Download Cumulative Sum (CUSUM) Control Schemes

CUMULATIVE SUM (CUSUM) CONTROL SCHEMES
James M. Lucas
E. I. du Pont de Nemours and Co., Inc.
Wilmington, DE
the production of nonconforming product.
Because CUSUM procedures give an early
ABSTRACT
Cumulative
Sum
(CUSUM)
quality
indication of process changes,
control
sistent
schemes are becoming widely used in industry
because they are powerful, versatile, and easy
to use. They cumul ate recent process data to
quickly
They
detect
also
tool.
out-of-control
serve
as
a
19898
There are now more
than
they are con-
philosophy
encourages doing it right the first time.
use of CUSUM procedures
situations.
powerful
a management
with
;s
that
The
also consistent
with a management by exception philosophy as
the CUSUM will point out areas needing
diagnostic
attention.
10,000 CUSUM
control schemes in use daily ;n Du Pont. This
talk
describes
design
and
implementation
procedures for CUSUM control schemes with
emphasis on properties that are recorded as
counts.
The talk will describe recent developments which make CUSUM procedures more
useful and more powerful.
Cumulative sum control schemes are current ly used more for the control of vari ab 1es
than for the control of counts (attributes).
To help remedy this, we will give counts more
emphasis here. We will show that the design
and implementation procedure for Counted Data
CUSUMs
(Lucas
1985)
is
very
similar
to
the
The recent developments described are:
procedure for CUSUMs for variables (Lucas
1976). We will use a Poisson distribution in
o
Fast Initial Response (FIR) CUSUM which
our examples.
gives extra sensitivity to out-ofcontrol situations at start up or after
a (possibly ineffective) control action.
is administratively convenient to record the
number of counts in a given sampling interval. When the number of counts per interval
follows a Poisson distribution, the time
between counts follows an exponential distribution. A time-between-events CUSUM which we
discuss elsewhere (Lucas 1985) should be used
if it ;s convenient to update the CUSUM with
each new count, and it is possible to record
the time since the last count.
o
A
Combined
Shewhart-CUSUM
which
com-
bines the key features of CUSUM schemes
and Shewhart Schemes by adding Shewhart
Control Limits to a CUSUM scheme.
o
ROBUST CUSUM schemes whi ch are not
unduly influenced by a few outliers or
A Poisson CUSUM is used when it
fl iers occurring in the stream of data.
We discuss the use of CUSUMs for the
detection of increases and/or decreases in
count rate; we discuss both one and two-sided
The philosophy of continual improvement of
a process is very compatible with CUSUM procedures. As CUSUM procedures gi ve much more
responsive control, a CUSUM signal does not
mean that the process is producing bad product. Rather it means that action should be
taken so that the process does not produce bad
product.
CUSUMs.
I NTRODUCTI ON
Cumulative
sum
(CUSUM)
quality
While
the most widely used applica-
tion of counted data CUSUMs will probably be
the control of nonconformances, they can also
be used to monitor changes in count level in
other situations such as accident rate or the
rate of occurrence of congenital malformations. We use the terminology counts, rather
than nonconformances, as counts has a broader
applicability.
control
schemes for variables are widely used in
industry. CUSUM procedures will usually give
tighter process control than classical quality
control schemes, such as Shewhart schemes.
With the tighter control that is available
with CUSUM schemes, there will be more emphasis placed on keeping the process on-aim
rather than allowing it to drift in limits.
Because of the tighter control, an out-ofcontrol (Signal will seldom indicate that the
process is. produci ng nonconforming product;
rather an out-of-control Signal will indicate
that control action should be taken to prevent
The recognition that a CUSUM control
scheme is a sequence of Wald Sequential
Probability
Ratio
Tests
(SPRT)
allows
the
optimal ity properties of CUSUM procedures to
be developed.
Johnson and Leone
(1962)
9ave
an early discussion of CUSUM procedures using
the relationship between SPRT1s and CUSUM.
Lorden (1971) showed the asymptotic optimality
of CUSUM procedures for detect i ng a change in
distribution.
Kenett and Pollak (1983) showed
the superiority of a CUSUM scheme for detecting a rare event over a non-CUSUM scheme
proposed by Chen (1978).
916
Yi
Recent enhancements to CUSUM quality
control schemes have included the Fast Initial
Response (FIR) feature (Lucas and Crosier
1982A). The FIR feature recognizes that when
will
the initial SPRT to test a different null
hypothesis than the following SPRTs.
