A Transition Matrix Representation of the
Algorithmic Statistical Process Control Procedure
with Bounded Adjustments and Monitoring
Changsoon Park
Department of Statistics
Chung-Ang University
Seoul, Korea
 Algorithmic Statistical Process Control (ASPC)
- Vander Wiel, Tucker, Faltin, Doganaksoy(1992)
- Integrated approach to quality improvement
- An approach that realizes quality gains through process adjustment &
process monitoring
Process adjustment ;
manipulate the compensating variables of a process to achieve the
desired process behavior ( e.g., output close to a target )
- adjustment scheme ( feedforward, feedback )
( e.g. repeated adjustment, bounded adjustment )
Process mornitoring ;
monitor a process so as to detect and remove root causes of variability
- control chart ( e.g. Shewhart, CUSUM, EWMA )
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 We consider
Disturbance Model – IMA(0,1,1) with a step shift
Z t  Z t 1  at  at 1   a I{U } (t )
IMA(0,1,1)
Step shift
due to noise
due to special cause
~ Pr(U  t )  (1  p)t 1 p,
t  1, 2,
ASPC procedure – Bounded Adjustments & EWMA Monitoring
Derive properties by a transition matrix representation
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IMA(0,1,1) with A Step Shift
6
5
4
3
2
1
0
0
5
10
15
20
25
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40
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-1
-2
-3
-4
Z t  Z t 1  at  0.3at 1  3I{30} (t )
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 Control Procedure
1. Bounded Adjustments
Zˆ t 1 : one-step ahead forecast
Zˆt 1  Zt  Zˆt ,
  1
Yt : total output compensation
Ot  Z t  Yt : observed deviation
Ft  Zˆt 1  Yt : predicted deviation
If Ft  L , then adjust the process (Yt  Zˆt 1 )
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A(1), A(2),  : adjustment time
yt ( Yt  Yt 1 ) : one-step output compensation
◦ observed deviation :

Ot  Ot 1  at  at 1   a I{U } (t )   y A(t )1 I{ A(t )1} (t )
t  1
◦ Recurrence relation
 Ot  Ot 1  at  at 1   a I{U } (t )

 Ft  Ot  Ft 1
If Ft  L , adjust by  Ft , i.e.
Restart with
 Ot  Ot  Ft

 Ft  0
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Bounded Adjustments
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L
2
1
A(1)
A( 4 )
0
0
5
10
15 A( 2)
20
25
30A(3)
35
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-1
L
-2
-3
-4
O
계열1
t
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F
계열2
t
Y
계열3
t
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◦ Random shock representation of Ot and Ft
r (t ) : adjustment time immediately before t
N 0 : no. of adjustments before a special cause occurs
t 1

at    a j , t  U
j  r ( t ) 1

t 1

Ot  at    a j   a , U  t  A( N 0  1)
j  r ( t ) 1

t 1
a  
a j  a r (t ) U 1 ,
A( N 0  1)  t

t

j  r ( t ) 1

t

  a j , t  U
 j r (t t ) 1

Ft    a j  a (1   t U 1 ) , U  t  A( N 0  1)
 j r (t t ) 1

r ( t ) U 1
t r (t )

a



(
1


), A( N 0  1)  t
a
  j
 j r (t ) 1
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2. EWMA Monitoring
et  Ot  Ft 1 : Forecast error
EWMA statistic : Et  ret  (1  r ) Et 1 ,
0  r 1
If Et  c , signal
When a signal is false, restart with Et 1  0
at , t  U
et  
t U
a



