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DOI: ./qre.
RESEARCH ARTICLE
Multivariate cumulative sum control chart for
compositional data with known and estimated process
parameters
Muhammad Imran1Jinsheng Sun1Fatima Sehar Zaidi2
Zameer Abbas3Hafiz Zafar Nazir4
School of Automation, Nanjing
University of Science and Technology,
Nanjing, China
Nanjing University of Aeronautics and
Astronautics, Nanjing, China
Government Ambala Muslim Graduate
College, Sargodha, Pakistan
Department of Statistics, University of
Sargodha, Sargodha, Pakistan
Correspondence
Jinsheng Sun, Nanjing University of
Science and Technology, Nanjing, China.
Email: jssun@.com
Abstract
This article uses the classic multivariate cumulative sum (MCUSUM) chart
scheme proposed by Crossier () to present a new modified MCUSUM chart
for compositional data (CoDa). For this purpose, the data are first transformed
using isometric log-ratio (ilr) coordinates representation to eliminate the con-
stant sum constraint of CoDa.TheMCUSUM-CoDa control chart has been
defined along with the performance measures of the proposed chart using the
average run length (ARL). Besides, the Markov chain method has been used to
study the ARL performance of the proposed chart. Assuming that the ilr trans-
formed data are normally distributed, the proposed MCUSUM-CoDa charts have
been compared with existing competitors such as 𝑇2-CoDa and MEWMA-CoDa
charts. The comparison shows that the proposed chart has better performance
than the 𝑇2-CoDa control charts, while the performance of the proposed chart
is comparable with the MEWMA-CoDa chart. The effect of the estimated mean
vector and variance-co-variance matrix on run-length characteristics of the pro-
posed MCUSUM-CoDa control chart has also been studied in this paper. For the
ARL performance of MCUSUM-CoDa with estimated parameters Monte Carlo
simulation has been adopted. The effect of the number of variables 𝑝, sample size
𝑛, and subgroup size 𝑚has also been studied on the data’s upper control limit
(UCL)andARL. In the end, two illustrative examples of the particle size distri-
bution of plants and production of muesli are provided to represent the practical
implementation of the MCUSUM-CoDa chart.
KEYWORDS
CoDa, Markov chain, MCUSUM, parameter estimation, performance
1 INTRODUCTION
The goal of statistical process control (SPC) is to identify a shift in the parameters of the underlying process as quickly
as possible after it has occurred. Control charts are the most vital SPC tools; Walter Shewhart initially invented them in
the s and s. Since then, several modern control charts have been proposed for different data structures. (See for
instance,–)
Qual Reliab Engng Int. ;–. © John Wiley & Sons Ltd. 1wileyonlinelibrary.com/journal/qre
2IMRAN .
A control chart aims to detect assignable causes of the shifts so that they can be detected and removed before many
nonconforming units have been produced. However, during the process, the main goal is to detect the shift as soon as
possible, no matter if the size of the shift is large or small, and to find out the main cause of the shifts. Thus, control charts
have better performance if they identify the shifts in the process when it is out-of-control (OOC) and generate fewer false.
Ali et al.examine that the parametric control charts produce more false alarms and unacceptable OOC signals when
the underlying distribution of the process is not normal. A new progressive mean (PM) control chart is proposed for
monitoring drift in the proportion of nonconforming products. The proposed chart’s design is evaluated and compared
using various properties of run-length distribution suggested by Abbas et al.
It is sometimes difficult to identify which control chart is suitable for specific data in practice. It can be determined
by studying the distribution of the underlying process data. The compositional data (CoDa) vectors are vectors having
positive components presented as percentages, ratios, proportions, or parts of some whole. CoDa can be used in several
areas, including chemical research surveys, engineering sciences, and econometric data analysis. Some of the latest articles
that deal with statistical methods and processing of CoDa are discussed here. For instance, Egozcue et al.studied the
linear association in compositional data analysis. Further, Morais et al.studied that an automobile market application is
presented in which we model brand market share as a functionof media investments, controlling for the brand’s price and
scrapping incentive. Egozcue and Pawlowsky-Glahnstudied that the sample space’s algebraic–geometric structure was
created to adhere to these principles at the turn of the millennium, and sample space and the structure of compositional
data. Blasco-Duatis et al.worked on the agenda-setting theory, priming, and the spiral of silence in political party Twitter
accounts are explored. Carreras-Simo and Coenders studied Principal Component Analysis of Financial Statements
using compositional data. Recently, Coenders and Ferrer-Rosell analyzed compositional data for tourism. The log-ratio
approach to the analysis of compositional data has matured. Many applied problems have been solved using J. Aitchison’s
principles and statistical tools from the s, and for more detailed information about CoDa, refer to Aitchison and
Pawlowsky-Glahn et al. The CoDa variable aggregates are restricted to constant values, so treating them in the same
context as standard multivariate data is impossible. There are just a few articles in the SPC literature that investigate and
address CoDa control processes. Boyles can be cited as having been examined to monitor compositional process results.
He proposed using a chi-square control chart.
In multivariate control charts, a graphical technique for evaluating OOC signals is proposed by Vives-Mestres et al.,
and a 𝑇2chart for the composition vector consists of three parts by Vives-Mestres et al. They suggested a chart for
specific CoDa aspects focused on the isometric log-ratio (ilr)𝑇2
𝐶control chart transformation. After deleting one element,
they compared it to the traditional 𝑇2control chart. Guevara-González et al. analyzed 𝑇2charts for CoDa by monitoring
the profiles using a Dirichlet regression methodology. Recently, Zaidi et al. worked on the effect of measurement error
on the ARL performance of Hotelling 𝑇2-CoDa control charts. After, Zaidi et al. also studied the effect of measurement
error on the ARL performance of MEWMA-CoDa control charts.
CUSUM charts are well-known to detect smaller and more frequent shifts than Shewhart charts, and they are supposed
to be one of the most frequent and extensively applied in practice. Furthermore, these charts are better for process con-
trol because of the sequence’s nature of the data processing. Hawkins and Olwell provide general details of CUSUM
control charts to the reader. By performing multi-objective optimization, it is necessary to simultaneously optimize sev-
eral different objective functions, and the multivariate control chart method is indifferent to small and moderate variable
shifts examined by Hotelling. Since, the literature includes many multivariate CUSUM procedures and one multivari-
ate EWMA procedure, both of which utilize additional information. Multivariate charts are divided into two categories:
direction-dependent and directionally invariant. The difference between the off and on target mean describes the average
run length (ARL) performance of directionally invariant control charts. In the direction of a multivariate process, multiple
univariate methods are used. These multivariate charts are typically used to detect changes in process parameters around
their respective axes, and they must be indicated in a specific direction. If you are observed only in one direction by a
difference in the mean variable, in a multivariate situation, Healy discovered that a univariate CUSUM chart focusing
on a linear variable simultaneous analysis could be useful.
