The Monitoring of Linear Profiles
Keun Pyo Kim
Mahmoud A. Mahmoud
William H. Woodall
Virginia Tech
Blacksburg, VA 24061-0439
(Send request for paper, submitted to JQT, to bwoodall@vt.edu)
1
th
j
We assume that for the
random sample collected over
time, we have the observations
(xi , yij), i = 1, 2, …, n.
2
Applications include…
• Calibration problems in analytical
chemistry (Stover and Brill, 1998)
• Semiconductor manufacturing
(Kang and Albin, 2000)
• Automobile manufacturing (Lawless
et al., 1999)
• DOE applications (Miller, 2002 and
Nair et al. 2002)
3
It is assumed that when the process is
in statistical control, the underlying
model is
Yij  A0  A1 X i   ij
i = 1, 2, …, n, where the  ij’s are
independent, identically distributed
2

(i.i.d.) N(0, ).
4
The least squares estimators a0 j and
have a1 j have a bivariate normal
distribution with the mean vector
μ  ( A0 , A1 )
T
and the variance-covariance matrix
  

Σ 2
  2 
 01 1 
2
0
2
01
5
1 x 

    
 n S xx 
2
2
0
2
1
 
S xx
2
1

2
01
2
x
 
S xx
2
6
Phase II
First we consider the Phase II case
involving process monitoring with incontrol values of the parameters
assumed to be known.
7
The first control strategy of
Kang and Albin (2000) is a T2
chart based on the estimated
regression coefficients
Z j  (a0 j , a1 j )
T
1
T  (Z j  μ) Σ (Z j  μ)
2
j
T
8
Their second control strategy is
to apply an EWMA - R chart
combination scheme to the
residuals obtained with each
sample.
9
The residuals for the
are
th
j
sample
eij  yij  A0  A j xi
i = 1, 2, … , n.
10
Instead, we propose scaling the
X-values to obtain the model
Yij  B0  B1 X i   ij
B0  A0  A1 X
B1  A1
X i  ( X i  X )
11
Since now the least squares
estimators are independent, we
recommend three EWMA charts in
Phase II to detect sustained shifts
in the parameters.
There is a chart for each regression
coefficient and one for the
variation about the line.
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ARL Comparisons
We use the in-control model
Yij  3  2 X i   ij
with error terms i.i.d. N(0, 1). The values
for X are 2, 4, 6, 8.
13
Figure 1. ARL Comparisons Under Intercept Shifts 
200
EWMA/R
T2
EWMA_3
ARL
150
100
50
0
0.0
0.5
1.0

1.5
2.0
14
Figure 2. ARL Comparisons Under Slope Shifts 
200
EWMA/R
T2
EWMA_3
ARL
150
100
50
0
0.00
0.05
0.10
0.15

0.20
0.25
15
Figure 3. ARL Comparisons Under Standard Deviations Shifts 
200
EWMA/R
T2
EWMA_3
ARL
150
100
50
0
1.0
1.5
2.0

2.5
3.0
16
Figure 4. ARL Comparisons Under Slope Shifts 
200
T2
ARL
150
EWMA/R
T2
EWMA_3
100
50
0
0.0
0.2
0.4
0.6

0.8
1.0
17
Table 1. ARL Comparisons Under Slope Shifts in Model (10) FromB1
To B1  
(Xi -values are 1, 2, 3, and 4 and In-control ARL = 200)

Chart
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
EWMA/R 149.06 110.07 75.50
50.65
33.29
22.26
15.03
10.53
7.53
EMWA_3
8.87
6.63
5.27
4.38
3.78
3.32
49.07
22.85
13.13
18
Our proposed method (EWMA_3)
has better ARL performance than
competing methods.
The interpretation is also much
easier.
19
Phase I
In Phase I, one has k sets of bivariate
observations.
One checks for stability of the linear
profiles over time and estimates
parameters.
20
We recommend Shewhart type charts
for each regression parameter and
change-point methods.
EWMA charts are not recommended
in Phase I.
21
Relationship to Regressionadjusted Control Charts
Monitoring linear profiles is a generalization
of regression-adjusted methods studied by
Mandel (1969), Zhang (1992), Wade and
Woodall (1993), Hawkins (1991, 1993), and
Hauck et. al (1999).
22
Suppose X is an input quality
variable and Y is the output quality
variable with k = 1 and n = 1.
Then we have the simplest
regression-adjusted chart, sometimes
referred to as the cause-selecting
chart. (Note X-values are random.)
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Conclusions
• Monitoring linear profiles seems to be
quite useful.
• Regression-adjusted methods deserve
wider application since usual methods
can be misleading if output quality is
affected by input quality as is often the
case.
• Methods can be extended to more
complicated models.
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