The Effect of Sample Size On (Cusum and ARIMA) Control Charts

International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-2, Issue-5, May- 2016]
Infogain Publication (Infogainpublication.com) ISSN: 2454-1311
www.ijaems.com Page | 397
The Effect of Sample Size On (Cusum and
ARIMA) Control Charts
Dr. Kawa M. Jamal Rashid
Ass. Prof , Department of Statistics,College of Administrations and Economics, Sulamani University, Sulamani, Kurdistan, Iraq
Abstract— The purpose of this paper is to study Statistical
Process Control (SPC) with a cumulative sum CUSUM
chart which shows the total of deviations, of successive
samples from the target value and the Average Run Length
(ARL) is given quality level is the average number of
samples (subgroups) taken before an active signal is given.
Sample size has a good effect on the quality chart. The
average run length of the cumulative sum control chart is
the average number of observations that are entered before
the system is declared out of control. Control limits for the
new chart are computed from the generalized ARL
approximation, The Autocorrelation of the observation
increasing by the sample size of the cumulative value
distributed by Manhattan diagram. The new chart is
compared to other distribution-free procedures using
stationary test processes with both normal and abnormal
marginal.
Keywords— Cumulative sum CUSUM Chart, ARIMA
Control chart, Average Run Length, Distribution-Free
Statistical Methods, Manhattan diagram.
I. INTRODUCTION
Quality control via the use of statistical methods is a very
large area of study in its own right and, is central to success
in modern industry with its emphasis on reducing costs
while at the same time improving quality, Statistical quality
control came from Dr. Walter Shewhart in 1924 during his
employment at Bell Telephone Laboratories. He recognized
that in a manufacturing process, there will always be
variation in the resulting products. He also recognized that
this variation can be understood, monitored, and controlled
by statistical procedures. Shewhart developed a simple
graphical technique - the control chart - for determining if
the product variable is within acceptable limits. In this case
the production process is said to be in "control" and control
charts can indicate when to leave things alone or when to
adjust or change a production process. In the latter cases the
production process is said to be out of control.’ Control
charts can be used (importantly) at different points within
the production process. [4]
The aim of statistical process control is to ensure that a
given manufacturing process is as stable. In short, the aim is
the reduction of variability to ensure that each product is of
a high a quality as possible. Statistical process control is
usually thought of as a toolbox whose contents may be
applied to solve production-related quality control
problems. [4] [3]
II. CUMULATIVE SUM CUSUM CHART
The basic Principles The Cusum control chart for
monitoring the process mean.
The development of cumulative sum (CUSUM) control
charts originally introduced by Page [1954]. The CUSUM
control chart is a procedure based on the CUSUM of the
deviations of the sample statistics from the target value [7].
Over the years, CUSUM control charts have proven to be
superior to the classical Shewhart control charts in the sense
that the CUSUM control charts tend to have smaller
Average Run Lengths (ARL’s) particularly when small
changes in the population parameters of the process have
occurred [1].
∑=
−=
i
j
ji XCUSUM
1
0 )( µ …. (1)
Where the initial value of the cumulative statistics is taken
to be zero [7].
To compute the upper and lower cumulative statistics, we
now need the value of the reference value of k or (slack
value). In our situation, ∆u is specified to be (0.01)
implying k is (=∆/2). Then k is computed by using σˆ
The tabular Cusum works by accumulating derivations from
µ0 that are above target with one statistic
+
Cusum and the
below target with other statistics. The statistics
+
Cusum and
−
Cusum are called one –sided upper and
lower Cusum. They are computed as follows:
{ }
{ }+
−
+
+
−
+
+−−=
++−=
10
10
)(,0maxCUSUM
)(,0maxCUSUM
ii
iii
CXk
CkX
µ
µ
…2

Where the starting the value are
000 == −+
CC
Adjustment to some manipulatable variable is required in
order to bring the process back to the target value 0µ this
can be computed as:[2][9]