Although the optimal ARL performance for this technique’s anticipated transition is provided, it does not identify the
shift if the mechanism has changed unwontedly or unexpectedly. Hawkins and Olwell applied this approach to the
case where several interest rules are defined. Wang and Huang proposed adaptive multivariate CUSUM control chart
for the shift in location. Further, Ajadi and Riaz studied improvement in process monitoring using a combination of
multivariate CUSUM and multivariate EWMA control charts. Adegoke studied the performance improvement of the
MCUSUM control chart by shrinking the variance covariance matrix. Zaman et al. proposed adaptive CUSUM location
control charts based on score function. Several researchers have proposed many adaptive and enhanced MCUSUM charts.
For instance, one can see, Refs. –
IMRAN . 3
Process parameters are unknown in most applications; that is why estimates are used to check the control charts’ per-
formance studied by Jensen et al. To obtain reasonable estimates in the multivariate setting is not an easy task, especially
in higher dimensions is proposed by Sain. As a result, control charts are compared using various methods to estimate
the co-variance matrix. If this variability is not considered, the control chart’s IC and OOC performance can be severely
harmed. A more robust procedure that takes advantage of the serial nature of the observations studied by Refs. , Zhang
et al. and Qiu et al. suggested that when the mean vector changes in a step, adjacent observations are more likely to be
the same or approximately the same mean vector. The mean vector must be taken into consideration when calculating the
sampling distribution. These new control chart’s parameters are optimized using genetic algorithms to match a required
in-control ARL and minimize the OOC ARL for a given mean shift while optimizing the gauge dimensions examined by
Ho and Aparisi. However, the effectof parameter estimation on CUSUM performance for normal observations has been
studied by Bagshaw and Johnson as well as Hawkins and Wu and Jones et al.
This paper proposes a Multivariate CUSUM chart for composition data. This article is arranged as follows: in Section .,
the introduction and some basic geometry for CoDa has been presented. In Section , the MCUSUM control chart is
introduced for CoDa with known and estimated parameters. In Section , the performance of MCUSUM-CoDa control
chart is examined. In Section , the comparison of the proposed chart with the previously defined charts has been studied.
Finally, two comprehensive examples of grit production and muesli production are given for the practical implementation
of the chart in Section . Section addresses the results and discussions.
1.1 Compositional data (CoDa)
Compositional data are defined as a 𝑝-part composition consisting of a row vector𝐲=(𝑦
1,…,𝑦
𝑝)defined on the simplex
space 𝑝. Where 𝑝can be defined as
𝑝=𝐲=(𝑦
1,𝑦
2,…,𝑦
𝑝)𝑦𝑖> 0,𝑖 = 1,2,…,𝑝 &
𝑝
𝑖=1
𝑦𝑖=𝜅
,()
Where 𝜅is a constant and is always greater than zero. 𝜅can take different values, such as 𝜅 = 100,ifwedealwithpropor-
tions, and 𝜅=1, if the composition components are in probabilities or proportions. In this paper, all the compositional
vectors are supposed to be row vectors. If two vectors carry the same relative information, they are said to be composi-
tionally equivalent. For example, 𝐲 = (0.75, 0.1, 0.15) and 𝐳 = (75, 10, 15) are not equal numerically but they convey same
information. So, in this case, we use a closure function that is defined as
(𝐲) = 𝜅𝑦1
𝑝
𝑖=1 𝑦𝑖
,𝜅𝑦2
𝑝
𝑖=1 𝑦𝑖
,…, 𝜅𝑦𝑝
𝑝
𝑖=1 𝑦𝑖.()
Using the closure mentioned above function, we can say that (𝐲) = (𝐳).
Because of the constant sum, we cannot use the standard Euclidean geometry used for real space (i.e., ℝ𝑝). For exam-
ple, if we have two compositional vectors, 𝐲 = (0.1, 0.65, 0.35) ∈ 𝑝and 𝐳 = (0.25, 0.5, 0.25) ∈ 𝑝then their sum using
Euclidean geometry will be 𝐲 + 𝐳 = (0.35, 1.15, 0.6) ∉ 𝑝and similarly, if we multiply a compositional vector with a scalar
such that 5 × 𝐲 = (0.5, 3.25, 0.0875) ∉ 𝑝. So, we can say that the Euclidean geometry operators are not suitable in the
case of CoDa. Aitchison proposed a specific geometry known as Aitchison’s geometry with new operators to overcome
this problem. These operators are defined as
∙the perturbation operator ⊕of 𝐲∈𝑝by 𝐳∈𝑝(substitute of “+”) defined as
𝐲⊕𝐳=(𝑦1𝑧1,𝑦
2𝑧2,…,𝑦
𝑝𝑧𝑝), ()
∙the powering operator ⊙of 𝐲∈𝑝by a constant 𝑐∈ℝ(substitute of multiplication with a scalar) defined as
𝑐⊙𝐲=(𝑦𝑐
1,𝑦
𝑐
2,…,𝑦
𝑐
𝑝). ()
4IMRAN .
CoDa can be dealt with in two ways; one way is to use the original data. But in that case, we have to deal with the
constraint of a constant sum. The other option is to transform the data into real data using the predefined log-ratio trans-
formations. One of them for compositional vector 𝐲∈𝑝is the centered log-ratio transformation that is defined as
clr(𝐲) = ln 𝑦1
𝑦𝐺
,ln 𝑦2
𝑦𝐺
,…,ln 𝑦𝑝
𝑦𝐺,()
where
𝑦𝐺is the component-wise geometric mean of 𝐲, that is,
𝑦𝐺=𝑝
𝑖=1
𝑦𝑖1
𝑝
=exp1
𝑝
𝑝
𝑖=1
ln 𝑦𝑖.()
Another log-ratio transformation for a compositional vector 𝐲∈𝑝is the isometric log-ratio transformation defined as
ilr(𝐲) = 𝐲∗= clr(𝐲)𝐁⊺,()
wherewehavemanypossibleoptionsfor𝐁. Where 𝐁is a matrix of size (𝑝 − 1, 𝑝). The one we used in this paper is given
below:
𝐵𝑖,𝑗 =
1
(𝑝−𝑖)(𝑝−𝑖+1) 𝑗≤𝑝−𝑖
−𝑝−𝑖
𝑝−𝑖+1 𝑗=𝑝−𝑖+1
0 𝑗>𝑝−𝑖+1
.()
Conversely, we can use the inverse isometric log-ratio to transform the ilr-coordinates 𝐲∗into the composition coordi-
nates 𝐲. The iilr transformation is defined as
ilr−1(𝐲∗)=𝐲=(exp(𝐲∗𝐁)). ()
In this paper, the compositional coordinates 𝐲∈𝑝are denoted as vectors or matrices without a “∗” sign. At the same
time, the ilr transformed coordinates 𝐲∗∈ℝ
𝑝−1 are denoted as vectors or matrices with a “∗.”