, U t
a
 t
( Et , Ft ) : Bivariate process control statistic
adjustment, true signal : action
A(1), A(2),  : action time
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EWMA Monitoring
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0
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-1
-2
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-4
et
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Et
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 Transition Matrix Representation
Calculate properties of ASPC procedure
(no. of false signals, no. of adjustments, sum of squared deviations)
1. A cycle
start
special cause
A( 2 )
A(1)
A( N 0 )
signal
A( N 0  1) A( N 0  2)
period Ⅱ
period Ⅰ
end
period Ⅲ
special cause signal
start
A(1)
A( 2 )
A( N 0  1)
A( N 0 )
end
period Ⅰ
: adjustment
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: false signal
period Ⅱ
: true signal
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2. Representation of ( Et , Ft ) by a finite states
- use of Gaussian quadrature points and weights
2.1 Partition of (c, c) : no signal interval
points : xi , weights : vi , i  1, 2,  , h (odd), (h  h  1)
(c, c) weight
c
0
c
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subinterval
points
vh
vh 1
Ih
I h 1
xh
xh 1
vh / 2
I h / 2
xh / 2
v2
v1
I2
I1
(,c]  [c, )
x2
x1
xh
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2.2 Partition of (  L, L ) : no adjustment interval
points : y j , weights : w j , j  1, 2,, g (odd), ( g   g  1)
(  L, L ) weight
L
wg
wg 1
0
L
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wg  / 2
w2
w1
subinterval
points
Jg
J g 1
yg
y g 1
J g/ 2
yg/ 2
J2
J1
(, L]  [ L, )
y2
y1
yg
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3. Transition Matrix Representation in each period
◦ bivariate process
states
( xi , y j )
( Et , Ft )
◦ Transition matrix
g h
i  1, 2,, h
j  1, 2,, g 
g h
Pt (ij , kl)
Pt (ij , kl)  Pr[( Et 1 , Ft 1 )  ( xk , yl ) ( Et , Ft )  ( xi , y j )]
i  1, 2,, h
,

 j  1, 2,, g 
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k  1, 2,, h

l  1, 2,, g 
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◦ Partition the whole states into classes according to the action
Denote each class by a character
“We are interested in each action, not in each state.”
“We can identify the classes involved by the dimension of the matrix
or vector.”
1a
0a
: one vector of dimension a
: zero vector of dimension a
1a ('class') : vector of dimension a whose elements corresponding
to ‘class’ are all 1’s and all the rests are 0 .
1a (state) : vector of dimension a whose element corresponding
to the state is 1 and all the rests are 0 .
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3.1 Period Ⅰ
◦ classes
class
no action
state
no. of states
‘ n’
{( xi , y j )}
i  1,2,, h
j  1,2,  , g
gh
‘ a’
{( xi , y g  )}
i  1,2,, h
h
{( xh , y j )}
j  1,2,  , g
g
adjustment only
false signal only
‘ f’
false signal & adjustment
‘1’
{( xh , y g  )}
1
‘*’
whole states
g h
‘ nf ’
‘ n’  ‘ f ’
gh
‘ a1’
‘ a’  ‘ 1’
h
‘ f 1’
‘ f ’  ‘ 1’
g
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◦ decomposition of the total transition matrix
‘ n’
‘ n’  Q nI , n
Q*,*
I
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

‘ a’  Q aI , n
 
‘ f ’  Q If , n


1, n
‘1’  Q I
‘ a’
‘ f’
‘1’
Q nI ,a
Q nI , f
Q aI ,a
Q aI , f
Q If ,a
Q If , f
Q1I,a
Q1I, f
Q nI ,1 


a ,1
QI 


Q If ,1 

Q1I,1 
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◦ Define transition matrix until U
- rather than only in period Ⅰ
- state of a special cause (occurrence or not) is added
nsc
nsc  (1  p )Q*,*
I

PI 

sc 
0g h
sc
p  1 g h 


1 
(1  p) Q*,*
: transient state transition matrix
I
Occurrence of a special cause : absorbing state
(there is no absorbing state in period Ⅰ)
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◦ s I : starting state vector of period Ⅰ
s I  1gh ( xh / 2 , yg / 2 )  1gh (0,0)
TF : time that a false signal occurs
' f 1'  false signal
t 1
Pr(TF  t )  sI [(1  p)Q*,*
[(1  p)Q*,I f 1 ]1g 
I ]
t
 sI [(1  p)Q*,*
I ] 1 g h (' f 1')
◦ Average no. of false signal

E ( F )   1  Pr(TF  t )
t 1

t
 sI  [(1  p )Q*,*
I ] 1 g h (' f 1')
t 1
*
I
I
*
I
I
 s R [(1  p )Q*,*
I ] 1 g h (' f 1')
 s R [(1  p )Q*,I f 1 ]1 g 
1
where R*I  [I  (1  p)Q*,*
I ]
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T AI : time that an adjustment occurs
'a1'  adjustment
t 1
*,a1
Pr(TAI  t )  sI [(1  p)Q*,*
I ] [(1  p )Q I ]1h
t
 sI [(1  p)Q*,*
]
1g h('a1')
I
◦ Average no. of adjustments

