On Some Applications of SPC Techniques on
Water Data
John Mitsakos and Stelios Psarakis
Abstract -- Statistical Process Control (SPC) is an important
tool for improving the quality of a process. It is achieved using
statistical methods, and the main tools that SPC uses are the
control charts.
Traditional SPC control charts are effective tools for
improving process quality. A basic assumption in standard
applications of control charts is that observations from the
process at different times are independent and identically
distributed (i.i.d) variables that follow a normal distribution.
However, the assumption of independence is often violated for
processes in many industries, such as chemical industries.
Under
this
scenario,
observations
are frequently
autocorrelated, and this autocorrelation has a serious impact
on the control charts. A typical effect of autocorrelation is the
decrease of the in-control average run length (ARL) leading to
a higher false alarm rate than that of an independent process.
In this study, we deal with autocorrelated data that
came from a chemical process (the quality characteristics of
water for human consumption). The variables that are under
investigation are the free chlorine and turbidity.
Index terms-- Statistical process control; Control charts;
Autocorrelation; Times series models; Autocorrelated process
control
I.
INTRODUCTION
Statistical Process Control uses statistical
techniques for checking, analyzing and finally improving
the quality of the processes reducing their variability. This
variability consists of two major parts: the common cause
and the special cause part. The common causes are inherent
part of the procedure, so, they cannot be avoided. On the
other hand, the existence of the special causes is not an
inherent part of the procedure, and the main objective of the
SPC techniques is the disaggregation between a common
cause and a special cause, in order to eliminate the latter
one.
The basic tools that SPC uses are control charts.
The traditional control charts are based on the principle that
the observations, that they examine, are independent and
identically distributed (i.i.d) observations. However, this
principle is often violated, due to the fact that there are a lot
of cases that the observations are not independent, because
of time dependence. Thus, if we apply the traditional SPC
control charts at these data, the effect of autocorrelation will
generate a smaller in control average run length (ARL),

Athens University of Economics and Business,
Department of Statistics.
which leads to a higher false alarm rate. Time dependence
usually appears in chemical procedures, medical processes,
processes where we have environmental changes e.t.c. As a
result, a systematic trend may be apparent in the procedure
that depends on the parameter time. So, the data are
autocorrelated, and some actions must be taken, in order to
eliminate the autocorrelation. Practically, if the amount of
the autocorrelation, that is present in the data, is small
enough, the traditional SPC control charts can be applied.
This can happen if there is an adjustment at the estimated
process parameters and at the control limits. However, if
there is a serious amount of autocorrelation present in the
data, other techniques must be applied.
An appropriate technique that is widely used has
been proposed by Alwan and Roberts [1]. They suggested
the use of time series analysis, in order to explain the
autocorrelation that is present in the data. The fitting of an
appropriate time series model to the data is verified by
checking the residuals that come from the model. If the
residuals are i.i.d, then they can be used at the traditional
SPC charts, in order to confirm whether the procedure is in
control or not.
In this paper we apply the SPC techniques on two
variables of water data the “Free CL” and the “Turbidity”.
There are 365 available data, which they come from
measurements of the Athens – Piraeus Water Company,
from 1/1/2001 to 31/12/2001. Here we present the results of
this analysis and some ideas for further research.
With the purpose of designing a control chart, we
must specify the sample size and the frequency of sampling.
When we are choosing the sample size, we must have in
mind the size of the shift that we are trying to detect. In
order to choose the sample size, we must take in account
the run length (RL) of the process. It is one of the most
important properties associated with any control chart. The
run length is the number of observations required to obtain
an observation outside the control limits for a given shift in
the mean. (Wadsworth et. al. [19]). The knowledge of the
run length distribution gives us the possibility to compute
the average run length (ARL). The average run length is the
average number of points until an observation falls outside
the established control limits. We desire the ARL to be
large enough when no special causes are present in a
process and small when they have occurred. Knowledge of
the ARL for a particular special cause allows us to design
more effective control charts.