2 MULTIVARIATE CUSUM CONTROL CHART FOR COMPOSITIONAL DATA
2.1 MCUSUM control chart for CoDa with known parameters
Assumes that, we have 𝑚measurements of the quality characteristics 𝐱𝑖,𝑗,𝐱𝑖,1,…,𝐱
𝑖,𝑡 where 𝑖 = 1, 2, … and 𝑗 = 1, 2, … , 𝑡 .
Let 𝐱∗
𝑡be the ilr transformed coordinates of 𝐱𝑡. Then, we have 𝐱𝑡∼MNOR
𝑝(𝝁∗
0,𝚺
∗)and 𝐱𝑡∼MNOR
𝑝(𝝁∗
1,𝚺
∗), where
the IC and OOC process mean is defined by 𝝁∗
0and 𝝁∗
1, respectively. Where 𝐱∗
𝑡=ilr(𝐱𝑡).AMCUSUM chart for CoDa may
be suggested using a similar method as in Tran et al. The MCUSUM chart monitors the following statistic:
𝐂𝑡=𝑚𝐬⊺
𝑡𝚺−1
0𝐬𝑡−11∕2,𝑡 = 1,2,… ()
where 𝐬𝑡is the vector in ℝ𝑝−1 defined as
𝐬𝑡=0 if 𝐐𝑡≤𝑘
𝐬𝑡=(𝐬
𝑡−1 +𝐱∗
𝑡−𝝁
∗
0)(1 − 𝑘∕𝐐𝑡)if 𝐐𝑡>𝑘, ()
IMRAN . 5
with 𝐬0=0and 𝑘>0.Let
𝐐𝑡=𝑚𝐬𝑡−1 +𝐱∗
𝑡−𝝁
∗
0⊺𝚺−1
0𝐬𝑡−1 +𝐱∗
𝑡−𝝁
∗
01∕2.()
The MCUSUM chart displays a signal when 𝐂𝑡>ℎ, where ℎis the UCL. A predefined IC ARL0is used to select the
suitable value of ℎ.
To assess the MCUSUM-CoDa control chart’s run-length efficiency, we implement a Markov chain approximation orig-
inally suggested by Crosier. The Markov chain model needed to measure the MCUSUM-CoDa chart’s ARL is given
below,
The potential values of 𝐂𝑡are expressed by 𝑓+1states, according to Brook and Evan. Each of the states is an absorbing
condition of 𝐂𝑡>ℎ.The𝑓transient states numbered 0, 1, 2, … , (𝑓 − 1) reflect 𝐂𝑡values between and ℎ. The Markov
chain should be seen as a discrete random variable (let us label it 𝐂⊺
𝑡) with values 0, 𝑡, 2𝑡, … , 𝑓𝑡, where
𝑤= 2ℎ
2𝑓 − 1 .()
To find the transition probabilities, we need the transient states. That are,
𝑃𝐂⊺
𝑡=𝑗𝑤𝐂⊺
𝑡−1 =𝑖𝑤
,where 𝑖,𝑗 ∈ {0,1,2,…,(𝑓− 1)}. ()
The above equation shows the transient state probabilities with 𝐂𝑡=max[0,𝐐
𝑡−𝑘], where
𝐐𝑡=𝑚𝐬𝑡−1 +𝐱∗
𝑡−𝝁
∗
0⊺𝚺−1𝐬𝑡−1 +𝐱∗
𝑡−𝝁
∗
01∕2,()
is used to find the transient state probabilities. Instead of being treated as a random variable, 𝐐𝑡is treated as a con-
stant because of transient state probabilities conditional nature. 𝐸(𝐬𝑡−1 +𝐱
𝑡−𝑎)=𝐬
𝑡−1 and Var(𝐬𝑡−1 +𝐱
𝑡−𝑎)=𝚺
for the on-target event. Note that 𝐐𝑡has a noncentral chi-square distribution with noncentrality parameter [𝐂𝑡=
[𝑚(𝐬⊺
𝑡−1𝚺−1𝐬𝑡−1)]1∕2 =𝐂
𝑡−1, where 𝐱∗
𝑡follows a multivariate normal distribution. When 𝑗=0.
𝑃𝐂⊺
𝑡=0𝐂⊺
𝑡−1 =𝑖𝑤
=𝑃
𝛘2
𝑝−1,𝑖𝑤 ⩽𝑘+𝑤∕2
,()
and when 𝑗>0
𝑃𝐂⊺
𝑡=𝑗𝑤𝑃𝐂⊺
𝑡−1 =𝑖𝑤
=𝑃
𝑘 + (𝑗 − 0.5)𝑤 < 𝛘2
𝑝−1,𝑖𝑤 ⩽ 𝑘 + (𝑗 + 0.5)𝑤,()
where 𝐐𝑛has a noncentral chi-square distribution with the noncentrality parameter 𝑖𝑤 with 𝑝−1degree of freedom.
Similar to Refs. ,,, we have also used zero state ARL to investigate the performance of the proposed chart. Accord-
ing to Brook and Evan, the zero-state ARL for many Markov chains having different sizes and then using extrapolation
of the continuous case using the formula
ARL(𝑚) = asymptotic(ARL)+𝐵∕𝑚+𝐶∕𝑚
2.()
2.2 MCUSUM control chart for CoDa with estimated parameters
When 𝝁∗
0,𝚺
∗
0are unknown, 𝑛ICsamples of size 𝑚have been used to estimate them. The estimated value of IC process
mean vector 𝝁∗
0can be found using the given formula,
𝑿∗=
Σ𝑛
𝑗=1
𝑋∗
𝑗
𝑛,()
6IMRAN .
where
𝑋∗
𝑗is the 𝑗th sample mean vector when 𝑗 = 1, 2, … , 𝑛. The estimated value of IC process variance covariance matrix
𝚺∗
0can be found using the given formula,
𝑺∗=
Σ𝑛
𝑗=1𝑆∗
𝑗
𝑛,()
where 𝑆𝑗is the within-sample variance covariance matrix. Three different cases can be considered for the estimation of
parameters.