In
A Robust CUSUM (Lucas and Crosier, 19828)
Robust
CUSUM
counts
in
the
ith
use
a
positive
starting
value.
We
for the initial SPRT.
has been recommended when isolated outliers or
extreme values occur for reasons other than a
A
of
the process is off-aim at start-up.
For a
process running at the desired level, the head
start will soon !Izera!!, so it has little
effect. It is instructive to view the choice
of So in terms of the II all and lIa n errors of
the equivalent SPRT.
The a-error is the
probabi 1ity of declaring the process off-aim
when it is not.
The a-error is the probability of declaring the process on-aim when it
is not. When the CUSUM is cons i dered to be a
sequence of Wald SPRTs, one finds that the
SPRTs have small a and large a errors. Using
a head start value approximately equal to h/2
is equivalent to equating the a and a errors
between-event CUSUMs see Lucas (1985).
shift.
number
generally recommend a head start (SO) value
approximately equal to h/2. With such a head
start, the CUSUM will more quickly Signal if
process control work, it is more likely that
the process is not at the desired aim value at
start-up than after the process has been
The FIR
running smoothly for some time.
feature gives a simple procedure for more
quickly detecting an out-of-control situation
at start-up. If the process is initially in
control, the Fast Initial Response feature has
little effect while if the process starts out
in an out-of-control condition, an out-ofcontrol signal is given much more quickly. In
this paper, we demonstrate the benefits of the
FIR feature for a Poisson CUSUM.
For a
discussion of the FIR feature for a time-
process
the
A standard CUSUM will have starting value
So = 0, while a Fast Initial Response CUSUM
a CUSUM is started, it may be appropriate for
true
is
interval.
A Robust CUSUM (Lucas and Crosier, 19828)
is obtained by using the Utwo-in-a-row rule".
To robustify the CUSUM, an outl ier 1 imit is
specified. A single observation outside this
limit does not enter the CUSUM. However, two
outliers in a row are an out-of-control Signal.
will
quickly detect true changes in level that
occur in the process; yet it will be insensit i ve to the occurrence of an occas i on a 1
outlier or flyer. In this paper, we discuss a
Robust Poisson CUSUM.
CUSUM
evaluated
The properties of a combined ShewhartCUSUM control scheme are described by Lucas
(I982).
For variables, the combined scheme
has proven of most value for measurement
control.
In this application a Shewhart
signal can indicate a bad sample while a CUSUM
Signal often indicates a calibration problem.
length (ARL).
bution of run lengths (Brook and Evans 1972).
For a standard CUSUM, the ARL distribution is
nearly geometric except that there ;s a lower
probability of extremely short run lengths.
When the FIR feature is used, the distribution
is nearly geometric except that there is an
increased probability of short run lengths due
A cumulative sum control scheme cumulates
the difference between an observed value Vi
and a reference value k. If this cumulation
equals or exceeds the decision interval value
h, an out-of-control signal is given.
The
CUSUM statistics are:
max (0, Yi - k + SH(i-l))
SLi
max (0, k - Yi + SL(i-l))
The ARL is the average number
of samples taken before an out-of-control
signal is obtained. The ARL should be large
when the process is at its aim level and short
when the process shifts to an undesirable
level.
In-control
ARLs
are
often well
approximated by a geometric distribution;
hence, the ARL also characterizes the distri-
IMPLEMENTATION EXAMPLE
SHi
contro 1
schemes
are
usually
by calculating their average run
to the head start (Lucas and Crosier 1982A).
Table 1 illustrates the implementation of
a Poisson CUSUM with parameter values k=5 and
h=lO.
A CUSUM with and without a head start
value SO=5 is shown.
where max (a,b) is the maximum of a and b.
The first formula is used to detect an
increase in level, while the second formula is
used to detect a decrease in level.
A two
sided CUSUM uses both formulas Simultaneously.
a
would
work
We wi11 find that such
well
if
the
acceptable
count 1eve 1 was about 4 and it was des ired to
quickly detect if the count level increased to
7. I n such a case, the i n-contro 1 average run
1engt would be approximately 397 or 422 and
the out-of-control ARL would be 3.35 or 5.59
for the FIR CUSUM and the standard CUSUM,
For the control of variables, the Vi in
the above formulas is a standardized variable:
Vi =
CUSUM
respectively. In Table 1, the first column is
the observation number, i=1. •• 15. The second
column gives the observed number of counts,
Vi. The th i rd co 1umn records an i ntermediate calculation step.
It calculates Vi-k.