In sections 2 and 3 of the paper we refer to
CUSUM and EWMA control charts. In section 4 we briefly
introduce the autocorrelation mechanism and some
appropriate methods in order to deal with the
autocorrelation. Finally, we present the statistical analysis
on two elements of the available water data the Free
Chlorine and Turbidity. Some conclusions are presented at
the final part of the paper.
II.
CUSUM CONTROL CHARTS
A major disadvantage of any Shewhart control
chart is the luck of ability to discover small persistent shifts
in a process. An effective alternative to the Shewhart
control chart in order to use when small shifts are of interest
is the Cumulative Sum control chart (CUSUM chart).
CUSUM charts were first proposed by Page [13].
The tabular CUSUM accumulates derivations from
μ0 that are above target with one statistic C+ and it
accumulates derivations from μ0 that are below target with
another statistic C-. These two statistics C+ and C- are called
one-sided upper and lower CUSUMs, respectively. The
formulas that allow us to compute these statistics are
C I = max[0,xi – (μ0 + K) + C+I-1]
C I = max[0, (μ0 - K) - xi + C-I-1]
The value K is called reference value and it is
often chosen about halfway between the target value μ 0 and
the out of control value of the mean, say μ1, that we are
interested in detecting quickly. Therefore, if the shift is
expressed in units of standard deviation as μ 1=μ0+δσ, where
δσ=|μ1-μ0|, then K is one half the magnitude of the shift:

2
 
| 1   0 |
2


If either C I or C I exceeds the decision interval
H, the process is said to be out of control.
A reasonable value of the decision interval H is
five times the process standard deviation σ. CUSUM charts
has been extensively discussed in the recent bibliography.
For an extended review of CUSUM control charts see
Maravelakis [8] as well as Montgomery [10].
III.
EXPONENTIALLY
WEIGHTED
AVERAGE CHART (EWMA CHART)
zt = λ x t + (1-λ)zt-1
where zt is the EWMA value at time t (t=0,1,2,3,..) and λ is
a constant 0<λ  1. In the case that λ=1, the EWMA chart is
equivalent to Shewhart chart. The starting value of z is the
target value μ0 so z0=μ0.
Sometimes as a starting value we use the average
of preliminary data so z0= x .
Therefore, the center line, the UCL and the LCL
for the EWMA control chart are:
UCL= μ0+Lσ
MOVING
The additional alternative method to the Shewhart control
chart when we are interested to detect small shifts in a
process is the exponentially weighted moving average chart
(EWMA chart). The performance of the EWMA control
chart is approximately equivalent to that of the CUSUM
control chart, and is some ways it is easier to set up and
operate.
Roberts first introduced the EWMA control chart in 1959.
The philosophy of EWMA procedure is to give to the most
recent observation the greatest weight and all previous
observations weights decreasing in geometric progression
from the most recent to the first. Since EWMA can be
viewed as a weighted average of all past and current
observations, it is very insensitive to the normality
assumption. The exponentially weighted moving average is

2
CL= μ0
LCL= μ0 - Lσ
where the starting values are C+0 = C-0 = 0
K
defined as:

2
EWMA control charts have been extensively
discussed in recent bibliography. For an extended review of
EWMA control charts see Monopolis [9] as well as
Montgomery [10].
IV. THE GENERATION OF AUTOCORRELATED PROCESSES
In the case that the value of a particular parameter
is dependent on previous values of this parameter then
autocorrelation is present in the data. Basically, all
manufacturing processes are driven by inertial elements,
and when the interval between samples becomes small, the
observations of the process will be correlated over time.
The properties of traditional control charts are
based on the assumption of independence. This results that
these control charts are not perform properly if the quality
characteristics, that are under investigation, exhibits various
levels of autocorrelation. Even in the case that small values
of autocorrelation are present, it can have serious effects on
the statistical properties of conventional control charts.
Many authors have dealt with this problem including
Berthouex et. al [3], Alwan and Roberts [1], Harris and
Ross [4], Montgomery and Mastrangelo [11], Alwan [2]
and others.