Case-I: When the process mean is known and the standard deviation is to be estimated. The 𝑀𝐶𝑈𝑆𝑈𝑀-CoDa control
chart statistics can be written as
𝐂𝑡=𝑚𝐬⊺
𝑡
𝑺∗−1𝐬𝑡−11∕2 , 𝑡 = 1,2,… ()
where 𝐬𝑡is the vector in ℝ𝑝−1 defined as
𝐬𝑡=0 if 𝐐𝑡≤𝑘
𝐬𝑡=𝐬𝑡−1 +𝐱∗
𝑡−𝝁
∗
0(1 − 𝑘∕𝐐𝑡)if 𝐐𝑡>𝑘, ()
with 𝐬0=0and 𝑘>0.Let
𝐐𝑡=𝑚𝐬𝑡−1 +𝐱∗
𝑡−𝝁
∗
0⊺
𝑺∗−1𝐬𝑡−1 +𝐱∗
𝑡−𝝁
∗
01∕2.()
Case-II: When the process standard deviation is known, and the process mean to be estimated. The 𝑀𝐶𝑈𝑆𝑈𝑀-CoDa
control chart statistics can be written as
𝐂𝑡=𝑚𝐬⊺
𝑡𝚺−1
0𝐬𝑡−11∕2, 𝑡 = 1,2,… ()
where 𝐬𝑡is the vector in ℝ𝑝−1 defined as
𝐬𝑡=0 if 𝐐𝑡≤𝑘
𝐬𝑡=(𝐬
𝑡−1 +𝐱∗
𝑡−
𝑿∗)(1 − 𝑘∕𝐐𝑡)if 𝐐𝑡>𝑘, ()
with 𝐬0=0and 𝑘>0.Let
𝐐𝑡=𝑚𝐬𝑡−1 +𝐱∗
𝑡−
𝑿∗⊺𝚺−1
0𝐬𝑡−1 +𝐱∗
𝑡−
𝑿∗1∕2
.()
Case-III: When the process standard deviation and the process mean both are to be estimated. The 𝑀𝐶𝑈𝑆𝑈𝑀-CoDa
control chart statistics can be written as
𝐂𝑡=𝑚𝐬⊺
𝑡
𝑺∗−1𝐬𝑡−11∕2 ,𝑡 = 1,2,… ()
where 𝐬𝑡is the vector in ℝ𝑝−1 defined as
𝐬𝑡=0 if 𝐐𝑡≤𝑘
𝐬𝑡=(𝐬
𝑡−1 +𝐱∗
𝑡−
𝑿∗)(1 − 𝑘∕𝐐𝑡)if 𝐐𝑡>𝑘, ()
with 𝐬0=0and 𝑘>0.Let
𝐐𝑡=𝑚𝐬𝑡−1 +𝐱∗
𝑡−
𝑿∗⊺
𝑺∗−1𝐬𝑡−1 +𝐱∗
𝑡−
𝑿∗1∕2
.()
IMRAN . 7
As the MCUSUM control chart possesses the property of directional invariant, so when the parameters are unknown,
the run-length distribution of the proposed chart depends on the reference parameter 𝑘, the number of variables 𝑝,sample
size 𝑛, and the subgroup size 𝑚.TheOOC ARL of the chart also depends on noncentrality parameter 𝛿equals to
𝛿=𝝁∗
1−
𝑿∗
𝑺∗−1𝝁∗
1−
𝑿∗⊺.()
It is solely dependent on the values of 𝑝,𝑚,𝑛,and𝑘that the distribution of a multivariate directionally invariant
CUSUM chart with estimated parameters is determined. The distribution of the OOC run-length depends on such con-
stants and the shift size measured by the noncentrality parameter 𝛿. As a result, it is possible to analyze the performance
of the multivariate CUSUM chart without knowing the IC values or estimates of the processing parameters.
Several researchers have used integral equations and Markov chain approximation to inspect multivariate control
charts’ run-length properties. When the parameters are estimated from 𝑛ICPhase I samples, there is no direct way to
analyze the run-length performance of the MCUSUM charts using the double integral equations or the approximation to
the Markov chain. As a result, Monte Carlo simulations have been used to study the zero-state ARL performance of the
defined chart.
While designing the MCUSUM-CoDa control chart’s run-length efficiency with estimated parameters, the main steps
involved are,
∙Determine the upper control limit for the desired combination of 𝑝,𝑚,𝑛,and𝑘according to the fixed value of ARL.
∙Generate a random vector
𝑋from a multivariate normal distribution with (𝜇0,Σ
0∕𝑚𝑛) and a matrix
𝑆∗from Wishart
distribution with (Σ0, 𝑚(𝑛 − 1)). These parameters are then used as estimated parameters for phase I.
∙Generate a random vector 𝑋𝑖to represent the phase II information observed at the time 𝑖.ComputeMCUSUM-CoDa.
∙Compare the values of MCUSUM-CoDa chart statistics with the corresponding UCL.
∙Repeat steps – to record run length.
∙Repeat steps – until , repetitions have been completed.
3 PERFORMANCE OF THE MCUSUM-CODA CONTROL CHART
3.1 When parameters are known
The ARL’s of multivariate CUSUM schemes procedures are based on the Hotelling 𝑇2statistics, where the noncentrality
parameter mainly depends on the mean vector and variance–covariance matrix. The MCUSUM chart has good ARL
properties, as the observations in opposite directions can cancel the effect of IC values. When dealing with the shift in
the process mean, the cancellation of the shift effects raises the IC ARL0and decreases the OOC ARL1’s so that the false
alarm rate is minimized.
While designing the MCUSUM chart scheme, the main step is to choose the value of 𝑘=𝑑∕2, where the shift in the
mean vector depends mainly on the noncentrality parameter 𝑑.Thevalueof𝑑is chosen in such a way to minimize the
OOC ARL1, when the IC ARL0is fixed on a specific value of shift. The value of ℎ=UCLis chosen according to the fixed
selected value of IC ARL0.
Table shows the values of the OOC ARL1’s for the MCUSUM-CoDa chart for selected values of the number of vari-
ables 𝑝having values (3, 5, 10, 20), the values of IC ARL0selected to be (200, 500) and the values of shift 𝛿that are
(0.25, 0.5, 0.75, … , 3).Tablealso presents the different chosen values of ℎaccording to the IC ARL0and the number
of variables 𝑝.
From Table , we can conclude that the values of ℎincrease as the value of 𝑝increases. For example, when 𝑝=3, the
value of ℎ=5.5but, when we increase the number of variables 𝑝=10,ℎ = 14.6. Also, when we increase the IC ARL0
from ARL0≈ 200 to ARL0≈ 500, keeping the values of 𝑝same, the values of ℎincrease with the increase in IC ARL0.For
instance, when ARL0≈ 200 and 𝑝=3, the value of ℎ=5.5but when ARL0≈ 500 and 𝑝=3remains same, the value of
ℎ = 6.75. Due to the values 𝛿and ARL0 ≈ 200,OOC values ARL1depend on the parameter 𝑝number value. The OOC
ARL1continues to increase, in particular with p. For example, when ARL0≈ 200,𝛿=2and 𝑝=3,wehaveARL = 4.3
but when we increase the value of 𝑝=5, the ARL = 5.48 increases. Also, when 𝑝=20, the ARL = 12.7.
8IMRAN .
