Stat 321 - Modeling of nonlinear responses Sections 11-1 and 11-2: Response surface
methods
How would you find the highest point in Minnesota? You
can use a vehicle that measures your altitude, but you can't
leave the ground (can't fly over).
Response surface methods - Including steepest ascent
Steepest Ascent:
1. Explore your neighborhood to find the
steepest slope.
2. Climb the hill until you go over a top.
3. Explore the new neighborhood to find the
steepest slope, or if you are at a curved
top.
4. If curved, look around carefully for the true
top. If not, climb the hill in the new
direction until you go over a top. Then go
back to 3.
More mathematically:
1. For k factors, Perform a 2k factorial experiment with n
center points to check for curvature.
Is curvature significant?
Yes- go to Step 4.
No- go to Step 2.
2. Determine the direction of steepest ascent in terms of
a vector representing relative changes in the control
variables’ settings.
3. Run experiment trials up the ascent path. Does the
latest response value improve over the previous trial?
Yes - go to Step 3.
No- go to Step 1.
4. Define a central composite design in the area of the
peak or last trials, and run the new trials.
5. Characterize the response surface with a quadratic
model (allows for curvature, has X2 terms ).
How do we find the steepest ascent path to the maximum
response?
An example:
Response variable AY41 is based on the Box and Draper
(1987) printing experiment.
The true response surface (unknown as we start the
experiment) for response variable AY41 is
AY41=(327.6+177X1+109.4X2+131.5X3 + 32X12 - 22.4X22 29.1X32 +66X1X2 + 75.5X1X3 + 43.6X2X3)/10
RSM Step 1 - Perform a 2k factorial experiment with n
center points:
This is done with a simple experiment to determine
regression coefficients for each of 3 X’s on AY41. With three
control variables (X’s), a 23 factorial is defined in a selected
area of the control variable space. Three center points are
included to test for curvature of the response. Replication at
the center allows for an estimate of experimental error to be
used in testing the curvature effect.
Say initial levels of X variables are low = -.5, high = 0.
Generate an experiment from Design Expert as a full 23 with
3 "center points per block."
Some simulated results are shown.
Trials for steepest ascent method on AY41
^
^
^
^
^
^
^
^
^
^
^
^ First
experiment
X1 X2
-0.5 -0.5
0 -0.5
-0.5
0
0
0
-0.5 -0.5
0 -0.5
-0.5
0
0
0
-0.25 -0.25
-0.25 -0.25
-0.25 -0.25
X3
-0.5
-0.5
-0.5
-0.5
0
0
0
0
-0.25
-0.25
-0.25
AY41 Trial
15.64
1
20.64
2
16.84
3
23.18
4
22.94
5
29.07
6
22.81
7
30.27
8
23.95
9
21.88 10
23.99 11
For analysis of this preliminary ascent experiment, it is
customary to include the main effects in the model, ignore
interactions, and test for curvature.
ANOVA for Selected Factorial Model
Source
Model
Curve
Resid
Lack Fit
Pure Err
Cor Tot
Sum Sq
184.19
0.78
5.33
2.42
2.91
190.30
DF
3
1
6
4
2
10
Mean Sq
61.40
0.78
0.89
0.60
1.46
-
F
69.14
0.88
0.41
-
Prob > F
< 0.0001
0.3836
0.7945
-
The high significance level for curvature (“Curve”) of .3836 means that curvature
of the response surface in this region of the X values is not significant.
Factor
Int
A-x1
B-x2
C-x3
CtrPt
Coef Est
22.67
3.12
0.60
3.60
0.60
DF
1
1
1
1
1
Std Err
0.33
0.33
0.33
0.33
0.64
t
Prob > |t|
9.35
1.80
10.80
0.94
< 0.0001
0.1212
< 0.0001
0.3836
The coded factor results from Design Expert are
AY41 = 22.67 +3.12*A +0.60*B +3.60*C
The actual prediction equation comes from converting the (-1, +1)
coded levels to the range of (-.5, 0) for each X. Since the coded
range was 4 times as wide as the actual range (2 vs. .5), the
coefficients are multiplied by 4 to get the prediction equation for
actuals below - based on the 11-trial experiment:
AY41 = 29.99 +12.47X1 +2.41 X2 +14.39 X3 .
The direction vector in the control variable space is (12.5,
2.4, 14.4). That is, steps taken in this (X1,X2,X3) direction
will find the response increasing with greatest slope – that’s
steepest ascent. Steps can be taken of any size, as long as
the ratio of (X1,X2,X3) of the direction or ascent vector is
preserved.
So any multiple of this vector can be added to the center
point of the initial design to determine the next experiment
point. The next point should be slightly outside of the initial
space.
We can normalize the ascent vector above to make the
biggest factor step at X3 = 1. We divide the whole vector by
the maximum component, 14.4. The normalized values for
X1 and X2 are 12.5/14.4 = 0.866 and 2.4/14.4 = 0.167,
respectively. Add this vector to the original center point,
leading to the result:
(0.616, -0.083, 0.75) from (-.25, -.25, -.25) + (.866, .167, 1)
The simulated value for the next trial is:
X1
X2
X3
AY41
Trial
0.62 -0.08 0.75 61.71
12
This result exceeds all previous trials, so we add the same
ascent vector to the previous trial location. That is (0.616, 0.083, 0.75) + (.866, .167, 1)
X1
X2
1.48 0.08
X3
AY41
1.75 89.63
Trial
13
Check the arithmetic for x1, x2, x3.
The step-size can be determined by the experimenter. When we
pass a maximum (we see the response decrease) or reach a physical
X variable boundary, we do another experiment like the first 11-trial
experiment.
In the case of this example, we should continue along the same
direction, but the next trial would put X1 and X3 out of their practical,
usable range, at X1=X3=2. Then the investigation would continue
along these boundaries to find a maximum response.
Suppose we could do the next trial along the ascent path at
2.346 0.247 2.75
Suppose the result is less than the last result, at AY41= 85. We would then return
to Step 1 and fill in another 11-trial experiment, with the settings for our best
result,
1.48 0.08
1.75
as the center point, and the settings from the last and previous trials as corner
points:
0.62 -0.08
0.75
2.35 0.247
2.75
Input these two rows as high and low levels to Design Expert to get the new 11trial experiment (including 3 center points):
Three of these trials have already been performed - now run the other
8. If we detect curvature in the response surface on a 11-trial
experiment, we assume we have surrounded a maximum, and we
“augment the design” to measure curved effects, as directed in Step
2 of the algorithm. A very efficient experiment (that Design Expert
easily generates) is the Central Composite (Rotatable) Design (CCD
or CCRD). If there is no curvature, we are directed to find a new
ascent direction.
To accomplish a response surface experiment under Design Expert,
with the previous 11-trial experiment active in Dex, select Design
Tools from the main menu, then Augment Design, and take the
defaults for the Central Composite design. Optimal trials will be
added to allow the fitting of a second order model to all of the
accumulated data.
Try homework problems 11-1, 11-2, and 11-4 on p. 500.