The main effect of autocorrelation in the process
data to SPC schemes is that it produces control limits that
are much tighter than desired. This causes a substantial
increase in the average false alarm rate and a decrease in
the ability of detecting changes on the process. Padgett et
al. [12] investigated Shewhart charts when the correlation
structure of the process can be described by an AR (1) plus
a random error model and found that this type of
autocorrelation affects the false alarm rate. Alwan [2]
discussed the masking effect of special causes by the
autocorrelation of the data, and demonstrated that in the
presence of even moderate levels of autocorrelation, an out
of control point of the chart did not necessarily indicates a
process change.
Additionally, the ARL performance of the control
charts is degraded. Schmid and Schone [16] proved
theoretically that the run length of an autocorrelated process
is larger than in the case of independent values provided
that all the autocovariances are grater than or equal to zero.
Early detection of the occurrence of assignable causes
ensures that necessary corrective actions can be taken
before a lot of nonconforming units are produced.
Therefore, when there is autocorrelation in the process data
standard control charts should not be applied.
V. THE MAIN APPROACHES OF
FOR AUTOCORELLATED PROCESSES
CONTROL CHARTS
There are two main approaches for constructing control
charts for autocorrelated data.
The first approach uses standard control charts, but
adjusts the control limits to account for the autocorrelation
and adjusts the method of estimating the process variance
so that the true process variance is being estimated (see e.g.
Vassilopoulos and Stamboulis [18], VanBrackle and
Reynolds [17], Schmid [15]).
The second approach fits time series model to the
process data so that forecasts of each observation can be
made using the previous observations and then applies to
the residuals traditional control charts or some slightly
modified versions of those (see e.g. Alwan and Roberts [1],
Harris and Ross[4], Montgomery and Mastrangelo [11],
Mastrangelo and Montgomery [7], Lu and Reynolds [6].
In the following we present the Alwan and Robert
approach as well as the approach of Lu and Reynolds.
Alwan and Roberts’ method
Alwan and Roberts [1] suggested the
implementation of two basic charts rather than one. They
introduced the Special Cause Control chart known as SCC
chart and the Common Cause Control chart, known as CCC
chart. The definitions of these control charts are the
following:
I. Common Cause Control chart is a plot of the
fitted values or forecasts obtained when the correlated
process is modeled with an ARIMA model.
II. Special Cause Control chart is a traditional
control chart of the residuals or one step ahead predictor
errors.
The Common Cause Control chart assumes that no
special causes have been occurred. Common cause control
chart is not an actually control chart because of the fact that
it has not any control limits. It was merely intended to give
a representation of the current and estimated or predicted
state of the process. This chart essentially accounts for the
systematic variation that exists in a process.
The Special cause control chart is merely a
Shewhart or individuals chart, but rather than plotting the
standard deviation of the residuals, σe. If the process is
fitted correctly, then σe = σα.
The control limits of the SCC chart are the
following:
LCL = -Lσα
UCL=Lσα
where L is a constant multiplier, and it is usually assumed
to be equal to 3. The rational of using residuals charts is
that assuming that the correct time series model is fitted to
the data, the residuals will be independently and identically
distributed random variables. Then all the assumptions of
the traditional control charts in Statistical Process control
are met. However many people seem to agree that the
residuals charts do not have the same properties as the
traditional charts as the charts on the original observations
and that the ability of a chart to detect a change in the
process mean depends on the right choice of the model that
describes the observed data. The time series models that
have been proposed in the literature in order to deal with
autocorrelated process are AR (1) model, AR (2) model,
ARMA (1,1) model etc.
Lu and Reynolds’ method
Lu and Reynolds [5] considered the behaviour of the
residuals when there is a change in the variance of the
process. To evaluate the performance of control charts
based on the residuals from a forecast, it is necessary to find
the distribution to the residuals when the change exists.
When the process is in control the residual at observation k
from the minimum mean square error forecast made at
observation k-1 is:
ek = Xk – μ0 – φ (Xk-1 – μ0) + θ ek-1
where φ is the autoregressive parameter and θ is
the moving average parameter of the ARMA (1,1) model.