TABLE 1 ARL performance of the MCUSUM-CoDa chart
𝐀𝐑𝐋 ≈ 𝟐𝟎𝟎 𝐀𝐑𝐋 ≈ 𝟓𝟎𝟎
𝒑=𝟑 𝒑=𝟓 𝒑=𝟏𝟎 𝒑=𝟐𝟎 𝒑=𝟑 𝒑=𝟓 𝒑=𝟏𝟎 𝒑=𝟐𝟎
𝜹 𝒉 = 𝟓.𝟓 𝒉 = 𝟖.𝟏𝟓 𝒉 = 𝟏𝟒.𝟔 𝒉 = 𝟐𝟐.𝟑 𝒉 = 𝟔.𝟕𝟓 𝒉 = 𝟏𝟎.𝟐𝟕 𝒉 = 𝟏𝟔.𝟕𝟐 𝒉 = 𝟐𝟔.𝟐
. . . . . . . .
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0 0.5 1 1.5 2 2.5 3
0
50
100
150
200
δ
ARL
p=20
p=10
p=5
p=3
00.511.522.53
0
50
100
150
200
250
300
350
400
450
500
δ
ARL
p=20
p=10
p=5
p=3
FIGURE 1 OOC ARL1of MCUSUM-CoDa chart, ARL0≈ 200 (left) and ARL0≈ 500 (right)
The same is the case when ARL0≈ 500.TheOOC values of ARL1depend on the number of variables of 𝑝. For Instance,
when 𝛿=2and 𝑝=3,wehaveARL = 4.9, but when we increase the value of 𝑝=5, the AR L = 6.18 increases. Also, when
𝑝=20, the ARL = 13.6.
We can see from all values mentioned above that the OOC ARL1is greater when we choose the IC ARL0≈ 500 than in
the other case when we select ARL0≈ 200. The output of the MCUSUM-CoDa chart for monitoring a p-part structure is
the same as the output of the MCUSUM chart for controlling multivariate normal data (with 𝑝1 variables). The different
ARL values of the MCUSUM-CoDa chart can also be seen in Figure for different values of 𝑝when IC ARL0≈200and
ARL0≈ 500.
From Figure , we can conclude that when we increase the number of variables involved 𝑝, the OOC ARL1also
increases. Also, as the value of shift increases, the OOC ARL1decreases for both IC ARL’s.
3.2 When parameters are unknown
In a multivariate CUSUM control chart, using estimators for the parameters introduces additional variability that should
be considered when computing the sampling distribution. Control charts’ IC and OOC performances can be significantly
IMRAN . 9
TABLE 2 UCL of the MCUSUM-CoDa chart with estimated parameters
𝐀𝐑𝐋𝟎≈𝟐𝟎𝟎 𝐀𝐑𝐋
𝟎≈𝟓𝟎𝟎
𝒏 𝒑 𝒎=𝟑 𝒎=𝟓 𝒎=𝟏𝟎 𝒎=𝟏𝟓 𝒎=𝟑 𝒎=𝟓 𝒎=𝟏𝟎 𝒎=𝟏𝟓
. . . . . . . .
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affected if this variability is ignored. One consequence of failing to account for this variability is a substantial increase in
false alarms, particularly when the Phase I data set sample size is small. This section evaluates the MCUSUM-CoDa chart
IC performance using estimated parameters obtained through the Monte Carlo simulation method. An IC reference sam-
ple of 𝑛subgroups of size 𝑚, where 𝑛1 and 𝑚(𝑛1) > 𝑝 is used to estimate when the mean vector and variance–covariance
matrices are unknown.
Then, the estimators used are 𝝁∗
0=
𝑋∗and 𝚺∗
0=
𝑆∗, where
𝑋∗and
𝑆∗are defined in Equations ()and(),
respectively.
Table shows the values of the UCL’s for the MCUSUM-CoDa chart with estimated parameters for selected values
of the number of variables 𝑝having values (3, 5, 10, 20), the subgroup size 𝑚having values (3, 5, 10, 15), the values
of IC ARL0selected to be (200, 500). The values of shift 𝛿that are (0.25, 0.5, 0.75, … , 3) and the number of samples
𝑛that are (,,,,,,). Ten thousand simulation runs have been used to estimate the UCL = ℎ val-
ues. The different values of new corrected UCL = ℎ were chosen according to the fixed IC ARL0using the estimated
parameters.
10 IMRAN .
From Table , we can conclude that
∙The values of ℎincrease as the value of 𝑝increases. For example, when 𝑛=30,𝑚=5,and𝑝=3, the value of ℎ = 6.38,
but when we increase the number of variables 𝑝=10, keeping the other parameters constant, the value of ℎbecomes
ℎ = 15.76.
∙The values of ℎdecrease with an increase in subgroup size 𝑚increases. For example, when 𝑛=30,𝑚=3,and𝑝=3,
the value of ℎ = 6.40, but when we increase the subgroup size 𝑚=15, keeping the other parameters constant, the value
of ℎbecomes ℎ = 6.35.
∙The values of ℎalso decrease with an increase in 𝑛increases. For example, when 𝑛=30,𝑚=3,and𝑝=3, the value
of ℎ = 6.40, but when we increase the sample size 𝑛 = 1000, keeping the other parameters constant, the value of ℎ
becomes ℎ = 4.99.
∙The values of ℎalso increase as the value of IC ARL0increases. For example, when ARL0≈ 200,𝑛=30,𝑚=3,and
𝑝=3, the value of ℎ = 6.40, but when we increase the IC ARL0≈ 500, keeping all the other parameters constant, the
value of ℎbecomes ℎ = 8.91.
Table shows the values of the ARL1’s that are OOC for the MCUSUM-CoDa chart with estimated parame-
ters for selected values of the number of variables 𝑝having values (3, 5, 10, 20), the subgroup size 𝑚having values
(3, 5, 10, 15), the values of IC ARL0selected to be (200, 500) and the values of shift 𝛿that are (0.25, 0.5, 0.75, … , 3)
and the number of samples 𝑛 = 500. Ten thousand simulation runs have been used to estimate the ARL
values.
∙The OOC ARL with estimated parameters are greater than the OOC ARL with known parameters.
∙The values of OOC ARL decrease as the value of 𝛿increases. For example, when 𝛿=0.5,𝑚=3,and𝑝=3, the value
of ARL1= 50.78, but when we increase the value of 𝛿=3, keeping the other parameters constant, the value of ARL1
becomes ARL1=2.6.
∙The values of OOC ARL also decreases as the value of subgroup size 𝑚increases. For example, when 𝛿=0.5,𝑚=3,and
𝑝=3, the value of ARL1= 50.78, but when we increase the value of 𝑚=15, keeping the other parameters constant,
the value of ARL1becomes ARL1= 50.71.
∙The values of OOC ARL increase as the value of the number of variables 𝑝increases. For example, when 𝛿=0.5,𝑚=3,,
and 𝑝=3, the value of ARL1= 50.78, but when we increase the value of 𝑝=20keeping the other parameters constant,
the value of ARL1becomes ARL1= 98.95.