>−−
>++
=
−
+
−
+
+
+
HCif
N
C
k
HCif
N
C
k
i
i
i
i
0
0
ˆ
µ
µ
µ …3
III. PRODUCT SCREENING AND PER-SELECTION
Cusum chart can be used in categorizing process output.
This may be for the purposes of selection for different
process or different assembly operation. The Cusum chart
has been divided into different sections of average process
mean by virtue of a change the slope of Cusum plot.
The average process calculated as:
)4......(/)( 1
1
nSST
n
X
in
n
i
i
−
=
−+=∑
Where S is a Cusum value this information may be
represented on a Manhattan diagram- [4]
IV. ARIMA CONTROL CHART
Classical Shewhart SPC concept assumes that the measured
data are not autocorrelated.
Even very low degree of autocorrelation data causes failure
of classical Shewhart control charts.
Failure has a form of a large number of points outside the
regulatory limits in control diagram.
This phenomenon is not unique in the case of continuous
processes, where the autocorrelation data given by the
inertia processes in time. Autocorrelation of data becomes
increasingly frequent phenomenon in terms of discrete
processes, a high degree of automation of production and
also in the test and control operations.
One of the ways to tackle autocorrelated data is the concept
of stochastic modeling of time series using autoregressive
integrated moving average models, the ARIMA model.
Linear stochastic autoregressive models (models AR),
moving average (model MA), mixed models (the ARMA
models), and ARIMA models, based on Box-Jenkins
methodology is seen as a time series realization of
stochastic process [8][9] These models have a characteristic
shape of the autocorrelation fiction (Autocorrelation
Fuction–ACF) and partial autocorrelation function (Partial
Autocorrelation Function – PACF). The original integrated
process is called an autoregressive integrated moving
average process of order p, d, q, ARIMA (p, d, q) where p
number of autoregressive terms, d is number of
nonseasonal differences, and q is a number of moving –
average terms.
Location of the mean value CL and upper and lower
control limits (UCL, LCL) for the ARIMA (p,d,q) chart for
individual values can be determined from the formula as:
ixRXLCLandUCL
xCL
128.1
3
0
m=
==
x : Is the average of residual value.
R : Is the average of moving rang.
Values CL, UCL and LCL can be calculated as follows.
0
267.3
=
=
=
LCL
RUCL
RCL
ix
To increase the sensitivity of control charts ARIMA is
recommended to use two-sided Cusum control chart with
the decision interval ± H. .[8][9][11]
V. AVERAGE RUN LENGTH - ARL
Any sequence of samples that leads to an out-of-control
signal is called a “run.” The number of samples that is taken
during a run is called the “run length.” The use of Average
Run Length (ARL) has considerable fire in recent years
.This is because the run length distribution is quite skewed
so that the average run length will not be a typical run
length, and the another reason that is the standard deviation
of the ARL is quite large, and as Geometric distribution, the
variance of geometric distribution.[10]
The term Average Run Length (ARL) is defined as the
average or expected number of sizes in the process level is
signaled by points that must be plotted before an out-of-
control.
If (p) is the probability of a single plotted point breaching
the predetermined control limits (signaling a lack of
statistical control), then the ARL is given by the mean of
the geometric distribution namely.
For attribute and variables Shewhart charts with only three
sigma action or control limits, the probability p is assumed
to be constant
For a Shwhart - type chart the ARL for a in control process
typically labeled 0ARL is given by;
α
1
0 =ARL