E ( AI )  1 Pr(TAI  t )
t 1
 sI R *I [(1  p)Q*,*
I ] 1 g h ('a1')
 sI R *I [(1  p)Q*,I a1 ]1h
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◦ H I ( xi ) : adjustment interval length given Et   xi , i  1,2,, h
Pr( H I ( xi )  t )  1gh( xi , y g / 2 ) [(1  p)Q nfI ,nf ]t 1[(1  p)Q nfI ,a1 ]1h

E[ H I ( xi )]   t  Pr(H I ( xi )  t )
t 1
 1gh( xi , y g / 2 ) [R nfI ]2 [(1  p)Q nfI ,a1 ]1h
where R nfI  [I  (1  p)Q nfI ,nf ]1
HI  ( H I ( x1 ), H I ( x2 ),  , H I ( xh ))
◦ S I ( xi ) : SSD in an action(adjustment) interval given Et   xi
t (t  1) 2 2
SSD H ( x ) t  { t 
 } a
I
i
2

t (t  1) 2 2
E[ S I ( xi )]  { t 
 } a  Pr( H I ( xi )  t )
2
t 1
  a2 1gh( xi , y g / 2 ) [R nfI ]2{Ι  2 R nfI [(1  p)Q nfI ,nf ]}[(1  p)Q nfI ,a1 ]1h
SI  (S I ( x1 ), S I ( x2 ),  , S I ( xh ))
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◦ sI R*I [(1  p)Q*,I a1 ] : average no. of visits to Et  xi & Ft  y g 
◦ Average period length
E( LI )  sI R*I [(1  p)Q*,I a1 ]H I  E[ H I ( xh / 2 )]  E[ H I ( EA( N0 ) )]
◦ Expected SSD
E(SS I )  sI R*I [(1  p)Q*,I a1 ]S I  E[S I ( xh / 2 )]  E[S I ( EA( N0 ) )]
special cause
start
A(1)
A( 2 )
period Ⅰ
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A( N 0  1)
A( N 0 )
period Ⅱ
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◦ Probability of E A( N 0 )
- the last adjustment time before a special cause occurs
Pr( A( N 0 )  t , E A( N 0 )  xi )
 Pr( Et  xi , Ft  y g  )  Pr( A( N 0 )  t Et  xi , Ft  y g  )
B
A
t 1
*,a1
A  sI [(1  p)Q*,*
I ] [(1  p)Q I ]1h( xi )

B   1gh( xi , y g ) [(1  p)Q nfI ,nf ]q 1 p1gh
q 1
 p  1gh( xi , y g ) R nfI 1gh
h
E[ H I ( E A( N 0 ) )]   H I ( xi )  Pr(E A( N 0 )  xi )
i 1
h
E[ S I ( E A( N 0 ) )]   S I ( xi )  Pr(E A( N 0 )  xi )
i 1
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3.2 Period Ⅱ
◦ merge {Et  xh } into {Et  xh / 2 }
' n  '  class of no action (' n' ) and false signal only (' f ' )
keep 'a' , '1'
◦ s II : starting state vector of period Ⅱ
For i  1,2, , h, j  1,2, , g
if i  h / 2, j  g  / 2
Pr( E A( N 0 )  xi )

s II ( xi , y j )  Pr( E A( N 0 )  xh / 2 )  Pr( E A( N0 )  xh ) if i  h / 2, j  g  / 2

otherwise
0
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 : the time, counted from the beginning of period Ⅱ, that a special
cause occurs
special cause
A( N 0  1)
A( N 0 )
start

period Ⅱ
U
adjustment or signal
◦ s (q) : state vector at immediately before time 
 sII [(1  p)Q
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n  ,n  q 1
I
]
p
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◦ classes after 
class
no action
‘ n’
adjustment
‘ a’
signal
‘ s’
‘*’
‘ as’
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state
no. of states
{( xi , y j )}
i  1,2,, h
j  1,2,  , g
gh
{( xi , y g  )}
i  1,2,, h
h
{( xh , y j )}
j  1,2,  , g 
g
whole states
‘ a’  ‘ s’
g h
h  g
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◦ decomposition of the total transition matrix
‘ n’
‘ n’  Q II ,t
n ,n

Q ‘a’  0

‘ s’  0
' a' , ' s ' : absorbing states
‘ a’
Q nII,,at
*,*
II ,t
Ih
0
‘ s’
Q nII,,st 