∙The values of OOC ARL also increase as the value IC ARL0increases. For example, when ARL0≈ 200,𝛿=0.5,𝑚=3,
and 𝑝=3, the value of ARL1= 50.78, but when we increase the value of ARL0≈ 500, keeping the other parameters
constant, the value of ARL1becomes ARL1= 74.42.
The results of Table are also presented in Figure , when 𝑚=3and 𝑛 = 500.
From Figure , we can conclude that when we increase the number of variables involved 𝑝, the OOC ARL1also
increases. Also, as the value of shift increases, the OOC ARL1decreases for both IC ARL’s.
4 COMPARISON WITH THE HOTELLING 𝑻𝟐-CoDa and the MEWMA-CoDa CHART
The IC ARL0for MCUSUM-CoDa, Hotelling 𝑇2-CoDa,andMEWMA-CoDa are set to be and in this subsection.
In Table , For several values of the shift (i.e., 𝛿 ∈ {0.25, … , 2}), we evaluate the OOC performances of all three charts.
The ARL values are presented in Table , and their percentage improvement indicators Δ𝑇2,MC =100(ARL𝑇2−ARLMC)
ARL𝑇2and
ΔMW,MC =100(ARLMW −ARLMC )
ARLMW
.
From Table , we can conclude the following results:
∙The MCUSUM-CoDa chart has a smaller ARL value in contrast to the 𝑇2-CoDa chart.
For example, while 𝑝=3and 𝑑𝑒𝑙𝑡𝑎 = 0.5, the ARL value for the 𝑇2-CoDa chart is (ARL = 76.86), whereas it is (ARL =
28.6) for the MCUSUM-CoDa chart.
IMRAN . 11
TABLE 3 ARL performance of the MCUSUM-CoDa chart with estimated parameters
𝐀𝐑𝐋 ≈ 𝟐𝟎𝟎
𝒎 𝒑 𝒉 𝜹 = 𝟎.𝟐𝟓 𝜹 = 𝟎.𝟓 𝜹 = 𝟎.𝟕𝟓 𝜹 = 𝟏 𝜹 = 𝟏.𝟐𝟓 𝜹 = 𝟏.𝟓 𝜹 = 𝟏.𝟕𝟓 𝜹 = 𝟐 𝜹 = 𝟐.𝟐𝟓 𝜹 = 𝟐.𝟓 𝜹 = 𝟐.𝟕𝟓 𝜹 = 𝟑
. . . . . . . . . . . . .
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ARL ≈ 500
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∙The MCUSUM-CoDa chart has a larger ARL than the MEWMA-CoDa chart. For example, where 𝑝=3and 𝛿=0.5are
ARL, the value of MEWMA-CoDa is (ARL = 26.4) and the value of MCUSUM-CoDa is (ARL = 28.6).
∙In terms of percentages, the MCUSUM-CoDa chart is between 69% and 84% more effective than the 𝑇2-CoDa
chart, depending on the number of variables 𝑝and the shift 𝑑𝑒𝑙𝑡𝑎. When the MCUSUM-CoDa chart is com-
pared to the MEWMA-CoDa chart, the MEWMA-CoDa chart is 8%–10% more effective than the MCUSUM-CoDa
chart.
More specifically (see the last row of Table , the MCUSUM-CoDa chart is 85.53% more efficient on average than the 𝑇2-
CoDa chart for 𝑝=3and 88.88% more efficient for 𝑝=5. However, the MEWMA-CoDa chart is 16.3% more efficient
on average than the MCUSUM-CoDa chart for 𝑝=3and 22.33%. The comparison of the charts mentioned above can
also be seen in Figure .
From Figure , we can conclude that the MCUSUM-CoDa chart performance is not very different from the MCUSUM-
CoDa chart, but both charts outperform the 𝑇2-CoDa control chart.
12 IMRAN .
0 0.5 1 1.5 2 2.5 3
0
50
100
150
200
δ
ARL
p=20
p=10
p=5
p=3
0 0.5 1 1.5 2 2.5 3
0
50
100
150
200
250
300
350
400
450
500
δ
ARL
p=20
p=10
p=5
p=3
FIGURE 2 OOC ARL1of MCUSUM-CoDa chart, ARL0≈ 200 (left) and ARL0≈ 500 (right)
TABLE 4 ARL comparison of the MCUSUM-CoDa chart with the 𝑇2-CoDa and the MEWMA-CoDa chart
𝐀𝐑𝐋𝟎≈𝟐𝟎𝟎
𝒑=𝟑 𝒑=𝟓
𝜹𝑻
𝟐𝐌𝐂 𝐌𝐖 𝚫𝑻𝟐,𝐌𝐂 𝚫𝐌𝐖,𝐌𝐂 𝑻𝟐𝐌𝐂 𝐌𝐖 𝚫𝑻𝟐,𝐌𝐂 𝚫𝐌𝐖,𝐌𝐂
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𝐀𝐑𝐋𝟎≈𝟓𝟎𝟎
𝒑=𝟑 𝒑=𝟓
𝜹𝑻
𝟐𝐌𝐂 𝐌𝐖 𝚫𝑻𝟐,𝐌𝐂 𝚫𝐌𝐖,𝐌𝐂 𝑻𝟐𝐌𝐂 𝐌𝐖 𝚫𝑻𝟐,𝐌𝐂 𝚫𝐌𝐖,𝐌𝐂
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5IMPLEMENTATION OF THE PROPOSAL
Two illustrated examples have been studied to implement the proposal, one related to particle size distribution data to
monitor the three particle sizes (i.e., large, medium, and small) in the grit production, and other monitor machines respon-
sible for the percentages of three components (i.e., whole-grain cereal, dried fruits, and nuts) in muesli production.
IMRAN . 13
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
0
50
100
150
200
δ
ARL
MCUSUM
T2
MEWMA
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
0
50
100
150
200
δ
ARL
MCUSUM
T2
MEWMA
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
0
50
100
150
200
250
300
350
400
450
500
δ
ARL
MCUSUM
T2
MEWMA
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
0
50
100
150
200
250
300
350
400
450
500
δ
ARL
MCUSUM
T2
MEWMA
FIGURE 3 ARL of MCUSUM-CoDa 𝑇2-CoDa and MEWMA-CoDa control chart for 𝑝=3(left) and 𝑝=5(right)
5.1 Application related to the implementation of grit production
For the sake of an illustrative example, we are using the particle size distribution data from a plant in Europe. The per-
centages of the particle size have been taken according to the weights. The study aims to monitor the three particle sizes
to ensure that the grit production used for the plant is well managed. The same example has also been used by Refs. ,,
As per Vives-Mestres et al., there are four OOC points (#1, #26, #45, and #52) in the three significant components
large (L), medium (M), and tiny (S), which shows that the distribution of particle size has been changed on these four
points. Many possible causes are responsible for the shift in the distribution (e.g., changes in the manufacturing process
or changes in raw material, etc.). Table shows the Phase I data set after the OOC mentioned above points had been
removed. The ilr co-ordinates 𝑥∗
𝑖=(𝑥
∗
𝑖,1,𝑥
∗
𝑖,2)are also presented in Table .