They supposed that there is a step change in the process
variance σ X , and this can be due to a special cause that
produces a change in any of the underlying process
2
parameters φ, σ α , σ ε . They investigated if an increase in
2
2
the process variance
σ 2X is caused by an increase in σ μ2 or
in σ ε . They assumed that the autoregressive parameter φ is
2
constant, thus, an increase in
σ μ2 is caused by an increase
in σ α .
2
They modeled this increase in σ α or σ ε by
2
2
supposing that between samples t-1 and t, σ α increases
2
actual observations, we plot the residuals et=Xt- X̂ t which
from the in control
are obtained after fitting the process with an ARIMA
model. The mean of the residuals is zero; therefore, the
centerline of the Special cause control chart is zero.
Similarly the standard deviation used in this case is the
in control
 20
 2 0 to  21 and σ ε2 increases from the
to   1 . Then, the residuals after the shift
2
are correlated normal random variables with mean 0 and
variance:
2
2
2
2
2
Var( et) = σ  0 + (σ  1 - σ  0 )+ ( σ  1 - σ  0 )
Thus the effect of an increase in σ α or σ ε is to
2
2
increase the variance of the residuals, while the means of
the residuals remain constant and equal to zero. After the
shift the smallest variance at t is given by:
σ  0 + (  1    0 ) + (  1    0 )
2
2
2
2
2
but then the variances continually increase to the
limit:
2
2
2
2
2
2
σ  0 +   2   1 (  1    0 ) + (  1    0 )
1 2
1 2
2
It is interesting to notice that an increase in σ  or
2
σ  results an increase in the variances of the residuals. At
the same time the overall mean remains unaffected. On the
other hand, a shift in the overall mean of the process
changes the means but not the variances of the residuals
with the largest change occurring at time t immediately
after the shift.
Another approach that Lu and Reynolds [5]
investigated was the application the EWMA of the logs of
the Squared Residuals chart. The control statistic of the
EWMA of the logs of the Squared Residuals chart is given
by:
2
2
Xt = max ((1-λ) Xt-1+λln (e t ), ln σ  0 )
2
where X0 = ln σ  0 and λ is a smoothing parameter
satisfying 0<λ1. They noticed that unlike the two-sided
EWMA chart that they used in the investigation of the
process mean. The EWMA chart that they considered for
the variance is an one-sided control chart and the control
2
statistic resets to the target value X0 = ln σ  0 .
VI. WATER’S ELABORATION METHODOLOGY
Water’s Elaboration Methodology at Athens &
Piraeus Water Company refineries is aimed at being the
water an excellent product for the human beings, salutary,
limpid, microbiologically secure, and free of every entity
that could have a dangerous affect at humans’ health.
The drinkable water is allocated through a
crisscross allotment. This check is incumbent upon the
current legal system. The water parameters that are been
checked every day in the refineries are the following:
Ion hydrogen’s concentration (PH), Ammonium, Salinitrites, Aluminium, Turbidity, Free Chlorine, Escherichia
Coli, Faecal Streptococcus and Heterotrophic bacilli.
In the following we explore two of the most
important water parameters the Free Cl and the Turbidity.
A. Fitting an appropriate Time Series
model for Free CL
First we have to check for the normality of our data. There
are three outliers in this variable, which constitutes
approximately the 1% of the whole population of the
variable. Normal Probability Plot and Shapiro-Wilk test for
Normality are suggested that this population follows the
normal distribution, due to the fact that p-value of ShapiroWilk test is equal to 0.1081, which is greater than 0.05, so,
we cannot reject the null hypothesis that our population is
following a normal distribution.
The fact that a serious amount of autocorrelation is present
in the data moved as to find a time series model for the
data. Firstly, we draw the Time Series plot of Free CL. The
Time series plot and the graphs of the ACF and PACF
showed that the procedure is not stationary, thus, it is
necessary to stabilize it. The slow decay that it is apparent
at the graphs of the ACF and PACF leads as to differencing
this procedure by one lag, in order to eliminate any linear
trend that it is present to the data. Any seasonal component
or a quadratic trend is not seemed to be present at the data.
The time series plot that arises from this action is illustrated
in Figure 1. The examination of Figure 1 indicates that only
the first coefficient of the ACF is statistically significant.