The fourth column calculates Si for a
Xi - AIM
S
where Xi is the observed sample average and
S is the standard deviation of X. For Counts
standard CUSUM (with SO=O) , and the fifth
column calculates Si for a FIR CUSUM with
917
SO=5.
In Table 1,
the first
AVERAGE RUN LENGTHS FOR
THE MOST WIDELY USED CUSUM PROCEDURES
ten observa-
tions are from a process at the desired mean
level while the last five are from a process
with a higher mean level. The process starts
out i nit i ally in contro 1, so the FI R feature
has little effect; the head start soon zeroes
out. The benefit of the FIR feature can be
Deviation From Aim
(Multiple of S)
seen by implementing the CUSUM using only the
last five observations.
This represents a
process that is out of control when the CUSUM
is started.
In this case, the CUSUM with the
FIR feature would give an out-af-control
signal at the first observation (Observation
11), while the standard CUSUM would not signal
until the fourth observation (Observation 14).
POISSON CUSUM
EXAMPLE
h=lO k=5
Y -k
S
No FIR
o
1
2
3
7
3
4
5
6
2
0
2
8
7
4
8
9
10
13
0
2
3
10
8
4
14
15
1l
1l
12
9
-2
2
-3
-5
-3
3
-1
-5
-3
-2
5
3
-1
4
6
0
0
2
0
0
0
3
2
0
0
0
FIR
5
3
5
2
0+
0
3
2
0
0
0
5
5
8
7
1l*
17*
8
7
1l*
17*
So
4
4
5
5
.5
.5
.5
.5
0.0
2.0
0.0
2.5
0
.5
168. 26.6
150 20.1
465. 38.0
432 28.7
1.
2.
4
8.4
5.3
10.4
6.4
3.3
2.0
4.0
2.4
1.7
1.0
2.0
1.2
should be selected to be close to:
This reference value is the same as the
reference value for an SPRT testing the null
hypotheSiS that the mean is ua and alternate
hypothes i s that the mean is ud. When kp 21, the kp value will usually be rounded to
the nearest integer. Then, the CUSUM computations require only integer arithmetic.
After k is selected, the decision interval
value (h)
*Out of Control Signal
+
k
We will give a more detailed discussion of
the design procedure for a Poisson CUSUM for
detecting an increase in count rate.
For a
Poisson CUSUM, the k value will be between the
accept ab 1e proces s mean (jJ a) and the mean
1eve 1 of counts ("d) whi ch the CUSUM scheme
is to detect quickly. While the desired value
(or goal value) for J.la is often zero, a ).Ia
value of zero requires that the CUSUM be
designed with h=l and k=O. Any occurrence of
a count gives a Signal; when count levels are
low and a decrease in count level indicates an
improved system, every count must followed
up. The circumstances surrounding every count
must be examined to find and remove assignable
causes.
In practice, ).I a is often chosen
near to the current mean 1eve 1; th i s represents
current
system
performance.
The
reference value for the Poisson CUSUM (k p )
TABLE 1
Y
h
is chosen using a table
look-up
procedure.
The value of h Should give an
appropriately large ARL when the counts are at
the
after the FIR CUSUM zeroes, the standard
acceptable
level.
It
should
also
be
chosen to give an appropriately small ARL
value when the process is running at the count
CUSUM and the FIR CUSUM are equivalent
level Which should be detected quickly.
The CUSUM parameter k is determined by the
a Poisson CUSUM
of counts is 4
7) is
("a = 4) and a count rate of 7 ("d
to be detected quickly. The Poisson CUSUM is
designed with a k value close to:
acceptable mean level (]Ja) and by the unacceptab 1e mean ("d) 1eve 1 whi ch the CUSUM
scheme is to detect quickly.
For variables
the k value is chosen half way between the
acceptable mean level and the unacceptable
mean level.
when
The CUSUM parameter h is the
("d-"a)/( In("d)-ln("a))
chosen by a table look up procedure to give an
acceptably long in-control ARL.
In practice
the parameter values h = 4 or 5 and k = .5 are
often used.
Consider the design of
the acceptable number
= (7-4)/(ln 7-ln 4)= 5.36
The FIR feature with S(O) = h/2
For ease in implementation using integer
arithmetic, a kp value of 5 is recommended.
usually should be used. With these parameters
you get the following ARL values for a two
With this kp
(Lucas 1985):
sided CUSUM procedure (Lucas and Crosier 1982):
918
value we find
for
the
ARL
THE FAST INITIAL RESPONSE FEATURE
AVERAGE RUN LENGTH
FOR THE POISSON CUSUM EXAMPLE
At start up, or following a possibly
ineffective control action, control of the
process is less certain. Extra sensitivity is
often des i red at these times.
