Multivariate Forecast Based Control Charting Schemes

Abstract

Recently, much research had been performed in the area of control charting techniques for monitoring autocorrelated processes, especially regarding forecast based monitoring schemes. Forecast based monitoring schemes involve fitting an appropriate time-series model to the process, generating one step ahead forecast errors, and monitoring the forecast errors with traditional control charts. Another recent suggestion involved generating both the one-step-ahead and two-step-ahead forecast errors, then monitoring them with multivariate control charts. This article investigates the suggested multivariate approaches in regards to various ARMA(1,1) and AR(1) processes and shows the performance of the control charts relative to their univariate counterparts

Full text

Journal of Business, Industry and Economics Volume 1, No 2, Spring 2002, pp 145-156
MULTIVARIATE FORECAST BASED CONTROL CHARTING SCHEMES
JOHN N. DYER
Georgia Southern University, Statesboro, GA 30460
B. MICHAEL ADAMS & MICHAEL D. CONERLY
University of Alabama, Tuscaloosa, AL 35487
J. DOUGLAS BARRETT
University of North Alabama, Livingston, AL 35632
Abstract
Recently, much research had been performed in the area of control charting techniques
for monitoring autocorrelated processes, especially regarding forecast based monitoring
schemes. Forecast based monitoring schemes involve fitting an appropriate time-series model to
the process, generating one step ahead forecast errors, and monitoring the forecast errors with
traditional control charts. Another recent suggestion involved generating both the one-step-
ahead and two-step-ahead forecast errors, then monitoring them with multivariate control charts.
This article investigates the suggested multivariate approaches in regards to various ARMA(1,1)
and AR(1) processes and shows the performance of the control charts relative to their univariate
counterparts.
Introduction
Most traditional control charts were designed to monitor output from independent and
identically distributed (iid) processes. When output data are identically distributed but correlated
over time, they are said to autocorrelated, and the traditional charts applied to the data have been
shown to be unreliable (Maragah and Woodall (1992), Harris and Ross (1991)). Recent
advances have provided forecast-based monitoring schemes to address this problem.
A forecast-based monitoring scheme involves identifying the proper time-series model
which characterizes the process, obtaining the appropriate Box-Jenkins one-step-ahead forecast

of process observations, then applying traditional control charts to forecast errors (Alwan and
Roberts (1988), Wardell, Moskowitz, and Plante (1994), Lin and Adams (1996), Lu and
Reynolds (1999a), and Lu and Reynolds (1999b), Dyer, et al. (2001)). If the assumed time-
series model is correct, the forecast errors are iid normal random variables. Hence, they should
perform in a manner predictable through traditional control charting techniques, enabling
monitoring for detection of step-shifts in the process mean level.
Performance Criteria
When control chart performance has been evaluated, the average run length (ARL) has
typically been used to quantify performance of the chart. The ARL is defined as the average
number of time periods until the control chart signals. An alternative performance criterion is
the cumulative distribution function (CDF). The CDF measures the cumulative proportion or
percent of signals given by the i
th
period following the shift. It should be noted that the CDF
completely characterizes the run length distribution, while the ARL is only the mean.
Additionally, the median run length (MRL) can be used in conjunction with the ARL and CDF
since it is a better measure of central tendency for skewed distributions such as the run length
distribution. The MRL is defined as the median (50
th
percentile) number of time periods until the
control chart signals.
Forecast Recovery
One unfortunate characteristic of forecast-based monitoring schemes is the phenomenon
of forecast recovery; that is, the process forecasts recover quickly from process disturbances.
Hence, the resulting forecast errors also recover quickly. The impact of forecast error recovery
on ARLs has been discussed (Dyer, et al. (2001), Adams, Woodall, and Superville (1994),
Superville and Adams (1994)) and the CDF technique has been recommended as a meaningful
criterion for evaluating the performance of charts based on forecast errors. Lin and Adams

(1996) found that when applied to AR(1) process forecast errors, the Individuals chart provides
relatively high ARLs and CDFs, the EWMA provides low ARLs and CDFs, and the Combined
EWMA-Shewhart (CES) borrows the best properties from both charts, low ARLs and high
CDFs. High (low) CDFs are defined as those exhibiting a high (low) probability of initial shift
detection relative to competing control charts. Initial shift detection refers to detection of a shift
in the first few periods following the shift. The Individuals chart is a Shewhart chart based on
forecast errors from individual observations. The EWMA chart is also based on forecast errors
from individual observations. The design of the EWMA and discussion of the smoothing
parameter  are addressed by Lin and Adams (1996), Crowder (1989), and Hunter (1986).
Article Focus
While other studies have focused on such techniques as removing correlation from
process data or correlation adjusted control limits for traditional charts, this article will focus on
the performance of various control charts applied to forecast errors generated from a number of
different time-series models. The article is in part an extension of recent research (Dyer, et al.
(2001), Lin and Adams (1996), and Wardell, Moskowitz, and Plante (1994)) investigating
forecast recovery and utilizing traditional control charting schemes, as well as an evaluation of
the Multivariate Exponentially Weighted Moving Average (MEWMA) control chart proposed by
Michelson, D.K. (1994). Performance criteria will be evaluated based on ARL, MRL, and CDF
measures for charts designed to detect a specified step-shift in a process mean level. The
standard error of the run length distribution (SRL) will also be provided. The following two
major issues will be addressed:

1. The relationship between the expected one-step-ahead (OSA) and two-step-ahead (TSA)
forecast errors and the control chart performance will be examined in regards to traditional
control charts and the MEWMA control charts.
2. Conclusions and suggestions will be made concerning the use of the traditional and
MEWMA control charts for situations involving autocorrelated processes.
The layout of this article is as follows: A description of the ARMA model with an
introduction to the notation associated with various time-series models is given. Two special
ARMA models, step shifts, and the concept of monitoring forecast errors are also discussed. An
introduction and description of MEWMA statistic will be provided. An evaluation and
discussion of ARL, MRL, and CDF performance of traditional schemes and the MEWMA
control charts applied to forecast errors arising from the ARMA(1,1) and AR(1) models will be
provided.
Models for Autocorrelated Data
Two ARMA(p, q) models have been found to have application in statistical process
control. The first model of interest is the ARMA(1,1). Wardell, Moskowitz, and Plante (1992)
address the ARMA(1,1) model, as it is a reasonable fit to data for some manufacturing processes.
The second model of interest is the ARMA(1,0), also known as the AR(1). Montgomery and
Mastrangelo (1991) and Alwan and Roberts (1988) have addressed the importance of the AR(1)
model in manufacturing processes. The next section briefly discusses process shifts associated
with the various time-series models before description of the models.
Process Shifts
Two types of shifts in the process can be considered. A step (level) shift is said to occur
if the process suddenly changes to a new level. A trend occurs when there is a step shift in the

mean of the white noise process, resulting in a gradual (trend) shift to a new process level. Step
shifts alone will be considered in this article.
The magnitude of the shift in the process can be measured either in terms of the white
noise variance, 

2
, or in terms of the process variance, 
p
2
. Process shifts will be reported in
terms of the white noise variance since it remains constant, whereas the process variance varies
depending upon the assumed model. For a given stationary model and white noise variance, the
shift size in terms of the process variance can be determined if one so chooses.
The ARMA(1,1) and AR(1) Models
In building an empirical model of an actual time-series process, the inclusion of both
autoregressive and moving average terms sometimes leads to a more parsimonious model than
could be achieved with either the pure autoregressive or pure moving average alone. This results
in the mixed autoregressive-moving average. When both terms are mixed in first order, the
resulting model is the ARMA(1, 1). The model for an in-control ARMA(1,1) is given by
Equation (1),
t
1
t-1
t
1
t-1
Y
= +
Y
+
-


 

(1)
To better understand the application of control charts based on forecast errors, one may
first consider the response of both the OSA and TSA expected forecast errors resulting from the
general ARMA(1,1) model given in Eq. (1). Extensive discussion of the behavior of OSA
forecast errors arising from various ARMA(1,1), AR(1) and MA(1) processes can be found in
Dyer, et al. [2001]. Suppose a shift of size c occurs in the ARMA(1,1) process between time
period r-1 and r. If monitoring the process with both the OSA and TSA forecast as given by
e -
Y
=
Y
ˆ
1-1
1-
1
(1)
t
t
t


(2)
e
Y
Y
ˆ
(1)
)2(
2
1
2
2
1 1



t
t
t


(3)

and considering the OSA and TSA forecast errors as
Y
ˆ
Y
(1)
t
t
)1(

e
t
(4)
Y
ˆ
Y
(2)
t
t
)2(

e
t
(5)
the expected forecast errors for Eqs. (4) and (5) can be described mathematically as
























....2,1 ,
)1(
)1)((
1
1,...2,1 0
)(
1
1
1
1
)1(
kkrtc
rtc
rt
e
E
k
t



(6)


































,...4,3
)1(
)1)((
11
2)1(
1 ,
1...2,1 0
)(
1
2
1
1
1
1
1
2
1
1
1
2
1
)2(
kkrtc
rtc
rrtc
rt
e
E
k
t







(7)
It should be noted that Eq (6) reduces to
c)1(
1


and Eq. (7) reduces to
( )1
1
2


c
for the AR(1)
model for all time periods t > r +1. Eq. (7) reduces to c for all time periods in the case of the
MA(1) model since there exists no TSA forecast. Hence, these findings will not be applied to
the MA(1) process. Additionally, in the in-control case, the OSA forecast errors are distributed
iid N(= 0, 
2
= 
2

) while the TSA forecast errors are distributed as N( = 0, 
2
= 
2

[1 + (

1
-

1
)
2
] ), but are not independent.
Tables 1 (a) and 1 (b) portray a realization of the expectation of both the OSA and TSA
forecast errors at time periods t < r, t = r, and t > r, for a c = 1


step shift in various positively
autocorrelated ARMA(1,1), and AR(1) models. Since positively autocorrelated ARMA(1,1) and
AR(1) processes are the only models where forecast error recovery is encountered, negatively

autocorrelated models will not be considered. It can be shown that the sustained level of the
original shift of size c for the OSA forecast errors is;
for t > r, as k 

,
E(e
t
(1)
) 
1
1
1
1
1











( )
( )



c
= E(e
t
*(1)
),
and for the TSA forecast errors;
for t > r, as k 

,
).(
)1(
)(
1)(
)2()2( *
1
1
1
1
e
Ec
e
E
tt














Tables 2 (a) and 2 (b) contain the sustained OSA and TSA expected forecast error values,
E(e
t
*(1)
) and E(e
t
*(2)
), for various combinations of

1
and

1
from ARMA(1,1) models, given a
c = 1


shift in the process mean level. Only parameter combinations resulting in forecast
recovery are included.
The OSA and TSA forecast errors are found to exhibit some very interesting
characteristics when the positively autocorrelated cases are studied. In all positively
autocorrelated processes, the forecast errors recover to a value smaller than the original shift of
size c at time t = r. But, the OSA and TSA forecast errors do not experience the same degree of
recovery. In many cases, the TSA forecast errors recover at a much slower rate than the OSA
errors, and some converge to a value over twice that of the sustained OSA value as can be
observed in Tables 1 (a) and (b). The most interesting characteristic can be realized in the case
where

1
and

1
are both negative (Model 8). While the OSA errors recover in this case, the
TSA forecast errors increase and converge to a value of E(e
t
*(2)
) which is greater than c.
Examples of this phenomenon can be seen in the italicized region of Table 2 (b) (lower left area).