-0.800
-0.700
-0.600
-0.500
-0.400
-0.300
-0.200
-0.100
0.000
0.100
0.200
0.300
0.400
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53
n=3
n=4
n=5
n=6
And the probability of getting an out-of-control signal if the
process has shifted is β, then the processes is out of control
as 1ARL is:
βα
αα
−
=
−
=−= ∑
∞
=
−
1
1
)(1
1
)](1[)(
1
1
1
P
PiPARL
i
i
Where α is the probability of a Type I error and β the
probability of a Type II error. [6][7]
VI. TABULAR CUSUM PROCEDURE FOR
MONITORING THE PROCESS MEAN
The Tabular Cusum is designed by choosing values for the
reference value K, and the decision interval H. These
parameters be selected to provide a good average run length
performance. The parameter H defines as H=hσ and K=
hσ where σ is the standard deviation of the samples used in
forming the Cusum. Using h=4 or h=5 and k=0. 5 a custom
that has a good ARL property against a shift of about 1σ in
the process mean.[2] -
ARL Approximation for upper-side CUMUM
+
ARL
and lower –side CUMSE
−
ARL are given by:
[ ]
)(2
1)166.1)(2)166.1)((2exp
k
hkhk
ARL u
uu
−
−+−++−−
=+
δ
δδ
…4
And
[ ]
)(2
1)166.1)((2)166.1)((2exp
k
hkhk
ARL u
uu
−−
−+−−++−−−
=−
δ
δδ
.… (5)
And the ARL for two sided Cusum can be obtained by
using the following as:
−+−+
+=
ARLARLARL
111
,
…(6)
From Eq. (5) If the value of (k) is equal (0.5) then the value
of (h) is equal (4.767), and ARL is(370) [6][7]
VII. NUMERICAL STUDIES
Numerical illustration: In this section an application is
considered to highlight the features of the above proposed
Cusum control charts. In this paper through a rea illustrative
data from ALA-Company for bottle water.The application
was made in controlling the proportion of (Power of
Hydrogen) PH component in the water. Thirty two samples
with a sample size of 5 (the total measurement number is
160) were taken from the production process in the ALA
Company as shows in table (4). These measurement data
are converted into trapezoidal Cusum numbers and given in
Table (1). Designing and determine calcium control chart
for a process average of variable quality. Table (2) shows
that the value of the Cusum stat chart for the sample size n=
(3, 4, 5, 6) as shows in Fig (1, 2, 3 and 4)
Table (1) Cusum Value of PH For different sample size n
Sample
Size n=3
Sample
size n=4
Sample
size n=5
Sample size
n=6
-0.024 -0.040 -0.024 -0.029
-0.059 0.040 0.063 0.055
0.077 0.082 0.057 -0.008
0.109 0.044 -0.023 -0.019
0.095 -0.028 -0.030 0.168
-0.016 -0.029 0.202 0.081
. . . .
. . . .
. . . .
-0.347 -0.329 -0.259 -0.214
-0.305 -0.354 -0.282 -0.215
-0.239 -0.289 -0.278 -0.233
-0.140 -0.219 -0.274 -0.236
-0.118 -0.179 -0.175 -0.173
-0.185 -0.094 -0.129 -0.120
-0.146 -0.120 -0.071 -0.059
0.003 -0.110 -0.106 -0.073
-0.005 0.000 0.000 0.037
Fig.1:Cusum distribution of sample size (n, 3,4,5,6)
Fig.2:Cusum stat distribution sample size n= 3
Samplie size n-3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52

Table (2) Basic statistic table
Sample
size n=3
Sample
size n=4
Sample
size n=5
Sample
size n=6
Minimum -0.194 -0.14 -0.122 -0.111
Maximum 0.396 0.262 0.232 0.187
Range 0.59 0.403 0.354 0.298
Standard
Deviation
0.09 0.073 0.075 0.068
Variance 0.008 0.005 0.006 0.005
Fig.6:Manhattan diagram for the average process mean
From Fig. (1,2,3,4 and5) and the basic statistic table shows
that the Cusum value has a different range of Cusum value,
each case depends upon the sample size n, the range of
Cusum stat is equal (0.298) if sample size is (n=3) and the
range is equal (0.59) if sample size is (n=6), with variance
(0.005, 0.008) it means that decreasing the deviation
between our observations or the rang value of Cusum by
increasing the sample size. In designing a control chart, we
must specify both the sample size and the frequency of
sampling. In general, larger samples will make it easier to
detect small shifts in the process, The Cusum rang value
depend upon the sample size of these deviations, Cusum is
particularly helpful in determining when the assignable
cause has occurred, as we noted in the previous example,
just count backward from the out-of-control signal to the
time period when the Cusum lifted above zero to find the
first period following the process shift. We conclude that
sample size has a good effect on a control chart. In some
situations where an adjustment to some main table variable
is required in order to bring the process back to the target
value m0, it may be helpful to have an estimate of the new
process mean following the shift. The counters N+ and N-
are used in adjustment.
This can be computed from by equation (3) we would
estimate the new process average of
4.654.7 99 == −+
CandC
We would conclude that mean shifted from 7 to 7.54, then
we would need to make an adjustment in PH rate by 0.54
units, to estimating the mean of any we would conclude that
mean. By using the Average process for Cusum for
Sample Size 5
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52
Sample size 6
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52
Sample size 4
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52