The examination of the PACF also gives us the results that
only the first and the second coefficient are statistically
significant. All the other coefficients are lying between the
confidence intervals, so we can conclude that this process is
stationary. A rough guide indicates that an ARIMA (2,1,1)
or ARIMA (1,1,1) or an IMA (1,1) are the advisable models
in order to explain this procedure. The final selection of the
model will be based on AICC, BIC and Gaussian
Likelihood statistics.
The AICC, BIC and Gaussian Likelihood criteria
between ARIMA (1,1,1) and IMA (1,1) models suggest that
the proper model is the IMA (1,1) model. The maximum
Likelihood estimation gives the proper coefficients for the
IMA (1,1) model:
X(t) = Z(t) - .6883 Z(t-1)
For concluding the appropriateness of this model we check
if the residuals (named ResdiffCL) are uncorrelated and
normal distributed. The normal probability plot as well as
Shapiro –Wilk test indicate that the residuals are following
a normal distribution.
The check of randomness of the residuals is based
on Ljung – Box statistic and on the ACF and PACF of the
residuals. The p-value of the Ljung – Box statistic is equals
to 0,12546 that signifies that the null hypothesis that the
autocorrelations for all lags up to lag k equal zero is not
rejected.
The examination of the ACF and PACF graphs of the
residuals does not differ from the conclusions of Ljung –
Box Statistic. The residuals are i.i.d distributed, so we have
found a proper model for the data. The fact that the
residuals can be considered as a white noise allows us to
work on the traditional SPC control charts.
Control charts for Free CL
The Special Cause Control, EWMA and CUSUM
charts, are illustrated in figures 2, 3 and 4.They show that in
SCC there is one point out of the control limits (298 th
observation). The EWMA chart of the variable ResdiffCL,
the value of λ is λ = 0.2 and the control limits are the
traditional 3-sigmas control limits. EWMA chart gives
points on the control limits at the beginning of the
procedure that signifies that the process is out of control.
Additionally, there is one point on the lower limit nearby
the 350th observation. The CUSUM chart gives three points
out of the upper control limit at the beginning of the
procedure, one point out of the lower CUSUM between
150th and 200th observation, one point on the lower
CUSUM nearby the 300th observation and one point out of
the lower CUSUM nearby the 350th observation. From the
above it is concluded that the EWMA chart behaves
similarly to the CUSUM chart and they act much faster than
SCC chart in order to detect transitions at the mean of the
process. Ones again the SCC’s performance is very poor
compared with the performance of the other two charts.
B. Fitting an appropriate Time Series
model at variable Turbidity
For finding the most appropriate time series model
to explain sufficiently the autocorrelation in case of
turbidity, we draw the time series plot and then we draw the
graph that it represents the ACF and PACF of the data (see
figure 5).
The examination of figure 6 puts across that the
time series model is not stationary, so, we have to take
action, in order to stabilize it. There are statistically
significant autocorrelations present at the data. This is the
reason that the traditional SPC control charts are improper
to investigate if the process is in-control. A pattern for
achieving a stationary process would be the differencing of
the process by one lag. The examination of figure 5
indicates that only the first coefficient of the ACF is
statistically significant. The examination of the PACF gives
us the results that only the first and the second coefficient
are statistically significant. All the other coefficients are
lying between the confidence intervals, so we can conclude,
due to the fact that at least the 95% of the coefficients are
not statistically significant, that this process is stationary. A
rough guide that comes from figure 6 indicates that an
ARIMA (2,1,1) or ARIMA (1,1,1) or an IMA (1,1) are the
advisable models in order to explain this procedure. The
final selection of the model will be based on AICC, BIC
and Gaussian Likelihood statistics.
The AICC, BIC and Gaussian Likelihood criteria between
ARIMA (1,1,1) and IMA (1,1) models show that the proper
model is the ARIMA (1,1,1) model. The maximum
likelihood estimation gives the proper coefficients for the
ARIMA (1,1,1) model:
X(t) =0.2678 X(t-1)+ Z(t) -0.6836 Z(t-1)
For concluding the appropriateness of this model we check
if the residuals (named ResdiffTH) are uncorrelated and
normal distributed. The normal probability plot as well as
Shapiro –Wilk test indicate that the residuals are following
a normal distribution.
