Plan for the Session
? Questions?
? Complete some random topics
? Lecture on Design of Dynamic Systems
(Signal / Response Systems)
? Recitation on HW#5?
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Dummy Levels and μ
?Before
?After
– Set SP2=SP3
– μ rises
– Predictions
unaffected
Factor Effects on the S/N Ratio
10
12
14
16
18
S
P
1
SP
2
SP
3
DA
1
DA
2
DA
3
CU
P
1
C
U
P2
C
U
P3
P
P1
P
P2
P
P
3
S
/
N Ra
t
i
o
(
d
B
)
Factor Effects on the S/N Ratio
10
12
14
16
18
S
P
1
SP
2
SP
3
DA
1
DA
2
DA
3
CU
P
1
C
U
P2
C
U
P3
P
P1
P
P2
P
P
3
S
/
N Ra
t
i
o
(
d
B
)
μ
μ
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Number of Tests
? One at a time
– Listed as small
? Orthogonal Array
– Listed as small
?White Box
– Listed as medium
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Linear Regression
? Fits a linear model to data
Y
i
β
0
β
1
X
i
.
ε
i
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Error Terms
? Error should be independent
– Within replicates
– Between X values
6
4
2
0
0 0.5 1 1.5 2
Population regression line
2
Population data points
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Error terms
MIT
Least Squares Estimators
? We want to choose values of b
o
and b
1
that
minimize the sum squared error
, SSE b
0
b
1
i
2
y
i
b
0
b
1
x
i
.
? Take the derivatives, set them equal to zero
and you get
b
1
i
x
i
mean x() y
i
mean y()
.
i
x
i
mean x()
2
b
0
mean y ()() b
1
.
mean x
MIT
Distribution of Error
? Homoscedasticity
? Heteroscedasticity
4
2
0
2
4
6
0 0.5 1 1.5 2
Population regression line
Population data points
Error terms
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Cautions Re: Regression
2 10
4
.
? What will result
210
4
if you run a linear
y
expo
k
1 10
4
regression on
0
0
0 2 4 6 8 10
these data sets?
0.012684 x
k
9.885085
28.521501
30
20
y
quad
k
1.369099
10
0
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1 2 3
0
50
Scatterplot of data
Estimated regression line
0 2 4 6 8 10
0.012684 x
k
9.885085
MIT
Linear Regression
Assumptions
1. The average value of the dependent variableY is a linear
function ofX.
2. The only random component of the linear model is the
error term
ε
. The values of X are assumed to be fixed.
3. The errors between observations are uncorrelated. In
addition, for any given value ofX, the errors are are
normally distributed with a mean of zero and a constant
variance.
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If The Assumptions Hold
? You can compute
confidence intervals on β
1
0.8
? You can test hypotheses
– Test for zero slope
β
1
=0
0.6
– Test for zero intercept
β
0
=0
0.7 0.75 0.8 0.85 0.9
Scatterplot of data
Estimated regression line
Upper prediction band
Lower prediction band
Prediction band for the mean of x
? You can compute
prediction intervals
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Design of Dynamic Systems
(Signal / Response Systems)
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Dynamic Systems Defined
“Those systems in which we want the system
response to follow the levels of the signal factor in
a prescribed manner”
– Phadke, pg. 213
Noise Factors
Response
Product / Process
Response
Signal Factor
Signal
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Control Factors
MIT
Examples of Dynamic Systems
? Calipers
? Automobile steering system
? Aircraft engine
? Printing
?Others?
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Static versus Dynamic
Static
? Vary CF settings
? For each row, induce
noise
? Compute S/N for each
row (single sums)
Dynamic
? Vary CF settings
? Vary signal (M)
? Induce noise
? Compute S/N for each
row (double sums)
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S/N Ratios for Dynamic Problems
Signals
Continuous
Digital
Continuous
Responses
Digital
C-C C-D
D-C D-D
Examples of each?
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Continuous - continuous S/N
? Vary the signal among
Response
discrete levels
y
β
? Induce noise, then compute
m n
∑∑
y
ij
M
i
β =
i=1 j=1
m n
2
∑∑
M
i
i=1 j=1
m n
σ
e
2
=
1
∑∑
( y
ij
?βM
i
)
2
M
mn ? 1
i=1 j=1
Signal Factor
β
2
η = 10log
10
σ
2
e
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C - C S/N and Regression
C-C S/N Linear Regression
() y
i
()x
i
β =
m n
∑∑
y
ij
M
i
mean x
.
mean y
∑∑
M
i
2
b
1
mean x
2
i=1 j=1
i
i= 1 j=1
i
m n
()x
i
m n
σ
e
2
=
1
∑∑
( y
ij
?βM
i
)
2
SSE b
0
b
1
, y
i
b
0
b
1
x
i
.
2
mn ? 1
i =1 j=1
i
β
2
η = 10log
10
σ
2
e
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Non-zero Intercepts
? Use the same formula
Response
for S/N as for the zero
y
α{
β
intercept case
? Find a second scaling
factor to
independently adjust β
and α
M
Signal Factor
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Continuous - digital S/N
Response
β
β
y
d
=1
y
d
=0
? Define some
on
y
continuous response y
? The discrete output y
d
is
off
a function of y
M
Signal Factor
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Temperature Control Circuit
? Resistance of
Response
β
β
Current in
R
T
=0
R
T
>0
Current in
on
thermistor decreases
R
T
with increasing
temperature
off
? Hysteresis in the
circuit lengthens life
R
3
Signal Factor
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System Model
? Known in closed form
R
T_ON
R
3
R
2
.
E
z
R
4
.
E
o
R
1
..
R
1
E
z
R
2
.
E
z
R
4
.
E
o
R
2
..
R
T_OFF
R
3
R
2
.
R
4
.
R
1
R
2
R
4
.
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Problem Definition
Noise Factors
R
3
Product / Process
Response
Signal Factor
R
T-ON
R
1
, R
2
, R
4
, E
o
, E
z
What if R
3
were also a noise?
What if R
3
were also a CF?
Control Factors
R
T-OFF
R
1
, R
2
, R
4
, E
z
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R
1
Results (Graphical)
R
2
R
4
E
z
35
35
30
25
35
30
25
35
30
25
,m η
ON
A
mean η
ON
level
,m η
OFF
A
mean η
OFF
level
30
25
20
20 20 20
1 2 3
1 2 3 1 2 3 1 2
level
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m 20 log β
ON
.
A,
level
mean 20 log β
ON
.
m 20 log β
OFF
.
A,
level
mean 20 log β
OFF
.
level
1 2 3
0
10
20
1 2 3
0
10
20
1 2 3
0
10
20
1 2 3
0
10
20
3
Results (Interpreted)
?R
T
has little effect on either S/N ratio
– Scaling factor for both βs
– What if I needed to independently set βs?
? Effects of CFs on R
T-OFF
smaller than for
R
T-ON
? Best choices for R
T-ON
tend to negatively
impact R
T-OFF
? Why not consider factor levels outside the
chosen range?
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Next Steps
? Homework #8 due 7 July
? Next session Monday 6 July 4:10-6:00
– Read Phadke Ch. 9 -- “Design of Dynamic
Systems”
– No quiz tomorrow
? 6 July -- Quiz on Dynamic Systems
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