The synthetic control chart based on two sample variances for monitoring the covariance matrix

Machado, Marcela Aparecida Guerreiro; Costa, Antônio Fernando Branco; Rahim, M. A.

doi:10.1002/qre.992

Cited by 43 publications

(23 citation statements)

References 35 publications

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“…This chart is suitable for detecting shifts in the ratio

μ_{X_{1}} / μ_{X_{2}}

as well as on ρ . Other multivariate synthetic charts reported in the literature include the S1 chart based on the VMAX statistic by Machado et al, which can be used for monitoring the covariance matrix Σ of a bivariate normal process. The proposed charting statistic utilizes the sample variances of two correlated random variables, ie,

VMAX = max ({},, S_{x}^{2}, S_{y}^{2})

, where

S_{x}^{2} = \sum_{i = 1}^{n} x_{i}^{2} / n

S_{y}^{2} = \sum_{i = 1}^{n} y_{i}^{2} / n

and ( x i , y i ), i = 1, 2, … , n , is a sample of size n from a bivariate normal process with covariance matrix Σ .…”

Section: Multivariate Synthetic Chartsmentioning

confidence: 99%

An overview of synthetic‐type control charts: Techniques and methodology

Rakitzis

Chakraborti

Shongwe

et al. 2019

Quality & Reliability Eng

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In this paper, we provide an overview of a class of control charts called the synthetic charts. Synthetic charts are a combination of a traditional chart (such as a Shewhart, CUSUM, or EWMA chart) and a conforming run‐length (CRL) chart. These charts have been considered in order to maintain the simplicity and improve the performance of small and medium‐sized shift detection of the traditional Shewhart charts. We distinguish between different types of synthetic‐type charts currently available in the literature and highlight how each is designed and implemented in practice. More than 100 publications on univariate and multivariate synthetic‐type charts are reviewed here. We end with some concluding remarks and a list of some future research ideas.

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“…This chart is suitable for detecting shifts in the ratio

μ_{X_{1}} / μ_{X_{2}}

VMAX = max ({},, S_{x}^{2}, S_{y}^{2})

, where

S_{x}^{2} = \sum_{i = 1}^{n} x_{i}^{2} / n

S_{y}^{2} = \sum_{i = 1}^{n} y_{i}^{2} / n

and ( x i , y i ), i = 1, 2, … , n , is a sample of size n from a bivariate normal process with covariance matrix Σ .…”

Section: Multivariate Synthetic Chartsmentioning

confidence: 99%

An overview of synthetic‐type control charts: Techniques and methodology

Rakitzis

Chakraborti

Shongwe

et al. 2019

Quality & Reliability Eng

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show abstract

“…Using the same notion, Ghute and Shirke 2 integrated the Hotelling's T 2 and CRL charts to form the multivariate synthetic T 2 chart, where it was shown that the synthetic T 2 chart increases the sensitivity of the Hotelling's T 2 chart in detecting shifts in the mean vector, for the zero-state mode. Other works on multivariate synthetic charts reported in the literature are as follow: Machado et al 3 presented the VMAX statistic, based on the sample variances of two variables to monitor the covariance matrix of a bivariate process. Ghute and Shirke 4 proposed the multivariate synthetic chart for process dispersion by integrating the generalized sample variance |S| chart and the CRL chart.…”

Section: Introductionmentioning

confidence: 99%

Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

Khoo

Wong

et al. 2011

Quality & Reliability Eng

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The average run length (ARL) is usually used as a sole measure of performance of a multivariate control chart. The Hotelling's T 2 , multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts are commonly optimally designed based on the ARL. Similar to the case of univariate quality control, in multivariate quality control, the shape of the run length distribution changes in accordance to the magnitude of the shift in the mean vector, from highly skewed when the process is in-control to nearly symmetric for large shifts. Because the shape of the run length distribution changes with the magnitude of the shift in the mean vector, the median run length (MRL) provides additional and more meaningful information about the in-control and out-of-control performances of multivariate charts, not given by the ARL. This paper provides a procedure for optimal designs of the multivariate synthetic T 2 chart for the process mean, based on MRL, for both the zero and steady-state modes. Two Mathematica programs, each for the zero state and steady-state modes are given for a quick computation of the optimal parameters of the synthetic T 2 chart, designed based on MRL. These optimal parameters are provided in the paper, for the bivariate case with sample sizes, n ∈{4, 7, 10}. The MRL performances of the synthetic T 2 , MEWMA and Hotelling's T 2 charts are also compared.

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“…Subsequently, Aparisi and de Luna (2009) developed a synthetic T 2 chart that does not detect shifts in a region of admissible shifts but that detects only important shifts. Machado et al (2009) introduced a synthetic chart based on two sample variances for monitoring the covariance matrix of bivariate processes. Khilare and Shirke (2010) and Pawar and Shirke (2010) presented nonparametric synthetic charts.…”

Section: Introductionmentioning

confidence: 99%

Some Comments Regarding the Synthetic Chart

Machado

Costa

2014

Communications in Statistics - Theory and Methods

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The steady-state average run length (ARL) is a function of the in-control probabilities of being in each nonabsorbing state. Davis and Woodall (2002) tabulated values that are significantly smaller than the steady-state ARLs, because they used the out-of-control probabilities. The synthetic chart signals when a second sample point falls beyond the control limits, no matter whether one of them falls above the centerline and the other falls below it. The side-sensitive version of the synthetic chart does not signal when the points beyond the control limits are on opposite sides. With this rule, the chart detects mean changes more quickly.

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The synthetic control chart based on two sample variances for monitoring the covariance matrix

Cited by 43 publications

References 35 publications

An overview of synthetic‐type control charts: Techniques and methodology

An overview of synthetic‐type control charts: Techniques and methodology

Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

Some Comments Regarding the Synthetic Chart

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