0 
I g  
◦ Define
'
g 'IIclass'  Q nII,,n1 Q nII,,n2 Q nII,,nt 1 Q nII,',class
t
 Pr({time to visit ' class ' after a special cause}  t )

G
'class'
II
(k )   t k g II'class' (t )
for k  0,1, 2
t 1
 k  th monent of the time
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H II : action(adjustment or siganl) interval length
Pr( H II  t ,  q)  s (q)g as
II (t  q  1)1h  g 
0
1
q 1
q
1
s (q)
q 1
2
t 1
t
t  q t  q 1
period Ⅱ
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LII : length of period Ⅱ ( H II )
AII : no. of adjustment in period Ⅱ
◦ Average period length

t
E ( LII )   t  s (q )g as
II (t  q  1)1h  g 
t 1 q 1



  s (q ) (t   q  1)g as
II (t )1h  g 
q 1
t  1
n
as
 psII R [G as
(
1
)

(
R

I
)
G
II
I
II (0)]1h  g 
n
I
◦ Average no. of adjustments
h
E ( AII )   Pr( E A( N 0 1)  xi , FA( N 0 1)  y g  )
i 1

 psII R nI G aII (0)1h
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◦ Expected SSD
SSD H
II
( xi ) t
{ t 

t
t (t  1) 2
   2 (t  q  1) } a2
2
E ( SS II )   { t 
t 1 q 1
t (t  1) 2
2
   2 (t  q  1) } a
2
 s (q)g as
II (t  q  1)1h  g 




3
 psII R nI {[(1  2 )R nI  2 (1  p)Q nI (R nI ) 2  (2  1)]G as
II (0)
2
3 2
2 as
2
2 n
as
 [(1     )   R I ]G II (1)  G II (2)}
2
2
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◦ Final state probability of period Ⅱ
Pr( E A( N 0 1)  xi , FA( N 0 1)  y g  )

t
  s (q)g aII (t  q  1)1h ( xi )
t 1 q 1


q 1
t  1
  s (q) g aII (t )1h ( xi )
n
I
 psII R G aII (0)1h ( xi )

where R nI  [I  (1  p)QnI
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
, n  1
]
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3.3 Period Ⅲ
◦ decomposition of the total transition matrix
Q*,*
III ,t
 Q nIII,n,t
 a ,n
  Q III ,t
 0

Q nIII,a,t
Q aIII,a,t
0
Q nIII, s,t 

a ,s
Q III ,t 
I g  
' s ' : absorbing state
◦ s III : starting state vector of period Ⅲ
For i  1,2, , h, j  1,2,, g 
Pr( E A( N 0 1)  xi , FA( N 0 1)  y g  ) if j  g 
s III ( xi , y j )  
if j  g 
0
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◦ Define
, na na, na
na, na
na,'class'
g 'IIIclass' (t )  Q na
Q

Q
Q
III ,1
III , 2
III ,t 1 III ,t
 Pr({time to visit ' class '}  t )

G
'class'
III
'class'
(k )   t k g III
(t )
for k  0,1, 2
t 1
 k  th moment of the time
◦ Average period length
E( LIII )  sIII G sIII (1)1g
◦ Average no. of adjustments
E ( AIII )  sIII G aIII (0)1h
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◦ Expected SSD
For t0  0,1,
Pr( A(t )  t0 , Et0  xi , A(t   1)  t0  t )
 Pr( A(t )  t0 , Et0  xi )  Pr( A(t   1)  t0  t A(t )  t0 , Et0  xi )
 sIII g aIII (t0 )1h ( xi )
 1gh( xi , y g / 2 ) Q nIII,n,t0 1 Q nIII,n,t0 t 1Q nIII,as,t0 t 1h  g 
SSD H
III
( xi ) t
 [t{1  ( r (t )U 1 ) 2 } 

t (t  1) 2 2
 ] a
2

t (t  1) 2 2
 ] a
2
i 1 t0 1 t 1
 Pr( A(t )  t0 , Et0  xi , A(t   1)  t0  t )
h
E ( SS III )   [ t{1  ( r (t )U 1 ) 2 } 
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 Expected Cost Per Unit Time (ECU)
CM : cost per monitoring
C A : cost per adjustment
CT : off-target cost per SSD
C F : cost per false signal
ECU  CM 
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C A  E ( AI  AII  AIII )  CT  E ( S I  S II  S III )  CF  E ( F )
E ( LI  LII  LIII )
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