The check of randomness of the residuals is based
on Ljung – Box statistic and on the ACF and PACF of the
residuals. The p-value of the Ljung – Box statistic is equals
to 0.87397 that indicates that the null hypothesis that the
autocorrelations for all lags up to lag k equal zero is not
rejected. The residuals are i.i.d distributed, so we have
found a proper model for the data. The fact that the
residuals can be considered as a white noise allows us to
work on the traditional SPC control charts.
Control charts for Turbidity
The Special Cause Control, EWMA and CUSUM charts,
are illustrated in figures 6,7 and 8.
They show that in SCC there are thirteen points
out of the control limits the points 28, 146, 147, 309, 310,
311, 316, 317, 318, 320, 358, 361, 362 and it is noticed that
the process is out of control, because beside of the fact that
there are points out for the control limits, there are also
some values laying nearby the control limits. The EWMA
chart gives one point on the upper control limit between the
first observation and the 50th observation, a point out of the
control limits also indicates that there is special cause
apparent nearby the 150th observation, a group of points out
of the control limits nearby the 300th observation, there are
points in a sequence towards to the upper limit between the
300th and the 340th observation, another sequence of points
nearby the 350th observation out of the control limits, and
four points out of the control limits at the end of the
process. There are also points nearby the 350 th observation
towards to the lower limit. All that signifies that the process
is out of control. The CUSUM chart shows two points out
of the control limits between the first and the 50 th
observation, eight points out of the upper and the lower
CUSUM nearby the 150th observation, a group of points out
of the control limits between 300th and 350th observation
and three points out of the upper CUSUM nearby at the end
of the procedure. There are also points close to the bounds.
It is concluded that the EWMA chart behaves similarly to
the CUSUM chart and they act much faster than SCC chart
in order to detect transitions at the mean of the process. The
SCC’s performance is very poor compared with the
performance of the other two charts. On the other hand, the
SCC chart seems to act more accurately than the other two
charts in detecting large shifts.
C. Detection of Shifts in Variability
Free Chlorine
In order to detect shifts in the process variance, we compare
two control charts that they are proper to investigate shifts
in the process variance. The first one (figure 9) is the
traditional S-chart and the second one (figure 10) is the
Shewhart chart of the squared residuals.
The examination of theses charts indicates that the
Shewhart chart for the Squared Residuals performs better
than the traditional S-chart. They have indicated that there
are shifts much faster than the traditional S chart. The latter
chart is performed quite well in the case of large shifts. The
fact that the Shewhart chart for the Squared Residuals had
not any common shifts with the Traditional S-chart, it is
reasonable to believe that the S-chart has marked large
shifts and the other small shifts.
Turbidity
In order to detect shifts in the process variance, we compare
two control charts that they are proper to investigate shifts
in the process variance. The first one is be the traditional Schart (figure 11) and the second (figure 12) the Shewhart
chart of the squared residuals.
The examination of these charts indicates that the
Shewhart chart for the Squared Residuals performs much
better than the traditional S-chart. The Shewhart chart for
the Squared Residuals is on the alert with points out of the
bound nearby at the beginning of the procedure The S-chart
has pointed out that there is one observation out of control
at the 75th observation.
same like the corresponding charts about the mean.
Finally, in this paper we do not conclude the part
of the adjustment method. The performance of this method
was very poor due to the fact that there was a serious
amount of autocorrelation present to the data and this
technique is advisable only for small values of
autocorrelation.
Another approach in order to investigate if there is
a shift in these data would be the usage of the multivariate
SPC methods. The fact that there is a serious amount of
autocorrelation on these data will lead us to use a similar to
Alwan and Roberts approach that it is proposed by Pan and
Jarrett (2004). They suggested using the traditional MSPC
control charts on the uncorrelated residuals.
