Consider the change point model (11.1). For fixed denote
and
the corresponding probability measure and
expectation, respectively. Hereby,
stands for the case of no change,
i.e. the so called in-control case. Then the Average Run Length (ARL)
(expectation of the run length
) is defined as
![]() |
(11.7) |
Thus, the ARL denotes the average number of observations until signal for a
sequence with constant expectation. or
stands for no
change,
and
mark, that just at the first time point (or
earlier) a change takes place from
to
. Therefore, the ARL
evaluates only the special scenario of
of the SPC scheme. Other measures,
which take into account that usually
, were introduced by
Lorden (1971) and Pollak and Siegmund (1975), Pollak and Siegmund (1975).
Here, we use a performance measure
which was firstly proposed by Roberts (1959). The so called (conditional) Average
Delay (AD, also known as steady-state ARL) is defined as
where is the value of
in (11.1), i.e. the
expectation after the change. While
measures the delay for the
case
,
determines the delay for a SPC scheme which ran a
long time without signal. Usually, the convergence in (11.8) is very fast.
For quite small
the difference between
and
is very small already.
and
are average
values for the random variable
. Unfortunately,
is characterized by a
large standard deviation. Therefore, one might be interested in the whole
distribution of
. Again, we restrict on the special cases
and
. We consider the probability mass function
(PMF) and the
cumulative distribution function
(CDF). Based on the CDF, one is
able to compute quantiles of the run length
.
For normally distributed random variables it is not possible to derive exact solutions for the above characteristics. There are a couple of approximation techniques. Besides very rough approximations based on the Wald approximation known from sequential analysis, Wiener process approximations and similar methods, three main methods can be distinguished:
Here we use the first approach, which has the advantage, that all considered
characteristics can be presented in a straightforward way. Next, the Markov
chain approach is briefly described. Roughly speaking, the continuous statistic
is approximated by a discrete Markov chain
. The transition
is approximated by the transition
with
and
. That is,
given an integer
the continuation region of the scheme
,
zreflect
, or
is separated into
or
intervals of the
kind
(one exception is
as the first subinterval
of
). Then, the transition kernel
of
is approximated by the
discrete kernel of
, i.e.
for all
and
. Eventually, we
obtain a Markov chain
with
or
transient states and
one absorbing state. The last one corresponds to the alarm (signal) of the
scheme.
Denote by
the matrix of transition probabilities of the Markov
chain
on the transient states,
a vector of ones, and
the ARL vector.
stands for the ARL of a SPC scheme
which starts in point
(corresponds to
). In the case of a one-sided
CUSUM scheme with
the value
approximates the original
ARL. By using
we generalize the original schemes to schemes with
possibly different starting values
. Now, the following linear equation
system is valid, Brook and Evans (1972):
![]() |
(11.9) |
where denotes the identity matrix. By solving this equation system we get
the ARL vector
and an approximation of the ARL of the considered
SPC scheme. Remark that the larger
the better is the approximation. In the
days of Brook and Evans (1972) the maximal matrix dimension
(they considered
cusum1) was 15 because of the restrictions of the available computing
facilities. Nowadays, one can use dimensions larger than some hundreds. By
looking at different
one can find a suitable value. The quantlet
XFGrarl.xpl
demonstrates this effect for the Brook and Evans (1972) example. 9
different values of
from 5 to 500 are used to approximate the in-control ARL
of a one-sided CUSUM chart with
and
(variance
). We
get
>
![]() |
5 | 10 | 20 | 30 | 40 | 50 | 100 | 200 | 500 |
![]() |
113.47 | 116.63 | 117.36 | 117.49 | 117.54 | 117.56 | 117.59 | 117.59 | 117.60 |
The true value is 117.59570 (obtainable via a very large or by using the
quadrature methods with a suitable large number of abscissas). The computation
of the average delay (AD) requires more extensive calculations. For details see,
e.g., Knoth (1998) on CUSUM for Erlang distributed data. Here we apply the
Markov chain approach again, Crosier (1986). Given one of the considered
schemes and normally distributed data, the matrix
is primitive, i.e. there
exists a power of
which is positive. Then
has one single eigenvalue
which is larger in magnitude than the remaining eigenvalues. Denote this
eigenvalue by
. The corresponding left eigenvector
is strictly positive, i.e.
It can be shown, Knoth (1998), that the conditional density
of both the continuous statistic
and the Markov chain
tends
for
to the normalized left eigenfunction and eigenvector,
respectively, which correspond to the dominant eigenvalue
. Therefore,
the approximation of
can
be constructed by
![]() |
Note, that the left eigenvector
is computed for the
in-control mean
, while the ARL vector
is computed for a
specific out-of-control mean or
again.
If we replace in the above quantlet (
XFGrarl.xpl
) the phrase arl by ad, then we obtain the following output which demonstrates the
effect of the parameter
again.
>
![]() |
5 | 10 | 20 | 30 | 40 | 50 | 100 | 200 | 500 |
![]() |
110.87 | 114.00 | 114.72 | 114.85 | 114.90 | 114.92 | 114.94 | 114.95 | 114.95 |
Fortunately, for smaller values of than in the ARL case we get good
accuracy already. Note, that in case of cusum2 the value
has to be
smaller (less than 30) than for the other charts, since it is based on the
computation of the dominant eigenvalue of a very large matrix. The
approximation in case of combination of two one-sided schemes needs a
twodimensional approximating Markov chain. For the ARL only exists a more
suitable approach. As, e.g., Lucas and Crosier (1982) shown it is possible to use
the following relation between the ARLs of the one- and the two-sided schemes.
Here, the two-sided scheme is a combination of two symmetric one-sided schemes
which both start at
. Therefore, we get a very simple formula for the ARL
of the two-sided scheme and the ARLs
and
of the upper and lower one-sided CUSUM scheme
Eventually, we consider the distribution function of the run length itself.