Table 1 (a) One-Step-Ahead Forecast Error Expectation for Positively Autocorrelated
ARMA(1,1) and AR(1) Processes with a Shift of c = 1


at Time Period t = r.
Model
1
2
3
4
5
6
7
8
9
10



.950
.950
.950
.950
.475
.475
.475
-.475
0.950
0.475



.900
.450
-.450
-.900
.450
-.450
-.900
-.900
0.000
0.000



.072
.824
.971
.975
.025
.689
.737
.255
.950
.475
t
OSA Expected Forecast Errors, E(e
t
(1)
)
< r
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
r
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
r + 1
0.95
0.50
-0.40
-0.85
0.98
0.08
-0.38
0.58
0.05
0.53
r + 2
0.91
0.28
0.23
0.82
0.96
0.49
0.86
0.96
0.05
0.53
r + 3
0.86
0.17
-0.05
-0.68
0.96
0.30
-0.25
0.61
0.05
0.53
r + 4
0.83
0.13
0.07
0.67
0.96
0.39
0.75
0.92
0.05
0.53
r + 5
0.80
0.11
0.02
-0.55
0.96
0.35
-0.15
0.64
0.05
0.53
.
.
.
.
.
.
.
.
.
.
.
r + 44
0.50
0.09
0.03
0.04
0.95
0.36
0.28
0.78
0.05
0.53
r + 45
0.50
0.09
0.03
0.02
0.95
0.36
0.27
0.77
0.05
0.53
Table 1 (b) Two-Step-Ahead Forecast Error Expectation for Positively Autocorrelated
ARMA(1,1) and AR(1) Processes with a Shift of c = 1


at Time Period t = r.
Model
1
2
3
4
5
6
7
8
9
10



.950
.950
.950
.950
.475
.475
.475
-.475
0.950
0.475



.900
.450
-.450
-.900
.450
-.450
-.900
-.900
0.000
0.000



.072
.824
.971
.975
.025
.689
.737
.255
.950
.475
t
TSA Expected Forecast Errors, E(e
t
(2)
)
< r
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
r
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
r + 1
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
r + 2
0.95
0.53
-0.33
-0.76
0.99
0.56
0.35
1.20
0.10
0.77
r + 3
0.91
0.31
0.27
0.82
0.98
0.76
0.93
1.02
0.10
0.77
r + 4
0.87
0.22
0.00
-0.60
0.98
0.67
0.41
1.18
0.10
0.77
r + 5
0.84
0.17
0.12
0.68
0.98
0.71
0.88
1.04
0.10
0.77
.
.
.
.
.
.
.
.
.
.
.
r + 44
0.53
0.14
0.08
0.07
0.98
0.70
0.65
1.11
0.10
0.77
r + 45
0.53
0.14
0.08
0.08
0.98
0.70
0.66
1.11
0.10
0.77

Table 2 (a) Sustained OSA Expected Forecast Errors E(e
t
*(1)
), for Combinations of

1
and

1
from ARMA(1,1) Models, given a c = 1


Shift in the Process Mean.


\



-.95
-.85
-.75
-.65
-.55
-.45
-.35
-.25
-.15
-.05
.00
.05
.15
.25
.35
.45
.55
.65
.75
.85
.95
.03
.03
.03
.03
.03
.03
.04
.04
.04
.05
.05
.05
.06
.07
.08
.09
.11
.14
.20
.33
.85
.08
.08
.09
.09
.10
.10
.11
.12
.13
.14
.15
.16
.18
.20
.23
.27
.33
.43
.60
.75
.13
.14
.14
.15
.16
.17
.19
.20
.22
.24
.25
.26
.29
.33
.38
.45
.56
.71
.65
.18
.19
.20
.21
.23
.24
.26
.28
.30
.33
.35
.37
.41
.47
.54
.64
.78
.55
.23
.24
.26
.27
.29
.31
.33
.36
.39
.43
.45
.47
.53
.60
.69
.82
.45
.28
.30
.31
.33
.35
.38
.41
.44
.48
.52
.55
.58
.65
.73
.85
.35
.33
.35
.37
.39
.42
.45
.48
.52
.57
.62
.65
.68
.76
.87
.25
.38
.41
.43
.45
.48
.52
.56
.60
.65
.71
.75
.79
.88
.15
.44
.46
.49
.52
.55
.59
.63
.68
.74
.81
.85
.89
.05
.49
.51
.54
.58
.61
.66
.70
.76
.83
.90
.95
.00
.51
.54
.57
.61
.65
.69
.74
.80
.87
.95
-.05
.54
.57
.60
.64
.68
.72
.78
.84
.91
-.15
.59
.62
.66
.70
.74
.79
.85
.92
-.25
.64
.68
.71
.76
.81
.86
.93
-.35
.69
.73
.77
.82
.87
.93
-.45
.74
.78
.83
.88
.94
-.55
.79
.84
.89
.94
-.65
.85
.89
.94
-.75
.90
.95
-.85
.95
Table 2 (b) Sustained TSA Expected Forecast Errors, E(e
t
*(2)
), for Combinations of

1
and

1
from ARMA(1,1) Models, given a c = 1


Shift in the Process Mean.


