Matrix Experiments
Using Orthogonal Arrays 16.881
Robust System Design
MIT
Comments on HW#2 and Quiz #1
Questions on the Reading
Quiz
Brief Lecture
Paper Helicopter Experiment
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Robust System Design
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Learning Objectives
• Introduce the concept of matrix experiments
• Define the balancing property and
orthogonality
• Explain how to analyze data from matrix
experiments
• Get some practice conducting a matrix
experiment
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Static Parameter Design and the P-Diagram
Noise Factors
Induce noise
Product / Process
Signal Factor
Hold constant
for a “static”
experiment
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Response
Optimize
Control Factors
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Vary according to
an experimental plan
MIT
Parameter Design Problem
• Define a set of control factors (A,B,C…)
• Each factor has a set of discrete levels
• Some desired response η (A,B,C…) is to be
maximized
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Full Factorial Approach
• Try all combinations of all levels of the
factors (A1B1C1, A1B1C2,...)
• If no experimental error, it is guaranteed to
find maximum
• If there is experimental error, replications
will allow increased certainty
• BUT ... #experiments = #levels#control factors
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Additive Model
• Assume each parameter affects the response
independently of the others
η ( Ai , B j , Ck , Di ) = µ + ai + b j + ck + d i + e
• This is similar to a Taylor series expansion
f (x, y) = f (xo , yo ) +
16.881
∂f
∂x
⋅ (x − xo ) +
x= xo
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∂f
∂y
⋅ ( y − yo ) + h.o.t
y = yo
MIT
One Factor at a Time
Expt.
No.
1
2
3
4
5
6
7
8
9
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Control Factors
A
B
C
D
2
1
3
2
2
2
2
2
2
2
2
2
1
3
2
2
2
2
2
2
2
2
2
1
3
2
2
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2
2
2
2
2
2
2
1
3
η1
η2
η3
η4
η5
η6
η7
η8
η9
MIT
Orthogonal Array
Expt.
No.
1
2
3
4
5
6
7
8
9
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A
1
1
1
2
2
2
3
3
3
Control Factors
B
C
1
2
3
1
2
3
1
2
3
1
2
3
2
3
1
3
1
2
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D
1
2
3
3
1
2
2
3
1
η1
η2
η3
η4
η5
η6
η7
η8
η9
MIT
Notation for Matrix Experiments
L9 (34)
Number of experiments
Number of levels
Number of factors
9=(3-1)x4+1
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Why is this efficient?
• One factor at a time
– Estimated response at A3 is η3=µ+a3+e3
• Orthogonal array
– Estimated response at A3 is
η3=µ+a3+1/3(e7+ e7+ e7)
– Variance sums for independent errors
– Error variance ~ 1/replication number
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Robust System Design
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Factor Effect Plots
Factor Effects on the Mean
Time (sec)
20
15
A1
A2
A3
B3
C1 C2 C3
D1 D2
D3
B1
10
5
A1 A2 A3
B1 B2 B3
C1 C2 C3
D1 D2 D3
Factor Effects on the Variance
25
20
15
10
5
0
A2
A1
A3
A1 A2 A3
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B2
0
Time (sec)
• Which CF
levels will
you choose?
• What is your
scaling
factor?
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B1 B2 B3
B1 B2 B3
C1
C2
C1 C2 C3
D1 D2
C3
D3
D1 D2 D3
MIT
Prediction Equation
Time (sec)
Factor Effects on the Variance
25
20
15
10
5
0
A2
A1
A1 A2 A3
A3
B1 B2 B3
B1 B2 B3
C1
C2
C1 C2 C3
D1 D2
C3
µ
D3
D1 D2 D3
η ( Ai , B j , Ck , Di ) = µ + ai + b j + ck + d i + e
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Robust System Design
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Inducing Noise
Expt.
No.
1
2
3
4
5
6
7
8
9
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Control Factors
A
B
C
D
1
1
1
2
2
2
3
3
3
1
2
3
1
2
3
1
2
1
1
2
3
2
3
1
3
1
2
1
2
3
2
1
2
3
3
1
η1
η2
η3
η
4
η5
η6
η7
η8
η9
Robust System Design
{
Expt.
No.
1
2
Noise
Factor
N
1
2
MIT
Analysis of Variance (ANOVA)
• ANOVA helps to resolve the relative
magnitude of the factor effects compared to
the error variance
• Are the factor effects real or just noise?
• I will cover it in Lecture 7
• You may want to try the Mathcad “resource
center” under the help menu
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