By using the Markov chain approach and denoting with
the
approximated probability of
for a SPC scheme started in
, such
that
, we obtain
The vector
is initialized with
for the starting
point
and
otherwise. For large
we
can replace the above equation by
The constant is defined as
![]() |
where
denotes the right eigenvector of
, i.e.
. Based on
(11.12) and (11.13) the probability mass and the cumulative distribution
function of the run length
can be approximated. (11.12) is used up to a
certain
. If the difference between (11.12) and (11.13) is smaller
than
, then exclusively (11.13) is exploited. Remark, that the same
is valid as for the AD. For the two-sided CUSUM scheme (cusum2) the
parameter
has to be small (
).
The
spc
quantlib provides the quantlets
spcewma1arl
,...,
spccusumCarl
for computing the ARL of the corresponding SPC scheme. All routines need the
actual value of
as a scalar or as a vector of several
, two scheme
parameters, and the integer
(see the beginning of the section). The
XploRe
example
XFGarl.xpl
demonstrates all ...arl routines for
,
, zreflect
,
,
, in-control and out-of-control
means
and
, respectively. The next table summarizes the ARL
results
chart | ewma1 | ewma2 | cusum1 | cusum2 | cusumC |
![]() |
1694.0 | 838.30 | 117.56 | 58.780 | 76.748 |
![]() |
11.386 | 11.386 | 6.4044 | 6.4036 | 6.4716 |
For the setup of the SPC scheme it is usual to give the design parameter
and
for EWMA and CUSUM, respectively, and a value
for the
in-control ARL. Then, the critical value
(
or
) is the solution
of the equation
. Here, the regula falsi is used with
an accuracy of
. The quantlet
XFGc.xpl
demonstrates the computation of the critical values for SPC schemes with
in-control ARLs of
, reference value
(CUSUM), smoothing
parameter
(EWMA), zreflect
, and the Markov chain
parameter
.
chart | ewma1 | ewma2 | cusum1 | cusum2 | cusumC |
![]() |
2.3081 | 2.6203 | 3.8929 | 4.5695 | 4.288 |
The parameter guarantees fast computation and suitable accuracy.
Depending on the power of the computer one can try values of
up to 1000 or
larger (see
XFGrarl.xpl
in the beginning of the section).
The usage of the routines for computing the Average Delay (AD) is similar to the
ARL routines. Replace only the code arl by ad. Be aware that the
computing time is larger than in case of the ARL, because of the computation of
the dominant eigenvalue. It would be better to choose smaller , especially
in the case of the two-sided CUSUM. Unfortunately, there is no relation between
the one- and two-sided schemes as for the ARL in (11.11). Therefore,
the library computes the AD for the two-sided CUSUM based on a twodimensional
Markov chain with dimension
. Thus with values of
larger than 30, the computing time becomes quite large. Here the results follow
for the above quantlet
XFGrarl.xpl
with ad instead of arl and
for
spccusum2ad
:
chart | ewma1 | ewma2 | cusum1 | cusum2 | cusumC |
![]() |
1685.8 | 829.83 | 114.92 | 56.047 | 74.495 |
![]() |
11.204 | 11.168 | 5.8533 | 5.8346 | 6.2858 |
The computation of the probability mass function (PMF) and of the cumulative
distribution function (CDF) is implemented in two different types of routines.
The first one with the syntax spcchartpmf returns the values
of the PMF and CDF
at given single points of
, where
chart has to be replaced by ewma1, ..., cusumC. The second one
written as spcchartpmfm computes the whole vectors of the PMF
and of the CDF up to a given point
, i.e.
and the similar one of the CDF.
Note, that the same is valid as for the Average Delay (AD). In case of the
two-sided CUSUM scheme the computations are based on a twodimensional Markov
chain. A value of parameter less than 30 would be computing time friendly.
With the quantlet
XFGpmf1.xpl
the 5 different schemes (
,
for cusum2
) are compared according their in-control PMF and CDF
(
) at the positions
in
. Remark,
that the in-control ARL of all schemes is chosen as 300.
chart | ewma1 | ewma2 | cusum1 | cusum2 | cusumC |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
0.00318 | 0.00272 | 0.00321 | 0.00307 | 0.00320 |
![]() |
0.00332 | 0.00324 | 0.00321 | 0.00325 | 0.00322 |
![]() |
0.00315 | 0.00316 | 0.00310 | 0.00314 | 0.00311 |
![]() |
0.00292 | 0.00296 | 0.00290 | 0.00294 | 0.00290 |
![]() |
0.00246 | 0.00249 | 0.00245 | 0.00248 | 0.00245 |
![]() |
0.00175 | 0.00177 | 0.00175 | 0.00176 | 0.00175 |
![]() |
0.00125 | 0.00126 | 0.00124 | 0.00125 | 0.00125 |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
0.01663 | 0.01233 | 0.02012 | 0.01675 | 0.01958 |
![]() |
0.05005 | 0.04372 | 0.05254 | 0.04916 | 0.05202 |
![]() |
0.08228 | 0.07576 | 0.08407 | 0.08109 | 0.08358 |
![]() |
0.14269 | 0.13683 | 0.14402 | 0.14179 | 0.14360 |
![]() |
0.27642 | 0.27242 | 0.27728 | 0.27658 | 0.27700 |
![]() |
0.48452 | 0.48306 | 0.48480 | 0.48597 | 0.48470 |
![]() |
0.63277 | 0.63272 | 0.63272 | 0.63476 | 0.63273 |
A more appropriate, graphical representation provides the quantlet
XFGpmf2.xpl
. Figure 11.4 shows the corresponding graphs.