The Fast
Initial Response feature for CUSUM control
schemes was developed to give this extra
sensitivity.
For a standard CUSUM scheme
SO, the initial CU5UM value, is set equal to
" (multiple of k p )
kp
hp
7
7
So
5
5
5
5
5
5
10
10
15
15
0
4
0
5
0
8
4(.8)
7(1.4)
108
95
422
397
3740
3630
4.09
2.37
5.59
3.35
8.09
4.36
zero.
The
FIR
CUSUM
uses
an
initial
CUSUM
value, SO, that is greater than zero.
Our
recommended starting value is h/2. With this
starting value,
a FIR CUSUM detects an
initially out-at-control situation about 40
percent faster than a standard CUSUM.
An appropriate Poisson CUSUM could be the
scheme with hp : 10, kp:5, SO=5.
TWO-SIDED CUSUMs
For the control
CUSUMs
The
of variables,
two-sided
are usually used even when
shifts
in
FIR features.
only one direction indicate a deterioration in
quality.
Different control actions may be
taken depending on which side of aim the CUSUM
scheme
is
usually
out-of-control
signal
indicate
need
the
on
for
recommended.
the
low
control
side
when
the
CUSUM
The CUSUM scheme with
the FIR
Shewhart charts provide rapid detection of
large deviations, while CUSUM schemes provide
rapid detection of small to moderate deviations.
The Shewhart-CUSUM scheme (Lucas,
1982) incorporates the good features of both.
The prime motive for Shewhart-CUSUM is to
obtain faster response to 1arge shifts with
only small changes in performance otherwise.
With the Shewhart-CUSUM scheme. Shewhart
Control Limits (SCL) are added to the CUSUM
scheme. An out-of-control signal is given if
the most recent sample ;s outside of the SCL
or if a regular CUSUM signal is given. Since
the control action is modified only for the
current observation, the combined scheme is
easy to implement.
The performance of the
combi ned Shewhart-CUSUM approach is comparab le
to that of more in olved modifications of
however.
for
seen
COMBINED SHEWHART-CUSUM CONTROL PROCEDURE
it is usually easier to obtain the properties
of a two-sided scheme by combining results for
two one-sided schemes.
When the parameter
values are such that one side gives an out-ofonly
be
to
For attributes, also we generally recommend two-sided control schemes where, for
example, a significant decrease in count rate
might trigger an investigation which could
lead to performance improvements. Properties
of a two-s i ded scheme may be evaluated
signal
can
An
increase the strength, while an out-af-control
control
CUSUM
is used both a 1arger i n-contro 1 ARL and a
quicker detection of an initial out-of-control
situation can be achieved.
would
action
and Cros i er. 1982A);
the FIR
The ARL tab les show that when the FIR feature
signal on the high side would indicate a need
to go into a detective mode to find why the
high strength was achieved. The investigation
could indicate ways to improve the product.
di rect ly (Lucas
of
feature usually should have a larger h value.
signals. For example, it may be important to
contro 1 the strength of a gi yen product
property. Even if there is no prob 1em when a
higher strength is achieved, a two-sided
control
value
from the ARL tables which we have given.
Compare ARLls for CUSUMls with and without the
the
other side has zeroed, the ARL for a two-sided
scheme can be obtained directly from the ARL
of two one-sided schemes. (Lucas and Crosier
CUSUMs such as Parabolic CUSUM (Lucas (1973)).
1982A. Lucas 1985) as:
-~
Des i gn
ARL (SOH. SOL)
easy.
a
a
Shewhart-CUSUM
CUSUM
is
scheme
designed
the
is
ARL
tables are examined. The Shewhart Limits are
included if they make a significant improvement in the ARL curve.
The combined scheme
should use the same k value and the same or
slightly larger h value to compensate for
decreases in in-control ARL.
We generally
recommend SCL of 3.5 or 4.0 rather than 3.0 as
the larger limits have less effect on the incontrol ARL. The following gives ARL values
LH(SOH) LL(O) + LH(O) LL(SOL) - LH(O) LL(O)
LH(O) + LdO)
where the subscript Hand L indicate parameter
values for the high and low side respectively,
and L *( SO*)
is the ARL for a one-s i ded
CUSUM scheme with head start SO* and L or H
may be substituted for
of
After
for h = 4 or 5 k = .5 CUSUM schemes.
ditional tables are given by Lucas (1982).