\



-.95
-.85
-.75
-.65
-.55
-.45
-.35
-.25
-.15
-.05
.00
.05
.15
.25
.35
.45
.55
.65
.75
.85
.95
.07
.08
.08
.08
.08
.08
.09
.09
.09
.10
.10
.10
.11
.11
.12
.14
.16
.19
.24
.37
.85
.22
.22
.22
.23
.23
.24
.24
.25
.26
.27
.28
.28
.30
.32
.35
.38
.43
.51
.66
.75
.35
.35
.36
.36
.37
.38
.39
.40
.41
.43
.44
.45
.47
.50
.54
.59
.67
.79
.65
.47
.47
.48
.49
.50
.51
.52
.53
.55
.57
.58
.59
.62
.65
.70
.76
.86
.55
.58
.58
.59
.60
.61
.62
.63
.65
.67
.69
.70
.71
.74
.78
.83
.90
.45
.68
.68
.69
.70
.71
.72
.73
.75
.77
.79
.80
.81
.84
.88
.93
.35
.77
.77
.78
.79
.80
.81
.82
.83
.85
.87
.88
.89
.92
.95
.25
.85
.85
.86
.86
.87
.88
.89
.90
.91
.93
.94
.95
.97
.15
.92
.92
.92
.93
.93
.94
.94
.95
.96
.97
.98
.98
.05
.97
.98
.98
.98
.98
.98
.99
.99
.99
1.00
1.00
.00
*
*
*
*
*
*
*
*
*
*
-.05
1.02
1.02
1.02
1.02
1.02
1.01
1.01
1.01
1.00
-.15
1.06
1.06
1.05
1.05
1.04
1.03
1.02
1.01
-.25
1.09
1.08
1.07
1.06
1.05
1.03
1.02
-.35
1.11
1.09
1.08
1.06
1.05
1.02
-.45
1.12
1.10
1.08
1.05
1.03
-.55
1.11
1.09
1.06
1.03
-.65
1.10
1.07
1.04
-.75
1.08
1.04
-.85
1.04

MEWMA Control Chart
Regardless of the control chart method used, performance has been shown to be very
poor in many cases (Alwan and Roberts (1988), Wardell, Moskowitz, and Plante (1994), Lin and
Adams (1996), Lu and Reynolds (1999a), and Lu and Reynolds (1999b), Dyer, et al. (2001)). In
an attempt to widen this window of opportunity and improve control chart performance,
Michelson [1994] suggests that the two-step-ahead forecast errors also be monitored. This
author suggests schemes involving placing a Multivariate EWMA (MEWMA) control chart on
the OSA and two-step-ahead (TSA) forecast error vectors. The design of the MEWMA control
chart (Lowry, Woodall, Champ, and Rigdon [1992]) adapted to the forecast errors can be defined
as follows:
t t
t
Z RX
I R Z
 

1
( )
(8)
for t = 1,2…,
0
0
Z

, R = diag(1, 2), 0 < 
j
 1, j = 1, 2, and
t
t
t
X
e
e







(1)
( )2
where
t
(1)
t
and
e e
( )2
are the OSA and TSA forecast errors at time t. The MEWMA control chart gives an
out-of-control signal whenever
t t t
Z
T Z Z
h
2
1





, (9)
where
Z

is the covariance matrix of Z, and h (>0) is chosen by simulation to achieve a
specified in-control ARL.
Michelson asserts that the weighting constants 
1
and 
2
need not be equal, and the
choice of 
1
and 
2
should be that set which provides the best univariate EWMA chart
performance. Another scheme proposed by Michelson is a combination chart, which combines
the univariate EWMA on the OSA forecast errors, with the MEWMA, much in the same way the
CES chart combines the EWMA with the Shewhart chart. The author contends that the “better”

performance of the MEWMA and Combination charts is the “fact” that the TSA forecast error
contains information about a shift in the process over that in the OSA forecast error. The better
performance claim is made relative to the ARL performance of the univariate EWMA control
chart placed on the OSA forecast errors. Michelson evaluated these control charts when placed
on the forecast errors arising from selected AR(2), AR(5) and ARMA(1,1) models.
Michelson’s research findings provide that the MEWMA chart alone is only marginally
better than the univariate EWMA, while the Combined chart works best on processes where the
lag one and lag two correlations are high, the second partial correlation is high, and the
correlation between the two forecast errors is low. The author recommends using the chart in
this situation. Michelson also finds that the performance of both charts relative to the AR(2) and
AR(5) is not significantly different than that of the univariate EWMA, and only marginally better
in the ARMA(1,1) case. The CDF performance on the charts is shown to be poor, as is the case
with the univariate EWMA control chart.
MEWMA Statistic
Regardless of the values of 
1
and 
2
chosen, the MEWMA statistic T
t
2
can be viewed in
its more general quadratic form, which may make it easier to determine situations in which the
MEWMA might perform best. Consider the following for the ARMA(1,1) case: From Eq. (8)
define
 
t t t
Z Z Z


(1) ( )2
(10)
where
t t
(1) (2)
Z
and
Z
are the EWMA values at time t for the OSA and TSA forecast errors. Further
define the asymptotic covariance matrix of the EWMA’s of the forecast errors as










Z
A C
C B[
( )
]1
2
1
1


(11)

where
A B C




 
1
1
2
2
1 2
1 2 1 2
2 2




 
   