Figure shows the Phase I IC data for CoDa in Simplex 𝑝using a ternary diagram, and the corresponding ilr trans-
formed values in real space ℝ𝑝using scatter plot. Figure also presents the 95% confidence ellipse for the IC parameters.
From Table , we can quickly get the parameters of the multivariate normal distribution that is
𝝁0=
0.892
0.056
0.052
,
14 IMRAN .
TABLE 5 The Phase I grit production data from Tran et al.
𝒊Medium Small Large 𝒙∗
𝒊,𝟏 𝒙∗
𝒊,𝟐
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(Continues)
IMRAN . 15
TABLE 5 (Continued)
𝒊Medium Small Large 𝒙∗
𝒊,𝟏 𝒙∗
𝒊,𝟐
. . . .
. . . . .
. . . .
. . . .
. . . .
. . . . .
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Medium Small
Large
123
1
2
3
4
x∗
1
x∗
2
FIGURE 4 Phase I grit production data in Simplex 𝑝(left) and corresponding ilr coordinates in real space ℝ𝑝(right)
or we can use the ilr transformed values, that is
𝝁∗
0=1.962
1.185,
and
𝚺∗=0.086 −0.0215
−0.0215 0.097 .
For the Phase II data set, samples of size 𝑛=3have been simulated using the above-mentioned variance–covariance
matrix and mean vector. This simulation is presented in Table , which shows the ilr coordinates, and the MCUSUM-
CoDa chart’s monitored statistics along with the 𝑇2
CoDa charts for the sake of comparative analysis. Figure also shows
the results of MCUSUM-CoDa and 𝑇2-CoDa and the upper control limits.
Table and Figure show that the process is IC up to sample #15, but sample #16 andsoontillsample#20 go OOC
(see the bold values) as the MCUSUM-CoDa chart’s value is greater than the 𝑇2-CoDa control charts, statistics show no
OOC point. We conclude that the MCUSUM-CoDa chart has better performance than the 𝑇2-CoDa charts.
5.2 Application related to the implementation of muesli production
Here we are using another illustrative example of muesli production where a company is producing muesli with almost
% of cereals, % of dried fruits and % of nuts. The percentages of each component of muesli have been taken according
to the weights. The study aims to monitor the machine that is responsible for the percentages of the three components.
16 IMRAN .
TABLE 6 A Phase II grit production data with the subgroup of size 𝑚=3with MCUSUM-CoDa and the 𝑇2-CoDa chart results
𝒊𝒋𝒙
𝒊,𝒋 𝒙𝒊𝒙∗
𝒊𝑪𝒊𝑻𝟐
𝑪,𝒊
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. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
(Continues)
IMRAN . 17
TABLE 6 (Continued)
𝒊𝒋𝒙
𝒊,𝒋 𝒙𝒊𝒙∗
𝒊𝑪𝒊𝑻𝟐
𝑪,𝒊
. . . . . . . . 7.2328 .
. . .
. . .
. . . . . . . . 9.2978 .
. . .
. . .
. . . . . . . . 10.816 .
. . .
. . .
. . . . . . . . 11.8359 .
. . .
. . .
. . . . . . . . 14.2795 .
. . .
. . .
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
i
MCUSUM Statistics
UCL = 6.75
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
i
T2Statistics
UCL
T2= 11.829
FIGURE 5 MCUSUM-CoDa chart (left) and 𝑇2-CoDa chart (right) for Phase II grit production data
The same example has also been used by Zaidi et al., As per Zaidi et al., there is only one OOC point (#15) in
the three components, which shows that the distribution of the three components has been changed on this point. As
per Zaidi et al., the cause responsible for the shift is “the level of whole-grain cereals dropped down suddenly due to a
malfunction of the hatch regulating the quantity of whole-grain cereals causing a shift.” Table shows the Phase I data
set after the OOC mentioned above point had been removed. The ilr co-ordinates 𝑥∗
𝑖=(𝑥
∗
𝑖,1,𝑥
∗
𝑖,2)are also presented in
Table .
Figure shows the Phase I IC data for both CoDa in Simplex 𝑝using a ternary diagram and the corresponding ilr
transformed values in real space ℝ𝑝using and so on till scatter plot. Figure also presents the 95% confidence ellipse for
the IC parameters.
From Table , we can quickly get the parameters of the multivariate normal distribution that is
𝝁0=
0.689
0.228
0.083
,
18 IMRAN .
TABLE 7 Phase I data for the muesli production from Zaidi et al.
𝒊𝒋Whole grains Dried fruits Nuts 𝐱𝒊𝐱∗
𝒊
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
(Continues)
IMRAN . 19
TABLE 7 (Continued)
𝒊𝒋Whole grains Dried fruits Nuts 𝐱𝒊𝐱∗
𝒊
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
. . . . . . . .
. . .
. . .
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Whole Grains Dried Fruits
Nuts
−2−112
−2
−1
1
2
x∗
1
x∗
2
FIGURE 6 Phase I muesli production data in Simplex 𝑝(left) and corresponding ilr coordinates in real space ℝ𝑝(right)
or we can use the ilr transformed values, that is
𝝁∗
0=1.281
1.782,
and
𝚺∗=0.0117 −0.0041
−0.0041 0.0765 .
For the Phase II data set, samples of size 𝑛=3have been simulated using the above-mentioned variance–covariance
matrix and mean vector. This simulation is presented in Table , which shows the ilr coordinates, and the MCUSUM-
CoDa chart’s monitored statistics along with the 𝑇2
CoDa charts for the sake of comparative analysis. Figure also shows
the results of MCUSUM-CoDa and 𝑇2-CoDa and the upper control limits.
Table and Figure show that the process is IC up to sample #12, but sample #13 and so on till sampling #20 go OOC
(see the bold values) as the MCUSUM-CoDa chart’s value is greater than the UCL = 6.75. While if we see the 𝑇2-CoDa
control charts, statistics show no OOC point. We conclude that the MCUSUM-CoDa chart has better performance than
the 𝑇2-CoDa charts.
20 IMRAN .
TABLE 8 A Phase II muesli production data with the subgroup of size 𝑚=3with MCUSUM-CoDa and the 𝑇2-CoDa chart results
𝒊𝒋𝒙
𝒊,𝒋 𝒙𝒊𝒙∗
𝒊𝑪𝒊𝑻𝟐
𝑪,𝒊
. . . . . . . . .
. . .
. . .