*
919
Ad-
AVERAGE RUN LENGTH
FOR A COMBINED SHEWHART-CUSUM
combined with a cluster distribution having
probability c.
If the cluster was sampled, a
large number of cou.nts (sufficient to trigger
a standard
Deviation From Aim
(Multiple of S1
CUSUM)
would
h
k
SCL
0
.5
1
2
4
4.
4.0
5.0
5.0
5.0
.5
.5
.5
.5
•5
3.0
3.5
3.0
3.5
4.0
125.
159.
223 •
391.
459 •
25.
26.
34.
37.
38.
8.0
8.3
9.8
10.2
10.4
3.0
3.2
3.5
3.8
4.0
1.2
1.3
1.2
1.3
1.6
We
to be 0, .001, and .01.
Continuing our design
example with h=10, k=5, 50=0,5 we find:
AVERAGE RUN LENGTHS FOR A ROBUST CUSUM
We often select a Shewhart-CUSUM scheme to
process control
observed.
row rule (with and without the FIR feature).
We considered the probability (c) of a cluster
Average Run Lengths
with 50=5(50=0)
contro 1 measurement processes (test 1aboratory
control)~ whereas we usually employ a standard
CUSUM to control production processes.
In
both cases we normally include the FIR
feature.
When the Shewhart-CUSUM is chosen
for measurement
be
compared the performance of a standard Poisson
CUSUM and a Poisson CUSUM using the two-in-a-
and a mean of:
Out 1 i er
Standard
a Shewhart
Cusum
signal often indicates a bad test sample while
a CUSUM signal indicates a calibration problem
or
instrument
offset.
If a measurement
process is prone to outliers, then a standard
Robust
Cusum
CUSUM or a Robust CUSUM scheme may be better.
Probabil ity
4
7
.000
.001
.010
397 (422)
281 (298)
77 (82)
3.35 (5.59)
3.34 (5.58)
3.28 (5.43)
.000
.001
.010
401 (425)
401 (425)
388 (412)
3.40 (5.72)
3.41 (5.73)
3.44 (5.78)
ROBUST POISSON CUSUM
More extens; ve compari son of the perforA Robust CUSUM
for
vari ab 1es
has
mance of a Robust CUSUM and a standard CU5UM
proven
valuable when isolated outliers or flyers
occur fairly often for reasons other than a
(Lucas and Crosier,
true process shift
1982B).
A
Robustification
procedure
for a representat; ve set of parameter values
are contained in (Lucas 1985).
for
variables is the IItwo-in-a-row rule ll in which
a single suspected outlier does not enter a
CUSUM, whi le two suspected outl iers in a row
are an out-of-control signal.
In evaluating
the properties of a Robust CUSUM for variables, we used the widely used contaminated
normal distribution (Andrews, et a1 1972; Chen
and Box 1979).
clusters occur (c = .001 or .01).
Using this distribution, we
CONCLUSION
We have discussed the design and use of
CUSUM procedures. We have shown that they are
simple to use, versatile in that they can be
specifically tailored to detect shifts in
level, and powerful in that they use all
i nformat; on in the data to qui ck 1y detect the
shift.
The design and implementation procedure for counts is essentially the same as
the procedure used
for
the
control
of
variables.
The FIR feature,
a combined
Shewhart-CUSUM scheme and robustification have
all proved to be valuable enhancements for
CUSUM control schemes.
It has been our experience that occasionally a cluster of nonconformances occurs in a
single sample and that it is sometimes valid
to ignore the occurrence of a single sample
having
a
cluster
of
nonconformances
(especially if this cluster could be due to
sampling or measuring instrument problems).
To model the Robust CUSUM, we assumed that the
underlying distribution was a mixture of a
distribution
(with
In these
cases, a Robust CUSUM gives a very large
percentage increase in the in-control ARLs and
on ly a Sma 11 percentage increase in the outof-control ARLs.
evaluated the performance of various CUSUM
schemes.
We demonstrated that the two-in-arow rule was a good robustification procedure
and showed that more complicated rules could
give little improvement. In this section, we
evaluate the properties of the two-in-a-row
rule for robustifying a Poisson CUSUM for
detecting an increase in count rate.
Poisson
The cost of
the robustification procedure can be seen in
the no outl ier case (c = 0). In these cases,
equivalent increases in in-control ARLs can be
achieved with less inflation of the out-ofcontrol ARLs by simply increasing the h
value.
The benefits of the robustification
procedures show up in the cases where outlier
probability I-c)
920
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"An
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Kenett,
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