, ,
.
From Eq. (9) rewrite T
t
2
as
 
t
t t
T
Z
B
AB
C
Z
B
Z Z
C
Z
A
AB
C
t
t t
2
2
1
1
2
1
1
2
2
2
1
1
2
2
2 2
2
1
2
1






 



( )
( )
[
( )
]
( ) ( )
[
( )
]
(1)
(1)
(1) ( )
( )






(12)
When 
1
= 
2
, Eq. (12) reduces to
t
T
Z
C
Z Z
C
t t t
2
2
2
1
1
2 2
 


( ) ( )
( )
(1) (1) ( )


(13)
When 
1
= 
2
= 1, Eq. (12) reduces to
t
T
e
e e
t
t t
2
2
2
2
1
1
2
 


( )
( )
( )
(1)
(1) ( )


(14)
For the sake of convenience, T
t
2
calculated according to Eqs. (12) and (13) will be referred to as
ME, while T
t
2
calculated according to Eq. (14) will be referred to as TSQ.
The advantage of viewing the T
t
2
(MEWMA) statistic in the forms of Eqs. (12) and (13)
is that it becomes more obvious that initial and subsequent period detection of shifts may be
troublesome for some models and choices of 
1
and 
2
. Consider TSQ, the case when

1
= 
2
= 1. For any time period after the shift when the expectation of OSA and TSA forecast
errors are equal or similar, each element (
ee
tt
(2)(1)
and
) on the left most side of Eq. (13) tends to
offset the effect of the shift on the other element.
For example, at the first period following the shift, both forecast errors have the same
expected value of c. Although the impact of the shift is manifest in the first part of the equation,
the elements in the second part tend to cancel one another. Hence, the statistic experiences only
a fraction of the impact of the shift. At the second period following the shift, the OSA forecast

error is expected to recover while the TSA forecast error remains constant at expectation c.
While the second part of the equation may manifest the shift, the first part of the equation has
begun to recover, again causing the statistic to experience only a fraction of the impact of the
shift. If both forecast errors experience similar degrees and magnitudes of recovery at all time
periods after the shift, the MEWMA statistic will not be an efficient control charting tool. Initial
detection of the shift will be poor in any case.
If both forecast errors behave as described above, then choosing values of 
1
and 
2
that
are equal (ME) or approximately equal will result in a T
t
2
(MEWMA) statistic behaving similar
to the case of the TSQ statistic, that is, when 
1
= 
2
= 1, as is apparent in Eq. (6). Since the
values (
1
= 
2
) are typically similar in magnitude, this problem may exist in many models.
Hence the MEWMA control charts may become even less desirable in cases of similar forecast
recovery characteristics. Some ARMA(1,1) and AR(1) processes are known to produce forecast
errors that exhibit similar recovery rates and characteristics as described above.
It should be noted that in the case of an iid normal process wherein 
1
= 
2
, the OSA and
TSA forecast errors are identical, and the resulting covariance matrix is singular hence the
inverse does not exist. In this case neither the ME (Eq. (13)) nor TSQ (Eq. (14)) statistic can be
calculated.
The goal of this research is to find if the TSQ and ME control charts outperform their
univariate counterparts, in hopes of combining them in a control chart similar to the CES control
chart. Hence, only models where all forms of the MEWMA statistic can be calculated will be
considered. Neither the TSQ nor ME control charts alone are intended to be considered a
reasonable alternative to the individuals or EWMA control charts. All MEWMA control charts
were designed to provide an in-control ARL of 300, while the choices of 
1
and 
2
for the ME

control charts are a function of the forecast recovery of both the OSA and TSA forecast errors.
For each level of

1
, a TSQ control chart is provided analogous to the individuals control chart.
An ME control chart analogous to the EWMA control chart (
1
= 
2
 1 or 
1
 
2)
is also
provided. The next sections relate the design and performance of the MEWMA control charts
applied in the case of the ARMA(1,1) and AR(1) models.
Control Chart Design
Tables 3 (a) through (c) provide the simulation results of the control chart performance of
the individuals, EWMA, TSQ, and ME control charts applied to the 10 ARMA(1,1) and AR(1)
processes identified in Table 1. A description of the simulation design and the FOTRAN
program code, as well as tables relating the chart parameters and control limits are available from
the author upon request.
The control charts presented in all tables are designed to provide in-control ARLs of 300.
The EWMA and ME control charts are designed to detect a shift of the magnitude of the
sustained expected forecast error for each model. Autocorrelated data forecast errors recover to a
value less than c; hence, the charts are designed to catch the sustained level of the shift.
Chart performance results are based primarily on Average Run Length (ARL) and
cumulative Distribution Function (CDF) measures, but the Median Run Length (MRL) is also
provided for each chart. Standard Error of the Run Length (SRL) measures are provided to
summarize the variability of each chart’s run length distribution, as well as to give the reader an
idea of the accuracy of each ARL measure. The performance of monitoring schemes has
traditionally been evaluated by considering the ARL associated with competing techniques. As a
result of forecast recovery, the run length distribution is no longer geometric; hence the ARL
measure is not necessarily the best or only comparison measure. Superville and Adams (1994)

recommend the use of the CDF measure as a meaningful criterion for comparing the
performance of monitoring schemes.
Control Chart Performance
As stated previously, the goal of the research related to the MEWMA control charts was
to examine the performance of the TSQ and ME charts relative to their univariate counterparts
(Individuals and EWMA control charts) in hopes of designing a combined multivariate control
chart such as the univariate CES control chart. The reasoning is that if both the TSQ and ME
charts outperform the individuals and EWMA charts, respectively, then a combined TSQ-ME
control chart should outperform the CES chart. In this article, these control charts are not
considered alternative techniques to their univariate counterparts.
The TSQ control chart does show performance improvement over the individuals control
chart, but the improvement is only marginal in most cases. The best TSQ chart performance can
be seen in models exhibiting low levels of sustained OSA and TSA forecast errors, that is,
sustained levels of forecast recovery greater than 0.60c. The percent improvement in ARL
performance in these models ranges from approximately 20% to 24% for a shift of size c = 1