. . . . . . . . .
. . .
. . .
. . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . . .
. . .
. . .
. . . . . . . . 7.8415 .
. . .
. . .
. . . . . . . . 11.5655 .
. . .
. . .
. . . . . . . . 15.0286 .
. . .
. . .
(Continues)
IMRAN . 21
TABLE 8 (Continued)
𝒊𝒋𝒙
𝒊,𝒋 𝒙𝒊𝒙∗
𝒊𝑪𝒊𝑻𝟐
𝑪,𝒊
. . . . . . . . 18.0829 .
. . .
. . .
. . . . . . . . 21.5873 .
. . .
. . .
. . . . . . . . 25.4513 .
. . .
. . .
. . . . . . . . 29.1647 .
. . .
. . .
. . . . . . . . 33.0022 .
. . .
. . .
0 2 4 6 8 101214161820
0
5
10
15
20
25
30
35
40
i
MCUSUM Statistics
UCL = 6.75
UCLT2 = 11.829
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
i
T2Statistics
FIGURE 7 MCUSUM-CoDa chart (left) and 𝑇2-CoDa chart (right) for Phase II muesli production data
6CONCLUSIONS
This paper defines the MCUSUM-CoDa chart for the log-ratio-transformed CoDa having 𝑝-part composition. The
MCUSUM-CoDa chart has been studied with known as well as estimated parameters. First, we have discussed the ARL
performance of the MCUSUM-CoDa chart when the parameters are known. In this case, different values of shift, involved
parameters, and the IC ARL0have been used to analyze the performance of the MCUSUM-CoDa charts using the Markov
chain approach. Second, the MCUSUM-CoDa chart with estimated mean vector and variance covariance matrix has been
examined to study its effects on the performance of the proposed chart. The extensive Monte Carlo simulation is used to
study the ARL performance of the MCUSUM-CoDa chart. The main conclusions are: (i) the proposed MCUSUM-CoDa
chart perform efficiently in the case of known process parameters as compared to the estimated case, (ii) the OOC ARL
decrease as 𝛿increases, (iii) the OOC ARL also decreases as subgroup size 𝑚increases, (iv) the OOC ARL increases as
the number of variables 𝑝increases, (v) the OOC ARL increases as the IC ARL0increases.
The effect of estimation on the UCL has also been studied in this paper. From UCL, we can conclude that: (i) the UCL
increase as the number of variables 𝑝increases, (ii) the UCL also increase when 𝐼𝐶 ARL0increases, (iii) the UCL decreases
with the increase in subgroup size 𝑚, (iv) the UCL also decrease when sample size 𝑛increases.
When comparing the ARL performance of the MCUSUM-CoDa control chart with the MEWMA-CoDa chart and 𝑇2-
CoDa chart. We found that the new proposed charts have significantly better statistical sensitivity than the 𝑇2-CoDa chart,
22 IMRAN .
while the proposed MCUSUM-CoDa chart has comparable ARL performance as the MEWMA-CoDa charts. Further-
more two illustrative examples of particle size composition of plants and muesli composition have been used to indicate
the MCUSUM-CoDa chart’s practical implementation. The main conclusion is that the MCUSUM-CoDa chart outper-
forms the 𝑇2-CoDa chart, but the MEWMA-CoDa chart is better in terms of performance than the MCUSUM-CoDa chart.
For future works, the proposed chart can be studied using steady-state ARL to investigate the ARL performance of the
MCUSUM-CoDa control chart. Also, the effect of estimated parameters can be seen on the two previously defined control
charts (i.e., 𝑇2-CoDa chart and MEWMA-CoDa chart).
ACKNOWLEDGMENTS
The authors are grateful to the editor and anonymous reviewers for their valuable suggestions that helped to improve the
manuscript’s initial version.
FUNDING
This work was supported by the National Natural Science Foundation of China (Grant number: ); Humanity and
Social Science Foundation of the Ministry of Education of China (Grant number: YJA).
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.
ORCID
Muhammad Imran https://orcid.org/---
Fatima Sehar Zaidi https://orcid.org/---
Zameer Abbas https://orcid.org/---X
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AUTHOR BIOGRAPHIES
Muhammad Imran obtained his MSc and MPhil in Statistics from the PMAS Arid Agriculture University Rawalpindi,
Pakistan, and he served as a visiting Lecturer at the PMAS Arid Agriculture University from to . Presently
he is a PhD. Scholar in the School of Automation, Nanjing University of Science and Technology, China. His research
interests include the development of multivariate Statistical Process Monitoring techniques. His email addresses are
imrankharal@njust.edu.cn and kharal@outlook.com.
Jin Sheng Sun is currently a Professor in the School of Automation, Nanjing University of Science and Technology. He
received the BS, MS, and PhD degrees from the Nanjing University of Science and Technology. His research activity
includes statistical process control, network congestion control, and distributed control of multiagent systems. His
email address is jssun@.com.
Fatima Sehar Zaidi obtained her MSc and M.Phil in Statistics from the PMAS Arid Agriculture University,
Rawalpindi, Pakistan and PhD in Mathematics and its interactions from the Universit de Nantes, France. She served
as a visiting Lecturer at the PMAS Arid Agriculture University from to . Currently a Post-Doctoral candidate
at the Nanjing University of Aeronautics and Astronautics, China. Her research is focused on the developments of
multivariate Statistical Process Monitoring techniques. Her email address is fatimaseharzaidi@gmail.com.
Zameer Abbas obtained his MSc in Statistics from the University of Punjab Lahore in with distinction and MPhil
in Statistics from the University of Sargodha, Pakistan, in . He joined Punjab Higher Education Department as a
Lecturer in and presently serves as an Assistant Professor at the Government Ambala Muslim College, Sargodha.
His research interests include statistical inference, statistical quality control, and nonparametric techniques. His email
address is zameerstats@gmail.com.
Hafiz Zafar Nazir obtained his MSc and MPhil degrees in Statistics from the Department of Statistics, Quaid-i-Azam
University, Islamabad, Pakistan, in and , respectively. He did his PhD in statistics from the Institute of Busi-
ness and Industrial Statistics University of Amsterdam, The Netherlands, in . He served as a lecturer in the Depart-
ment of Statistics, University of Sargodha, Pakistan, from to . He served as an Assistant Professor in the
Department of Statistics, University of Sargodha, Pakistan, from to . He is now serving as an Associate Pro-
fessor in the Department of Statistics, University of Sargodha, Pakistan, since October to date. His current research
interests include statistical process control, nonparametric techniques, and robust methods. His email address is hafiz-
zafarnazir@yahoo.com.
How to cite this article: Imran M, Sun J, Zaidi FS, Abbas Z, Nazir HZ. Multivariate cumulative sum control
chart for compositional data with known and estimated process parameters. Qual Reliab Eng Int. ;–.
https://doi.org/./qre.