Finally we have to notice that all the data that we
had in hands was in the bounds that the European Union
specification limits. So, there are no reasons for anxiety
about the quality of the water.
REFERENCES
[1]
VII. CONCLUSIONS
In this paper we analysed the behaviour of the charts under
some specific conditions. The traditional SPC control charts
work properly under the condition of independence. The
fact that there were present in the data a serious amount of
correlation, it has reduced dramatically the performance of
the charts, giving a huge number of false alarms. The
majority of the observations are drawn out of the control
limits. After the application of the Alwan and Roberts [1]
method and the elimination of the correlation of the data, in
the case of the Shewhart chart there are no points out of
control limits. The same conclusions can be also made in
the case of the EWMA chart. The same conclusions are
mined if we have reports on the other variables. The
presence of autocorrelation in the data reduces dramatically
the control limits and ARL of the procedures.
A point that requires investigation is the case of the outliers.
The presence of them seems to affect Shewhart chart more
than the other charts. This has as a result to signal points
out of control without knowing if these observations come
from the process. The alarms that produce this chart have a
meaning that they have been produced by a mistake at the
record of the data. That is a reason to check twice the data
before take actions against the procedure.
The performance of the charts in order to monitor
shifts in the processes that we have already studied is
generally satisfactory. The EWMA chart as well as
CUSUM chart act faster than Shewhart chart for detecting
small shifts in the process mean. On the other side
Shewhart chart acts better in large shifts than the other two.
The combination of these charts will give secure results
about the behaviour of the process.
In order to detect shifts in the process variance we
applied charts that have been recommended by Lu and
Reynolds [5]. The conclusions of these charts are almost the
Alwan, L.C. and Roberts, H.V. (1988). Time-Series
Modeling for Statistical Process Control. Journal of
Business and Economic Statistics 6, 87-95.
[2] Alwan, L.C. (1992). Effects of Autocorrelation on
Control Chart Performance. Communications in
Statistics – Theory and Methods 21, 1025-1049.
[3] Berthouex, P.M., Hunter, W.G. and Pallesen, L.
(1978). Monitoring Sewage Treatment Plants: Some
Quality Control Aspects. Journal of Quality
Technology, 10, 139-149.
[4] Harris, T.J. and Ross, W.H. (1991). Statistical
Process Control Procedures for Correlated
Observations. The Canadian Journal of Chemical
Engineering, 69, 48-57.
[5] Lu, C.W. and Reynolds, M.R. Jr (1999a). Control
Charts for Monitoring the Mean and Variance of
Autocorrelated Processes. Journal of Quality
Technology 31, 259-274.
[6] Lu, C.W. and Reynolds, M.R. Jr (1999b). EWMA
Control Charts for Monitoring the Mean of
Autocorrelated Processes. Journal of Quality
Technology 31, 166-188.
[7] Mastrangelo, C.M. and Montgomery, D.C. (1995).
SPC with Correlated Observations for the Chemical
and Process Industries. Quality and Reliability
Engineering International, 11 79-89.
[8] Maravelakis, P.E. (1998). CUSUM Procedures in
Statistical Process Control. Msc. Thesis. Athens
University of Economics and Business Department of
Statistics, ISBN: 960-7929-11-X
[9] Monopolis I.K. (1999). Exponentially Weighted
Moving Average (EWMA) Process in Statistical
Process Control (SPC). MSc Thesis Athens
University of Economics and Business Department of
Statistics ISBN:960-7929-38.
[10] Montgomery, D.C. (2005). Introduction to Statistical
Quality Control, 5th ed. John Wiley and Sons, Inc.,
N. York.
[11] Montgomery, D.C. and Mastrangelo C.M. (1991).
Some Statistical Process Control Methods for
Autocorrelated Data. Journal of Quality Technology
23, 179-193.
[12] Padgett, C.S., Thombs, L.A. and Pangett, W.J.
(1992). On the α-Risks for the Shewhart Control
Charts. Communications in Statistics-Simulation and
Computation, 21, 1125-1147.
[13] Page, E.S. (1954). Continuous Inspection Schemes.
Biometrica, 41, 100-115.