different sample size, from the fig. (6). It shows that the
variation in average process means in sample size n=6 is
approach to equal time –scale it shows that chart with
sample size n=6 have more stable distribution.
Fig.7:ARIMA control chart.
From Fig. (7, 8 , 9 and 10) it shows that the sample size has
an effective on the ARIMA control chart the Fig.(10) have
a more effective than Fig.(7) where the sample size (n=3)
Fig.8:ARIMA control chart
Table (3) UCL, LCL, Mean and variance If N= 3,4,5,6
Table (4) ARIMA Model Summary
In this paper determining the Average Run Length ARL
of PH parameter of water defining the ARL by different
type of parameter (K, and H) as shows in table (3) and ARL
distribution as Fig. (11). We must also determine the
frequency of sampling. The most desirable situation from
the point of view of detecting shifts would be to take large
samples very frequently; however, this is usually not
economically feasible. Define H= hσ and K= kσ, where σ is
the standard deviation of the sample. Calculating the ARL
for different value of a parameter (k (0.5, 0.4,) and h (5, 4))
with shift mean as shows in table and Fig. (12), although
determining the ARL with k=0.5 and changing the h from 1
to 5, if the value of parameter h= 4.77 this will provide
ARL0 = 370 samples as shows in table(5) and Fig. (11and
12).
ARIMA
Chart
N=3 N=4 N=5 N=6
UCL 0.158 0.1117 0.1025 0.0854
LCL -0.158 -0.1117 -0.1025 -0.085
Mean 7.0075 7.0076 7.0088 7.007
St dev 0.1567 0.14902 0.1702 0.1733
ARIMA
Parameter Estimate
Stand.
Error t
P-
value
Sample 3 0.0633 0.13831 0.458 0.648
Sample 4 -0.0096 0.16651 -0.057 0.954
Sample 5 -0.1691 0.18581 -0.910 0.37
Sample 6 -0.1691 0.18581 -0.910 0.37
ResidualAutocorrelationsforn=3
lag
Autocorrelations
0 4 8 12 16 20 24
-1
-0.6
-0.2
0.2
0.6
1
Residual Autocorrelations for samplesize4
lag
Autocorrelations
0 4 8 12 16 20 24
-1
-0.6
-0.2
0.2
0.6
1
Residual Autocorrelations for sample size 5
lag
Autocorrelations
0 4 8 12 16 20 24
-1
-0.6
-0.2
0.2
0.6
1
Residual Autocorrelations for sample size 6
lag
Autocorrelations
0 4 8 12 16 20 24
-1
-0.6
-0.2
0.2
0.6
1

