Supplemental Material for Chapter 8
S8-1. The Markov Chain Approach to Finding the ARLs for Cusum and EWMA
Control Charts
When the observations drawn from the process are independent, average run lengths or
ARLs are easy to determine for Shewhart control charts because the points plotted on the
chart are independent. The distribution of run length is geometric, so the ARL of the
chart is just the mean of the geometric distribution, or 1/p, where p is the probability that
a single point plots outside the control limits.
The sequence of plotted points on Cusum and EWMA charts is not independent, so
another approach must be used to find the ARLs. The Markov chain approach developed
by Brook and Evans (1972) is very widely used. We give a brief discussion of this
procedure for a one-sided Cusum.
The Cusum control chart statistic C  (or C  ) form a Markov process with a continuous
state space. By discretizing the continuous random variable C  (or C  ) with a finite set
of values, approximate ARLs can be obtained from Markov chain theory. For the upper
one-sided Cusum with upper decision interval H, the intervals are defined as follows:
(, w / 2],[ w / 2,3w / 2], ,[(k  1/ 2) w, (k  1/ 2) w], ,[( m  3 / 2) w, H ],[ H , ) where m
+ 1 is the number of states and w = 2H/(2m- 1). The elements of the transition
probability matrix of the Markov chain P  [ pij ] are
pi 0  
w/ 2

pij  
f ( x  iw  k )dx, i  0,1,..., m  1
( j  i / 2) w
( j i / 2) w
 i  0,1,..., m  1
f ( x  iw  k )dx 
 j  1, 2,..., m  1

pim   f ( x  iw  k )dx, i  0,1,..., m  1
H
pmj  0, j  0,1,..., m  1
pmm  1
The absorbing state is m and f denotes the probability density function of the variable that
is being monitored with the Cusum.
From the theory of Markov chains, the expected first passage times from state i to the
absorbing state are
m 1
i  1   pij  j , i  0,1,..., m  1
j 0
Thus, i is the ARL given that the process started in state i. Let Q be the matrix of
transition probabilities obtained by deleting the last row and column of P. Then the
vector of ARLs  is found by computing
   I  Q1
1
where 1 is an m1 vector of 1s and I is the m m identity matrix.
When the process is out of control, this procedure gives a vector of initial-state (or zerostate) ARLs. That is, the process shifts out of control at the initial start-up of the control
chart. It is also possible to calculate steady-state ARLs that describe performance
assuming that the process shifts out of control after the control chart has been operating
for a long period of time. There is typically very little difference between initial-state and
steady-state ARLs.
Let P(n, i ) be the probability that run length takes on the value n given that the chart
started in state i. Collect these quantities into a vector say
pn  [ P(n,0), P(n,1),..., P(n, m 1)]
for n = 1,2, …. These probabilities can be calculated by solving the following equations:
p1  (I  Q)11
p n  Qp n 1 , n  2,3,...
This technique can be used to calculate the probability distribution of the run length,
given the control chart started in state i. Some authors believe that the distribution of run
length or its percentiles is more useful that the ARL, since the distribution of run length is
usually highly skewed and so the ARL may not be a ‘typical” value in any sense.
S8-2. Integral Equations Versus Markov Chains for Finding the ARL
Two methods are used to find the ARL distribution of control charts, the Markov chain
method and an approach that uses integral equations. The Markov chain method is
described in Section S8-1 of the Supplemental Text Material. This section gives an
overview of the integral equation approach for the Cusum control chart. Some of the
notation defined in Section S8-1 will be used here.
Let P(n, u ) and R(u ) be the probability that the run length takes on the value n and the
ARL for the Cusum when the procedure begins with initial value u. For the one-sided
upper Cusum
P(1, u )  1  
H

 1 
f ( x  u  k )dx
w/ 2

m 1
f ( x  u  k )dx   
j 1
( j 1/ 2) w
( j 1/ 2) w
f ( x  u  k )dx
and
2
P(n, u )  P (n  1, 0) 
0
 P(n  1, 0) 
0


m 1

j 1
( j 1/ 2) w
( j 1/ 2) w
 P(n  1, 0) 
0

m 1
H
f ( x  u  k )dx   P (n  1, y ) f ( x  u  k )dx
0
f ( x  u  k )dx  
0
P(n  1, y ) f ( x  u  k )dx
P(n  1, y ) f ( x  u  k )dx
f ( x  u  k )dx  P(n  1,  0 ) 
w/ 2
0
  P(n  1,  j ) 
j 1
w/ 2
( j 1/ 2) w
( j 1/ 2) w
f ( x  u  k )dx
f ( x  u  k )dx
for n = 1,2,… and for  0  (, w / 2) and  j  [( j  1/ 2) w, ( j  1/ 2) w), j  1, 2,..., m  1. If
w is small, then  j is the midpoint jw of the jth interval for j = 1,2,…, m – 1, and
considering only the values of P(n, u ) for which u = iw results in
m 1
P(1, iw)  1   pij
j 0
m 1
P(n, iw)   P(n  1, iw) pij , n  2,3,...
j 0
But these last equations are just the equations used for calculating the probabilities of
first-passage times in a Markov chain. Therefore, the solution to the integral equation
approach involves solving equations identical to those used in the Markov chain
procedure.
Champ and Rigdon (1991) give an excellent discussion of the Markov chain and integral
equation techniques for finding ARLs for both the Cusum and the EWMA control charts.
They observe that the Markov chain approach involves obtaining an exact solution to an
approximate formulation of the ARL problem, while the integral equation approach
involves finding an approximate solution to the exact formulation of the ARL problem.
They point out that more accurate solutions can likely be found via the integral equation
approach. However, there are problems for which only the Markov chain method will
work, such as the case of a drifting mean.
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