[14] Pan and Jarrett (2004. Applying state space to SPC:
Monitoring multivariate time series. Journal of
Applied Statistics 31, 397-418.
[15] Schmid, W. (1997). On EWMA Charts for Time
Series. Frontiers in Statistical Quality Control, 5,
115-137.
[16] Schmid, W. and Schöne, A. (1997). Some Properties
of the EWMA Control Chart in the Presence of
Autocorrelation. Annals of Statistics, 3, 1277 – 1283
[17] VanBrackle, L.N. and Reynolds M.R. JR. (1997).
EWMA and CUSUM Control Charts in the Presence
of Autocorrelation. Communications in StatisticsSimulation and Computation, 26, 979-1008.
[18] Vasilopoulos, A.V. and Stamboulis, A.P. (1978).
Modification of Control Chart limits in the Presence
of Data Correlation. Journal of Quality Technology,
10, 20 – 30.
[19] Wadsworth, H.M., Stephens, K.S. and Godfrey,
A.B. (1986). Modern Methods for Quality
Control and Impovements. New York, John
Wiley.
EW MA Chart for RESDIFFC
EWMA
0,05
UCL=0,04557
Mean=4,79E-04
0,00
LCL=-0,04461
-0,05
0
100
200
300
400
Sample Number
Figure 3
CUSUM Chart for RESDIFFCL
Upper CUSUM
0,202531
Cumulative Sum
0,2
0,0
-0,2
-2,0E-01
Lower CUSUM
0
100
200
300
400
Subgroup Number
Figure4
Series
160.
Sample ACF
1.00
140.
120.
Sample PACF
1.00
.80
.80
.60
.60
.40
.40
100.
.20
80.
60.
Sample ACF
1.00
Series
.60
.40
.40
-.40
-.60
20.
.80
.60
.00
-.20
-.40
40.
Sample PACF
1.00
.80
.20
.00
-.20
-.60
-.80
0
50
100
150
200
250
300
-.80
-1.00
350
-1.00
0
5
10
15
20
25
30
35
40
0
5
10
15
20
.700
.650
.20
.20
.600
.00
.00
-.20
-.20
-.40
-.40
.400
-.60
-.60
.350
-.80
-.80
.550
.500
.450
-1.00
.300
100
150
200
250
300
-1.00
0
350
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
Figure 1
X-bar Chart for RESDIFFA
1
100
I Chart for RESDIFFC
0,2
1
UCL=0,1511
0,1
0,0
Mean=4,79E-04
Sample Mean
50
Individual Value
0
Figure 5
1
1
1
50
UCL=39,72
0
Mean=-0,03307
LCL=-39,79
-50
1
0
-0,1
1
100
200
300
Sample Number
LCL=-0,1502
-0,2
0
100
200
300
Observation Number
Figure2
400
Figure 6
400
25
30
35
40
S Chart for RESDIFFA
EW MA Chart for RESDIFFA
20
30
1
1
UCL=10,68
EWMA
10
0
Mean=-0,03307
-10
Sample StDev
20
UCL=14,69
10
S=4,497
LCL=-10,75
0
-20
0
100
200
300
LCL=0
400
0
Sample Number
50
100
150
Sample Number
Figure 7
Figure 11
CUSUM Chart for RESDIFFAL
Shewhart Chart for
SQRESAL
300
1
Upper CUSUM
42,8558
0
-42,8558
-100
100
200
300
400
1
1
Mean=32,62
0
LCL=-82,16
0
50
100
Figure 8
Figure 12
1
UCL=0,1267
0,10
0,05
S=0,03880
0,00
LCL=0
0
100
200
Sample Number
Figure 9
Shewhart Chart for
SQRESCL
1
0,02
1
1
1
UCL=0,01121
0,01
Mean=0,002731
0,00
LCL=-5,7E-03
100
Sample Number
Figure 10
200
UCL=147,4
100
Sample Number
0,15
Sample StDev
1
Subgroup Number
S Chart for RESDIFFC
0
1
-100
Lower CUSUM
0
Sample Mean
1
200
Sample Mean
Cumulative Sum
100
150