403632282420161284
500
400
300
200
100
0
k=0.5 and h =5 to 1
Fig.11:ARL Distribution with value K and H
Table (5) ARL value with different (K, and h)
ARL ARL ARL ARL ARL
k=0.5 k=0.4 k=0.5 k=0. 4 k=0.5
Mean h=5 h=5 h=4 h=4 1<h<5
0 469.1 207.4 169.0 89.4 464
0.01 467.6 206.8 168.7 89.2 424
0.02 463.0 205.2 167.7 88.8 383
0.03 455.6 202.6 166.1 88.1 346
0.04 445.5 199.0 163.9 87.2 312
0.05 433.2 194.5 161.2 86.0 282
0.06 419.0 189.3 157.9 84.7 255
0.07 403.2 183.5 154.3 83.1 230
0.08 386.3 177.2 150.2 81.3 208
. . . . . .
. . . . . .
0.9 12.3 10.3 9.8 0.9 20
0.91 12.1 10.2 9.7 0.91 17
0.92 11.9 10.0 9.5 0.92 15
0.93 11.6 9.9 9.3 0.93 14
0.94 11.4 9.7 9.2 0.94 12
0.95 11.2 9.6 9.0 0.95 11
0.96 11.0 9.4 8.9 0.96 9
0.97 10.9 9.3 8.7 0.97 8
0.98 10.7 9.1 8.6 0.98 7
0.99 10.5 9.0 8.5 0.99 6
1 10.3 8.9 8.3 1 6
Fig.12:ARL distribution k=0.5 and h=5
VIII. SUMMARY AND CONCLUSIONS
In this paper investigates the limitations of the traditional
concept of the Cusum and average run length, it is seen that
the Cusum chart is a good technique and have a small sift
between the observation and the sample size n have a good
estimate on the Cusum and ARL. The ARL distribution
affected by the parameter value of k and h, so in the best
ARL graph is if value of k=5, and h=5 Decreasing the
deviation and range between our observations by increasing
the sample size (subgroup).In designing a control chart, we
must specify the sample size in determining Cusum value.
In general, larger samples will make it easier to detect small
shifts in the process. The Cusum chart is particularly
helpful in determining when the assignable cause has
occurred, effective
Table (6) proportion of PH in Water for 32 days
PH
N. of
Day X1 X2 X3 X4 X5
1 6.97 6.88 6.76 6.8 7.02
2 7.02 6.86 7 6.82 6.92
3 6.96 6.8 7.05 6.82 7.03
4 6.92 6.97 6.9 7.03 6.96
5 7.05 6.96 6.67 6.92 6.97
6 6.95 6.97 6.91 6.99 7
7 7.34 7.17 7.22 7.11 7.1
8 7.01 6.88 7.1 7.05 7.22
9 7.08 7.03 7.06 7.17 7.07
10 7.09 7 7.04 7.08 7.06
. . . . . .
. . . . . .
. . . . . .
25 6.97 6.93 6.99 6.93 6.93

26 6.9 6.84 6.84 6.84 6.96
27 7.09 6.93 7.02 7 7.08
28 7.14 6.97 7.06 7.06 7.1
29 7.12 7.02 6.86 7.09 7.39
30 7.95 6.95 7.02 7.02 7.06
31 7 7.09 7.06 7.12 7.02
32 7.01 6.97 6.96 6.98 7
REFERENCES
[1] Arthur B. Yeh , Dennis K. J. Lin (2004) , Quality
Technology & Quantitative Management Vol.1, No.1
pp. 65-86, 2004 Unified CUSUM Charts for
Monitoring Process Mean and Variability”
[2] Douglas C. Montgomery 2009, Introduction to
Statistical Quality Control 6th edition.
[3] Getulio Rodrigues De Olivera Filho, MD (2000)
Economics, Education, And Health System Research
2002 . Construction of Learning Curves for basic skills
in Anesthetic Procedures An Application for the
Cumulative Sum Method
[4] John Oakland 2008 Statistical Process Control, Sixth
Edition.
[5] HELM (VERSION 1: April 8, 2004): Workbook Level
1 46.2: Quality Control
[6] KARAOGLAN A.D. and BAYHAN G. M. March
2012; ARL performance of residual control charts for
trend AR (1) process: A case study on peroxide values
of stored vegetable oil Department of Industrial
Engineering, Balikesir University, Cagis Campus,
10145, Balikesir -Turkey.
[7] Layth C. Alwan 2000 “Statistical Process Analysis
[8] M. Kovářík, Klímek Petr (2012) The Usage of Time
Series Control Charts for Financial Process Analysis,
Journal of Competitiveness Vol. 4, Issue 3, pp. 29-45,
September 2012 ISSN 1804-171X (Print), ISSN 1804-
1728 (On-line), DOI: 10.7441/ joc. 2012.03.03
[9] M. Kovářík, 2013: Volatility Change Point Detection
Using Stochastic Differential Equations and Time
Series Control Charts, Issue 2, Volume 7, 2013,
INTERNATIONAL JOURNAL OF
MATHEMATICAL MODELS AND METHODS IN
APPLIED SCIENCES
[10]Ryan T.P (2000) : ( Statistical Methods For Quality
Improvement), 2end Edition-
[11]Sermin E., Nevin U. And Mehemt S. Oct. 2008
Control Charts for qutocorrelated d claimant data
[12]Journal Of scientific & Industrial Research Vol.68
January 2009 pp11-17

The Effect of Sample Size On (Cusum and ARIMA) Control Charts

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