated Data

In this paper, synthetic T2 chart is developed to monitor bivariate process with correlated variables and autocorrelated observations. The proposed chart is a combination of the Hotelling's T2 chart and the conforming run length chart. The operation and design of the chart are described when observations are autocorrelated and cross correlated. The first‐order vector autoregressive process VAR (1) is used to model the bivariate data from an autocorrelated process of interest. Using an average run length as performance measure criterion in the VAR (1) model, it is observed that autocorrelation seriously impact the performance of the synthetic T2 chart. To reduce the effect of autocorrelation on the performance of the synthetic T2 chart, the skip and mixed sampling strategies are implemented to form rational subgroups in the construction of synthetic T2 chart. The average run length performance of the synthetic T2 chart implementing these strategies is compared with that of the standard strategy of formation of rational subgroups. It is observed that implementing skip and mixed sampling strategies within rational subgroup improves the performance of the synthetic T2 chart.

“…Leoni et al obtained the cross‐covariance matrix

Σ_{\overset{true¯}{X}}

of the sample mean vector

\overset{X}{true}

when sample items are collected according to rational subgroup of size n as

\sum_{\overset{true¯}{X}} = ([], \begin{array}{c} ξ_{11} & ξ_{12} \\ ξ_{21} & ξ_{22} \end{array}),

where ξ_{11} = \frac{σ_{1}^{2}}{n} ([], 1 + \frac{2}{n} {false\sum}_{j = 1}^{n - 1} ((), n - j) a^{j}) ξ_{22} = \frac{σ_{2}^{2}}{n} ([], 1 + \frac{2}{n} {false\sum}_{j = 1}^{n - 1} ((), n - j) b^{j}) ξ_{12} = \frac{σ_{12}^{2}}{n} ([], 1 + \frac{1}{n} {false\sum}_{j = 1}^{n - 1} ((), n - j) a^{j} + \frac{1}{n} {false\sum}_{j = 1}^{n - 1} ((), n - j) b^{j}) and ξ_{12} = ξ_{21} .

…”

Section: Var (1) Model and Cross‐covariance Matrixmentioning

confidence: 99%

“…Leoni et al 11 obtained the cross-covariance matrix Σ X of the sample mean vector X when sample items are collected according to rational subgroup of size n as…”

Section: Var (1) Model and Cross-covariance Matrixmentioning

confidence: 99%

\overset{X}{true}

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

New sampling strategies to reduce the effect of autocorrelation on the synthetic T² chart to monitor bivariate process

Dargopatil

Ghute

2018

“…The majority of works dedicated to this subject deal mainly with the monitoring of the mean vector, 1,2 the covariance matrix, [3][4][5][6] and autocorrelated observations. [7][8][9][10] The idea of using attribute charts to control the mean vector of bivariate processes has been already explored (see, for example, Ho and Costa 11 and Melo et al 12 ; however, we cannot say the same regarding the covariance matrix.…”

Section: Introductionmentioning

confidence: 99%

Attribute control charts for monitoring the covariance matrix of bivariate processes

Machado

Ho²,

Costa

2018

In this article, we consider the use of 3 attribute charts-the np xy , the np w and the Max D charts-to control the covariance matrix of bivariate processes. In comparison with the generalized variance |S| chart, the 3 attribute charts signal faster, with smaller samples, all kind of disturbances, except when the 2 variables are highly correlated. To compete with the VMAX chart, the Max D chart needs larger samples, but no more than twice bigger. An example illustrates the monitoring of the covariance matrix using the Max D and np w .

“…Abu‐Shawiesh, Kibria, and George proposed a new bivariate chart with the Hoteling's T 2 format but based on a robust estimator, which is insensitive to outliers. Leoni, Machado, and Costa proposed the use of 2

\overset{X}{true}

charts to control the mean vector of bivariate processes with autocorrelated data. When the assignable cause changes the position of the 2 means, the 2

\overset{X}{true}

charts tend to perform better than the T 2 chart as the autocorrelation and the cross‐correlation increase.…”

Section: Introductionmentioning

confidence: 99%

Monitoring bivariate and trivariate mean vectors with a Shewhart chart

Leoni¹,

Costa

2017

Self Cite

In this article, we propose the use of the mean chart to control multivariate processes. The basic idea is to control the mean vector of bivariate (X, Y) and trivariate (X, Y, Z) processes by alternating the charting statistic of the Shewhart chart. If the mean of X observations was the charting statistic to obtain the current sample point, then the mean of Y observations will be the charting statistic to obtain the next sample point (for the trivariate case, the mean of Z observations will be the charting statistic to obtain the sample point subsequent to the next one). As a Shewhart chart, the signal is given anytime a sample point is plotted beyond the control limits, independent of the charting statistic in use. A fair comparison between the proposed chart and the Hotelling chart is based on an equal number of measurements per sample. The Shewhart chart with alternated charting statistic (ACS) always outperforms the Hotelling chart, except for specific types of disturbances in quality characteristics highly correlated (ρ = 0.7). The ACS chart is substantially easier to operate and faster than the Hotelling chart in signaling changes in the mean vector of bivariate and trivariate processes. Even with fewer measurements per sample, the trivariate ACS chart outperforms the Hotelling chart.