.
The MRL performance improvements are about the same as for the ARL, implying improvement
in CDF rates. The TSQ chart CDFs always lag behind those of the Individuals chart for the first
period following the shift, but usually surpass them in subsequent time periods.
The percent increase in performance in ARLs for small shifts in models exhibiting
moderate levels of sustained OSA and TSA forecast errors, that is, sustained levels of forecast
recovery between 0.40c and 0.60c, range from approximately 10% to 16%. For models with
sustained levels of forecast recovery less than 0.40c, the range of ARL improvement is 0.4% to
7%. The gains in MRL and CDF performance are comparable in these models also.

The ME control chart is not found to outperform the EWMA control chart in regard to
any of the performance criteria. While in many models the two charts’ performances are nearly
identical, the ME chart often performs more poorly than the EWMA chart. It would seem that if
the TSQ chart outperforms the Individuals chart, then the ME chart should outperform the
EWMA in at least some models. Since this is not the case for the ME chart, it is assumed that
the choice of weighting constants (
1
and 
2
) are incorrect. Recall that Michelson [1994] seldom
found any significant improvement when utilizing the ME chart. No further attempt is made to
explain or improve the poor performance of the ME chart in this article.
Although the control limit for the TSQ chart is constant for a given in-control ARL,
regardless of the underlying process, the same does not hold true for the ME chart. Different
combinations of weighting constants require different control limits. In general, larger weighting
constants or differences in the size of weighting constants lead to larger control limits. In any
case, tedious simulation is required to find the appropriate control limits for all TSQ and ME
control charts.
Given the complication associated with designing the TSQ and ME charts relative to any
performance gain, this author does not recommend using these charts. It does not appear that the
complication involved in designing and understanding a combined TSQ-ME chart is worth any
marginal benefit that might be gained.
Conclusion
Two MEWMA control charts designed by Michelson [1994] were introduced, the TSQ
and ME control charts. To understand the design and logic of the MEWMA charts, a Box-
Jenkins based OSA and TSA forecast was developed, and a mathematical description of the
impact of forecast recovery on the TSA forecast errors arising from the ARMA(1,1) and AR(1)

processes was provided. Also, the TSQ and ME control charting statistics were defined and
rewritten in quadratic form to better understand how the statistics function. The stated goal of
this research was to find whether the TSQ and ME control charts outperformed their univariate
counterparts, in hopes of combining them in a control chart similar to the CES control chart.
A performance evaluation of various control charts applied to the forecast errors arising
from simulated ARMA(1,1) and AR(1) processes was conducted. Performance criteria was
based upon the ARLs, MRLs, and CDFs of each control chart for step shifts in the process mean
of size c = 1


, 2


, and 3


. Tables showed the performance results for a variety of the above
processes. It was found that the performance of the MEWMA charts relative to their univariate
counterparts was lacking, and it is recommended that these control charts not be utilized in
practice. When using the method suggested by Michelson, the TSA ahead forecast errors are not
helpful in detecting shifts over the OSA forecast errors alone.

Table 3 (a). ARLs , MRLs, and CDFs for the ARMA(1,1) Process with Step Shifts.
ARMA
Control
Cumulative Percentage of Signals Following Shift
Model
Shift, c
Chart
ARL
MRL
SRL
1
st
2
nd
3
rd
4
th
5
th
6
th
7
th
1
1
IND
115
70
128
2.52
4.67
6.81
8.61
10.06
11.70
13.10

1
= 0.95
EWMA
15
12
12
2.03
4.04
6.95
10.75
15.48
21.03
26.94

1
= 0.90
TSQ
97
58
113
2.19
5.60
8.15
10.48
12.38
14.23
15.98

1
= 0.072
ME
17
10
18
1.57
4.33
8.41
14.18
20.67
27.83
34.29
2
IND
16
5
27
17.43
29.98
39.19
45.92
51.24
55.71
59.32
EWMA
5
5
2
3.74
12.02
26.43
44.39
61.61
75.18
84.96
TSQ
9
4
16
13.81
37.49
49.56
58.32
64.93
69.81
73.51
ME
4
4
2
3.82
17.41
40.84
63.12
79.10
88.88
94.07
3
IND
2
1
3
51.74
73.98
84.65
90.34
93.62
95.44
96.74
EWMA
3
3
1
7.26
29.61
61.16
84.01
95.02
98.74
99.68
TSQ
2
2
1
45.13
84.08
92.68
96.61
98.23
98.96
99.40
ME
3
3
1
8.59
46.36
82.44
96.27
99.39
99.88
100
2
1
IND
279
191
284
2.52
3.23
3.62
3.98
4.28
4.62
5.00

