Plan for the Session
• Questions?
• Complete some random topics
• Lecture on Design of Dynamic Systems
(Signal / Response Systems)
• Recitation on HW#5?
16.881
MIT
Dummy Levels and µ
• Before
16
µ
14
12
16.881
PP
1
PP
2
PP
3
C
UP
1
C
UP
2
C
UP
3
D
A1
D
A2
D
A3
10
SP
1
SP
2
SP
3
S/N Ratio (dB)
18
Factor Effects on the S/N Ratio
18
16
µ
14
12
PP
1
PP
2
PP
3
C
UP
1
C
UP
2
C
UP
3
D
A1
D
A2
D
A3
10
SP
1
SP
2
SP
3
– Set SP2=SP3
– µ rises
– Predictions
unaffected
S/N Ratio (dB)
• After
Factor Effects on the S/N Ratio
MIT
Number of Tests
• One at a time
– Listed as small
• Orthogonal Array
– Listed as small
• White Box
– Listed as medium
16.881
MIT
Linear Regression
• Fits a linear model to data
Yi β 0
16.881
β 1 .Xi
εi
MIT
Error Terms
• Error should be independent
– Within replicates
– Between X values
6
4
2
0
2
16.881
0
0.5
1
Population regression line
Population data points
Error terms
1.5
2
MIT
Least Squares Estimators
• We want to choose values of bo and b1 that
minimize the sum squared error
SSE b 0 ,
b 1
yi
b0
b 1 .x i
2
i
• Take the derivatives, set them equal to zero
and you get
xi
b1
mean( x ) . yi
mean( y )
i
xi
i
mean( x )
b0
mean( y )
b 1 .mean( x )
2
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
16.881
MIT
Cautions Re: Regression
• What will result
if you run a linear
regression on
these data sets?
4
2 .10
y expo
2 10
k 1 10
0
4
4
0
0
0.012684
50
28.521501
2
4
6
x
k
8
10
9.885085
30
20
y quad
k
10
0
1
2
3
Scatterplot of data
Estimated regression line
16.881
1.369099
0
0
2
0.012684
4
6
x
k
8
10
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.
16.881
MIT
If The Assumptions Hold
• You can compute
confidence intervals on β1
• You can test hypotheses
– Test for zero slope β1=0
– Test for zero intercept β0=0
• You can compute
prediction intervals
16.881
0.8
0.6
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
MIT
Design of Dynamic Systems
(Signal / Response Systems)
16.881
MIT
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
Signal Factor
Response
Signal
16.881
Control Factors
MIT
Examples of Dynamic Systems
•
•
•
•
•
Calipers
Automobile steering system
Aircraft engine
Printing
Others?
16.881
MIT
Static versus Dynamic
Static
• Vary CF settings
• For each row, induce
noise
• Compute S/N for each
row (single sums)
16.881
•
•
•
•
Dynamic
Vary CF settings
Vary signal (M)
Induce noise
Compute S/N for each
row (double sums)
MIT
S/N Ratios for Dynamic Problems
Signals
Continuous
Continuous
Digital
C-C
C-D
D-C
D-D
Responses
Digital
Examples of each?
16.881
MIT
Continuous - continuous S/N
Response
• Vary the signal among
y
discrete levels
• Induce noise, then compute
m
β=
β
n
∑∑ y M
ij
i=1 j=1
m n
∑∑ M i
i
2
i=1 j=1
σe
16.881
2
m n
1
=
( yij − βM i ) 2
∑∑
mn − 1 i=1 j=1
β2
η = 10log10 2
σe
M
Signal Factor
MIT
C - C S/N and Regression
Linear Regression
C-C S/N
m
β=
n
∑∑ y M
ij
i=1 j=1
m n
∑∑ M
xi
i
2
b1
σ e2 =
mean( y )
i
xi
i
i=1 j=1
m
mean( x ) . yi
mean( x )
2
i
n
1
( yij − βM i ) 2
∑∑
mn − 1 i =1 j=1
SSE b 0 , b 1
yi
b0
b 1 .xi
2
i
β2
η = 10log10 2
σe
16.881
MIT
Non-zero Intercepts
• Use the same formula
for S/N as for the zero
intercept case
• Find a second scaling
factor to
independently adjust β
and α
16.881
Response
y
β
α
{
M
Signal Factor
MIT
Continuous - digital S/N
• Define some
continuous response y
• The discrete output yd is
a function of y
Response
y
βon
yd=1
βoff
yd=0
M
Signal Factor
16.881
MIT
Temperature Control Circuit
• Resistance of
thermistor decreases
with increasing temperature
• Hysteresis in the
circuit lengthens life
Response
RT
Current in
RT>0
βon
βoff
Current in
RT=0
R3
Signal Factor
16.881
MIT
System Model
• Known in closed form
R 3 . R 2 . E z. R 4 E o . R 1
R T_ON
R 1 . E z. R 2 E z. R 4 E o . R 2
R T_OFF
16.881
R 3. R 2. R 4
R 1. R 2
R4
MIT
Problem Definition
Noise Factors
What if R3 were also a noise?
R1, R2, R4, Eo, Ez
What if R3 were also a CF?
Product / Process
Signal Factor
Response
R3
RT-ON
Control Factors
RT-OFF
R1, R2, R4, Ez
16.881
MIT
Results (Graphical)
R1
35
m η ON, A
level
mean η ON
30
Ez
R4
R2
35
35
35
30
30
30
25
25
25
m η OFF, A
level
mean η OFF
25
20
1
2
level
20
3
20
1
2
20
3
20
1
2
20
3
1
20
20
10
10
2
3
m 20. log β ON , A
level
mean 20. log β ON
m 20. log β OFF , A
level
10
10
mean 20. log β OFF
0
16.881
1
2
level
3
0
1
2
3
0
1
2
3
0
1
2
3
MIT
Results (Interpreted)
• RT 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 RT-OFF smaller than for
RT-ON
• Best choices for RT-ON tend to negatively
impact RT-OFF
• Why not consider factor levels outside the
chosen range?
16.881
MIT
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
16.881
MIT