Comparison of joint control schemes for multivariate normal i.i.d. output

Morais, Manuel Cabral; Schmid, Wolfgang; Ramos, Patrícia Ferreira; Lazariv, Taras; Pacheco, Antònio

doi:10.1007/s10182-018-00331-3

Cited by 2 publications

(20 citation statements)

References 18 publications

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“…An extensive comparison study relying on numerical and simulation results led Morais et al to the conclusion that none of these three joint schemes for

bold-italicμ

and

𝚺

is uniformly better than the others. However, those same results also suggest that the joint scheme 1 has the best overall ARL performance under several out‐of‐control scenarios.…”

Section: Three Joint Schemes For Bold-italicμ and 𝚺mentioning

confidence: 99%

“…The joint distribution of

false(T^{false(1 false)}, U^{false(1 false)} false)

is known under the in‐control and any out‐of‐control scenario (see Morais et al and Morais et al). Regretfully, the one of

false(T^{false(i false)}, U^{false(i false)} false), i = 2, 3

, is only known when

bold-italicμ

is on target and

𝚺

is on target or off target (see once more Morais et al).…”

Section: Misleading Signals and Incorrect Diagnosticsmentioning

confidence: 99%

“…Let the mean vector and the covariance matrix of

boldX

be given by

bold-italicμ = {bold-italicμ}_{0} + \frac{1}{\sqrt{n}} 𝚺_{0}^{1 false/ 2} boldδ,

𝚺 = {false[σ_{i j} false]}_{i, j = 1, \dots, p},

where

boldδ = {false[δ_{i} false]}_{i = 1, \dots, p}

, as suggested by Morais et al If

boldδ \neq bold0

(resp.

𝚺 \neq 𝚺_{0}

), then we shall assume that a sustained shift in the mean vector (resp.…”

Section: Introductionmentioning

confidence: 99%

“…There are fundamentally two types of joint schemes for

bold-italicμ

and

𝚺

. On the one hand, the schemes make use of a single control chart to monitor both parameters; see, for example, Spiring and Cheng, Chen et al, Khoo, Chen and Thaga, Yeh and Lin, Zhang et al On the other hand, the most popular joint schemes that result from running simultaneously two individual charts, one for

bold-italicμ

and another one for

𝚺

, such as the ones discussed by Chen and Thaga, Reynolds and Cho, Hawkins and Maboudou‐Tchao, Machado and Costa, Reynolds and Stoumbos, Zhang and Chang, Costa and Machado, Reynolds and Cho, Ramos et al, Ramos, Ramos et al, Morais et al…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by Walsh, who studied the performance of the pairs of statistics

false(\overset{X}{true}, S^{2} false)

false(\sqrt{n} false(\overset{X}{true} - μ_{0} false) false/ S, S^{2} false)

, and

false(\overset{X}{true}, \sum_{i = 1}^{n} {false(X_{i} - μ_{0} false)}^{2} false/ n false)

for the location and spread of i.i.d. univariate Gaussian output (where

\overset{X}{true}

and

S^{2}

denote the sample mean and variance of the random sample

false(X_{1}, \dots, X_{n} false)

), Morais et al compared the average run length (ARL) performance of what could be considered the multivariate extensions of these three pairs of control statistics in the joint monitoring of

bold-italicμ

and

𝚺

.…”

Section: Introductionmentioning

confidence: 99%

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Misleading signals in joint schemes for the mean vector and covariance matrix

Morais

Schmid

Ramos

et al. 2020

Quality & Reliability Eng

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In multivariate statistical process control, it is recommendable to run two individual charts: one for the process mean vector and another one for the covariance matrix. The resulting joint scheme provides a way to satisfy Shewhart's dictum that proper process control implies monitoring both process location and spread. The multivariate quality characteristic is deemed to be out of control whenever a signal is triggered by either individual chart of the joint scheme. Consequently, a shift in the mean vector can be misinterpreted as a shift in the covariance matrix and vice versa. Compelling results are provided to give the quality control practitioner an idea of how joint schemes for the mean vector and covariance matrix are prone to trigger misleading signals that will likely lead to a incorrect diagnostic of which parameter has changed.

show abstract

“…An extensive comparison study relying on numerical and simulation results led Morais et al to the conclusion that none of these three joint schemes for

bold-italicμ

and

𝚺

is uniformly better than the others. However, those same results also suggest that the joint scheme 1 has the best overall ARL performance under several out‐of‐control scenarios.…”

Section: Three Joint Schemes For Bold-italicμ and 𝚺mentioning

confidence: 99%

“…The joint distribution of

false(T^{false(1 false)}, U^{false(1 false)} false)

is known under the in‐control and any out‐of‐control scenario (see Morais et al and Morais et al). Regretfully, the one of

false(T^{false(i false)}, U^{false(i false)} false), i = 2, 3

, is only known when

bold-italicμ

is on target and

𝚺

is on target or off target (see once more Morais et al).…”

Section: Misleading Signals and Incorrect Diagnosticsmentioning

confidence: 99%

“…Let the mean vector and the covariance matrix of

boldX

be given by

bold-italicμ = {bold-italicμ}_{0} + \frac{1}{\sqrt{n}} 𝚺_{0}^{1 false/ 2} boldδ,

𝚺 = {false[σ_{i j} false]}_{i, j = 1, \dots, p},

where

boldδ = {false[δ_{i} false]}_{i = 1, \dots, p}

, as suggested by Morais et al If

boldδ \neq bold0

(resp.

𝚺 \neq 𝚺_{0}

), then we shall assume that a sustained shift in the mean vector (resp.…”

Section: Introductionmentioning

confidence: 99%

“…There are fundamentally two types of joint schemes for

bold-italicμ

and

𝚺

bold-italicμ

and another one for

𝚺

Section: Introductionmentioning

confidence: 99%

“…Inspired by Walsh, who studied the performance of the pairs of statistics

false(\overset{X}{true}, S^{2} false)

false(\sqrt{n} false(\overset{X}{true} - μ_{0} false) false/ S, S^{2} false)

, and

false(\overset{X}{true}, \sum_{i = 1}^{n} {false(X_{i} - μ_{0} false)}^{2} false/ n false)

for the location and spread of i.i.d. univariate Gaussian output (where

\overset{X}{true}

and

S^{2}

denote the sample mean and variance of the random sample

false(X_{1}, \dots, X_{n} false)

), Morais et al compared the average run length (ARL) performance of what could be considered the multivariate extensions of these three pairs of control statistics in the joint monitoring of