1
= 0.95
EWMA
136
96
141
7.15
8.32
9.35
10.05
10.74
11.31
11.91

1
= 0.45
TSQ
276
188
293
2.19
3.69
4.20
4.64
4.95
5.21
5.61

1
= 0.824
ME
171
113
180
2.92
4.14
5.14
5.82
6.49
7.14
7.75
2
IND
209
124
255
17.43
19.56
20.28
20.74
21.03
21.38
21.77
EWMA
63
49
61
8.03
10.19
11.86
13.18
14.37
15.33
16.20
TSQ
193
113
245
13.81
21.48
22.39
22.96
23.29
23.56
23.92
ME
73
46
82
3.51
6.87
9.37
11.49
13.07
14.48
15.81
3
IND
98
1
185
51.74
55.39
56.17
56.56
56.81
57.07
57.33
EWMA
35
28
34
9.43
13.56
16.34
18.57
20.34
22.16
23.80
TSQ
80
2
161
45.13
60.73
61.67
62.14
62.41
62.60
62.77
ME
33
19
39
4.60
11.62
17.47
21.75
25.15
27.99
30.60
3
1
IND
290
199
295
2.52
3.23
3.60
3.93
4.23
4.53
4.88

1
= 0.95
EWMA
217
145
239
7.01
7.92
8.63
9.22
9.83
10.39
10.86

1
=-0.45
TSQ
288
196
307
2.19
3.49
3.94
4.30
4.58
4.86
5.23

1
= 0.971
ME
216
144
238
7.40
8.20
8.80
9.45
9.95
10.51
10.96
2
IND
237
143
286
17.43
18.81
19.39
19.70
19.97
20.22
20.54
EWMA
164
113
174
8.02
8.94
9.67
10.11
10.69
11.16
11.70
TSQ
236
136
296
13.81
20.12
20.70
21.11
21.34
21.60
21.90
ME
165
113
173
7.80
8.85
9.40
9.96
10.51
11.11
11.58
3
IND
130
1
236
51.74
53.66
54.22
54.38
54.55
54.70
54.90
EWMA
121
87
124
9.45
10.33
11.09
11.54
12.11
12.67
13.22
TSQ
119
2
234
45.13
56.87
57.56
57.81
57.95
58.08
58.24
ME
122
87
126
8.37
9.91
10.51
11.20
11.70
12.31
12.80
4
1
IND
270
177
293
2.52
4.34
5.97
7.14
8.10
8.93
9.75

1
= 0.95
EWMA
227
150
252
7.03
8.16
8.96
9.48
10.16
10.71
11.22

1
=-0.90
TSQ
263
167
300
2.19
5.10
7.28
8.78
9.97
11.01
11.92

1
= 0.975
ME
225
149
246
7.41
8.14
8.78
9.34
9.89
10.44
10.93
2
IND
156
15
259
17.43
26.50
33.68
37.66
41.13
43.11
44.92
EWMA
189
126
206
8.03
9.59
10.48
10.95
11.67
12.07
12.67
TSQ
122
5
247
13.81
32.85
42.28
48.06
52.20
54.88
56.93
ME
191
128
207
7.82
8.56
9.23
9.81
10.45
10.98
11.33
3
IND
29
1
119
51.74
68.79
78.14
82.37
85.53
87.09
88.43
EWMA
148
102
158
9.44
11.33
12.51
12.91
13.80
14.10
14.79
TSQ
11
2
71
45.13
79.37
87.73
91.77
93.85
94.97
95.64
ME
152
105
159
8.38
9.11
9.77
10.32
10.84
11.47
11.92

Table 3 (b). ARLs, MRLs, and CDFs for the ARMA(1,1) Process with Step Shifts.
ARMA
Control
Cumulative Percentage of Signals Following Shift
Model
Shift, c
Chart
ARL
MRL
SRL
1
st
2
nd
3
rd
4
th
5
th
6
th
7
th
5
IND
42
29
42
2.52
4.80
7.25
9.44
11.43
13.63
15.80

1
= 0.475
EWMA
9
8
5
1.51
4.28
8.73
15.38
23.33
32.56
41.75

1
= 0.45
TSQ
32
22
31
2.19
5.74
8.73
11.65
14.38
17.26
19.94

1
= 0.025
ME
10
8
7
1.61
5.39
11.55
19.29
27.67
36.19
43.79
IND
6
4
6
17.43
30.89
41.74
50.64
58.27
64.76
70.25
EWMA
4
4
2
4.24
18.82
44.32
68.66
85.34
94.00
97.80
TSQ
4
3
4
13.81
38.67
52.76
63.89
72.32
78.94
83.75
ME
3
3
2
5.49
28.38
58.56
79.50
91.00
96.06
98.33
IND
2
1
1
51.74
75.35
87.20
93.42
96.51
98.12
99.02
EWMA
3
2
1
11.04
51.05
86.69
97.91
99.79
99.99
100
TSQ
2
2
1
45.13
85.27
94.24
97.92
99.25
99.67
99.86
ME
2
2
1
15.49
70.18
95.46
99.56
99.95
100
100
6
IND
177
122
180
2.52
2.94
3.68
4.13
4.59
5.10
5.67

1
= 0.475
EWMA
35
30
26
2.59
3.38
4.34
5.13
6.09
7.28
8.46

1
= -0.45
TSQ
164
114
166
2.19
3.11
3.73
4.38
4.93
5.45
6.05

1
= 0.689
ME
38
30
29
1.84
2.77
3.46
4.39
5.35
6.55
7.77
IND
61
37
71
17.43
17.81
20.08
20.99
22.11
23.21
24.30
EWMA
14
13
8
4.11
5.36
7.89
10.34
13.87
17.61
22.03
TSQ
51
32
59
13.81
17.08
18.68
20.11
21.38
22.72
24.01
ME
14
12
8
2.35
4.82
6.81
10.18
14.00
18.53
23.85
IND
16
1
26
51.74
52.02
55.60
56.52
58.30
59.56
60.94
EWMA
8
8
5
7.03
8.72
14.63
20.22
28.43
37.38
46.64
TSQ
13
3
20
45.13
49.64
52.19
54.67
56.67
58.78
60.73
ME
8
7
4
3.39
8.82
13.53
22.17
31.07
41.75
51.9
7
IND
205
138
218
2.52
3.18
5.08
5.46
6.70
7.03
8.15

1
= 0.475
EWMA
48
41
37
4.46
5.19
6.27
6.72
7.80
8.31
9.32

1
= -0.90
TSQ
195
130
207
2.19
3.47
5.19
6.03
7.10
7.71
8.66

1
= 0.737
ME
51
41
41
2.73
3.42
4.10
4.89
5.55
6.34
7.05
IND
70
25
103
17.43
18.66
28.03
28.60
34.03
34.32
37.88
EWMA
21
19
13
5.70
6.36
8.84
9.30
12.02
12.65
15.73
TSQ
62
22
89
13.81
19.68
27.77
30.14
34.72
36.13
38.73
ME
20
18
13
3.03
4.60
5.84
7.56
9.11
11.76
13.82
IND
11
1
29
51.74
70.17
70.56
78.14
78.31
82.07
82.15
EWMA
13
12
8
8.32
13.10
13.45
19.38
20.24
27.95
29.23
TSQ
9
1
23
55.93
70.33
72.99
79.03
80.10
83.02
83.56
ME
12
10
7
6.57
8.69
12.88
16.20
22.25
26.82
34.53
8
IND
64
43
63
2.52
3.38
5.84
6.85
8.78
9.92
11.85

1
=-0.475
EWMA
13
11
8
1.61
3.06
5.99
9.03
14.30
19.20
26.22

1
= -0.90
TSQ
51
35
50
2.19
3.86
6.05
7.63
9.53
11.18
13.15

1
= 0.255
ME
13
11
9
1.20
2.87
5.83
10.02
15.46
21.43
28.17
IND
11
7
11
17.43
20.51
32.86
35.77
44.62
47.53
54.28
EWMA
5
5
2
3.87
9.35
26.65
40.40
62.21
73.96
86.39
TSQ
8
5
7
13.81
23.31
35.57
42.23
51.06
56.49
62.98
ME
5
5
2
2.37
10.27
28.08
48.69
68.03
80.96
89.77
IND
3
1
3
51.74
57.20
77.32
80.52
89.16
90.80
94.58
EWMA
3
3
1
9.18
24.42
63.99
81.61
95.87
98.72
99.82
TSQ
2
2
2
45.13
64.14
81.54
87.68
93.70
95.87
97.73
ME
3
3
1
4.76
28.31
67.22
89.77
97.73
99.47
99.93

Table 3 (c). ARLs, MRLs, and CDFs for the ARMA(1,1) Process with Step Shifts.
ARMA
Control
Cumulative Percentage of Signals Following Shift
Model
Shift, c
Chart
ARL
MRL
SRL
1
st
2
nd
3
rd
4
th
5
th
6
th
7
th
9
1
IND
290
199
295
2.52
2.93
3.24
3.56
3.84
4.14
4.51

1
= 0.95
EWMA
194
131
209
7.05
7.95
8.72
9.26
9.89
10.44
10.97

1
= 0.00
TSQ
288
197
306
2.19
3.12
3.46
3.77
4.05
4.33
4.72

1
= 0.95
ME
214
145
223
4.57
5.51
6.12
6.61
7.12
7.65
8.12
2
IND
235
143
277
17.43
17.81
18.10
18.39
18.64
18.92
19.27
EWMA
125
90
129
8.06
9.03
9.89
10.49
11.16
11.78
12.34
TSQ
237
144
286
13.81
17.01
17.29
17.60
17.88
18.14
18.49
ME
139
95
146
4.99
6.95
7.71
8.34
9.01
9.59
10.19
3
IND
130
1
232
51.74
52.00
52.17
52.35
52.53
52.72
52.96
EWMA
81
62
80
9.42
10.69
11.68
12.52
13.36
14.12
14.84
TSQ
128
6
225
45.13
49.45
49.56
49.74
49.93
50.09
50.31
ME
87
60
90
5.53
9.38
10.35
11.30
12.14
13.05
13.83
10
1
IND
119
81
120
2.52
3.27
4.08
4.91
5.60
6.41
7.21

1
= 0.475
EWMA
21
19
14
2.24
3.51
4.82
6.24
8.10
10.16
12.67

1
= 0.00
TSQ
106
73
107
2.19
3.71
4.51
5.44
6.28
7.13
8.07

1
= 0.475
ME
24
19
19
1.31
2.75
4.37
6.36
8.86
11.67
14.85
2
IND
29
18
33
17.43
19.80
22.27
24.47
26.67
28.87
31.00
EWMA
8
8
5
3.89
7.41
12.14
18.39
26.51
35.53
44.88
TSQ
22
13
24
13.81
22.03
24.68
27.76
30.57
33.58
36.33
ME
7
7
4
2.33
8.48
15.97
26.23
36.83
48.21
57.88
3
IND
6
1
9
51.74
55.91
59.79
63.25
66.35
69.36
72.07
EWMA
5
5
3
7.06
15.30
27.64
42.76
58.44
71.68
82.29
TSQ
4
2
6
45.13
61.94
65.81
70.31
74.32
77.77
80.65
ME
4
4
2
4.22
21.43
41.69
61.49
77.00
87.65
93.65

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Key Words: Autocorrelation, MEWMA, Multivariate SPC, Quality, Quality Control, Process
Control, SPC, Statistics, Statistical Process Control.
.