Response Surfaces
max(S(b))
Marco Lattuada
Swiss Federal Institute of Technology - ETH
Institut für Chemie und Bioingenieurwissenschaften
ETH Hönggerberg/ HCI F135 – Zürich (Switzerland)
E-mail: lattuada@chem.ethz.ch
http://www.morbidelli-group.ethz.ch/education/index
Response Surfaces
Object:
Response surface method is a tool to:
1. investigate the response of a variable to the changes in a set of
design or explanatory variables
2. find the optimal conditions for the response
Example
Consider a chemical process whose yield is a function of
temperature and pressure:
Y = Y(T,P)
Suppose you do not know the function Y(T,P) but you want
to achieve the maximum yield Y.
Marco Lattuada– Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 2
"COVT" Approach
"Change One Variable per Time" approach
Preliminary remark
Experimentation is often started in a region of the parameter values which is far
from the optimal.
Example
Suppose a chemist wants to maximize the yield (Y) of his reaction by varying
temperature (T) and pressure (P). He does not know the true response surface, that is
Y = Y(T,P), and he starts investigating first the effect of temperature and then the
effect of pressure.
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 3
"COVT" Approach
T
50
60
Contour curves for the yield (Y)
70
Design of experiments
80
Optimum ???
Starting point
Optimum !!!
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 4
COVT approach assumes the effect of
changing one parameter per time is
independent of the effect in changes of
the others. ThisP
is usually NOT true.
2k Factorial Design
T
50
60
Contour curves for the yield (Y)
Design of experiments
70
80
+1
-1
T
Y
-1
-1
40
-1
+1
78
+1
-1
59
+1
+1
58
Initial investigation
starts with a first order
approximation of the
response surface
+1
Optimum
-1
P
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 5
P
Example: Plastic Wrap
Description
An engineer attempts to gain insight into the influence of the sealing temperature (T)
and the percentage of a polyethylene additive (P) on the seal strength (Y) of a certain
plastic wrap.
Response function (unknown to the engineer...)
Y   20  0.85  T  1.5  P  0.0025  T  0.375  P  0.025  T  P
2
2
Objective
Maximize the strength of the plastic wrap
Suggested starting conditions:
Optimal conditions:
T = 140°C
T = 216°C
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 6
P = 4.0%
P = 9.2%
True Response Surface
300
60
20
280
70
50
30
260
75
40
70
70
60
75
220
75
78
50
78
70
200
180
60
o
Temperature ( C)
240
Optimum
70
75
60
160
70
140
60
120
100
0
50
50
Starting point
60
40
30
50
5
10
PE Additive (%)
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 7
15
22 Factorial Design
t 
T  140
p 
P4
20
T
2
P
Coded
t
p
120
2
-1
-1
120
6
-1
+1
160
2
+1
-1
160
6
Initial regression model:
+1
+1
Y  b0  b1 p  b2 t
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 8
+1
-1
+1
-1
22 Factorial Design
True Response Surface
75
Contour Curves of Y
70
Y
65
60
55
50
45
1
1
0
t
0
-1
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 9
-1
p
22 Factorial Design
Experimental Responses
75
70
Y
65
60
55
50
45
1
1.5
1
0.5
0
0
t
-0.5
-1
-1
-1.5
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 10
p
First Order Regression
Regressed Response
80
Y
70
60
50
40
1
t
0
-1
-1.5
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 11
-1
-0.5
0.5
0
p
1
1.5
22 Factorial Design with Center Point
t 
T  140
p 
P4
20
2
T
P
120
2
Coded
+1
t
p
-1
-1
-1
120
6
-1
+1
160
2
+1
-1
160
6
+1
+1
140
4
0
0
Initial regression model:
Y  b0  b1 p  b2 t
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 12
+1
-1
Central point does
not influence the
regression of the
slope
22 Factorial Design with Center Point
True Response Surface
80
Contour Curves of Y
Y
70
Experimental
Responses
60
50
40
1.5
1
1.5
1
0.5
0.5
0
t
0
-0.5
-0.5
-1
-1
-1.5
-1.5
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 13
p
First Order Regression
Regressed Response
80
Y
70
60
50
40
1.5
1
1.5
1
0.5
0.5
0
t
0
-0.5
-0.5
-1
-1
-1.5
-1.5
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 14
p
Curvature
Center points can give us an indication about the curvature
of the surface and its statistical significance
Hypothesis: it there is no curvature and the linear model is an appropriate
description of the response surface over the region of interest, then the average of
the experimental responses in the center point and in the corner points is roughly
equal (within the standard deviation)


s curv  t  , n center 
 2

 1
1 
var  Ycenter   
 2 
 n center 2 
C  E  Y center   E  Y corner   s curv
CAlessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 15
C+
Tukey-Ancombe Plot
3
2
Residuals
1
0
-1
-2
-3
-4
50
55
60
65
Y Regressed
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 16
70
75
Steepest Ascent Direction
Experimental Points
1.5
Contour Lines of the
Regressed 1st order Surface
1
0.5
Steepest Ascent Direction
t
0
Steepest Ascent Direction
-0.5
-1
-1.5
-1.5
-1
-0.5
0
0.5
p
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 17
1
1.5
Steepest Ascent Direction
80
75
70
Y
65
60
55
50
45
1
0
t
-1
-1.5
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 18
-1
-0.5
0.5
0
p
1
1.5
Monodimensional Search
300
280
60
50
20
40
30
70
70
260
75
60
75
78
Monodimensional search
200
75
70
60
T
220
78
240
50
70
70
180
Steepest Ascent Direction
75
60
160
70
140
50
60
60
120
100
0
50
40
50
5
10
P
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 19
30
20
15
Monodimensional Search
80
78
76
Y
74
72
Experimental points
70
68
True Response along the
steepest ascent direction
66
64
0
1
2
3
4
5
Step Number
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 20
6
7
8
9
22 Factorial Design with Center Points
1.5
Maximum of response
surface (unknown)
1
Maximum from the
monodimensional search
t
0.5
0
New 2k Factorial Design
-0.5
-1
-1.5
-1.5
-1
-0.5
0
p
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 21
0.5
1
1.5
22 Factorial Design with Center Points
True response surface
80
Experimental Points
78
76
74
72
70
1.5
1
0.5
0
t
-0.5
-1
-1.5
-1.5
-1
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 22
-0.5
0.5
0
p
1
1.5
First Order Regression
Regressed Response
80
78
76
74
72
70
1.5
1
0.5
0
t
-0.5
-1
-1.5
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 23
-1.5
-1
-0.5
0.5
0
p
1
1.5
Central Composite Design
2k Factorial Design
1.5
Central
Composite
Design
1
t
0.5
At least three
different levels are
needed to estimate a
second order function
0
-0.5
-1
-1.5
-1.5
-1
-0.5
0
p
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 24
0.5
1
1.5
Central Composite Design
Y  b 0  b 1 p  b 2 t  b 3 p  b 4 t  b 5 pt
2
2
80
78
Y
76
74
72
70
1
0
t
-1
-1.5
-1
-0.5
0
0.5
1
1.5
p
Check Jacobian of the regression to
verify the nature of the stationary point
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 25
Central Composite Design
Tukey-Ancombe Plot
2.5
2
1.5
Residuals
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
73
74
76
75
Regressed Y
Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 26
77
78
79
Principal Component Analysis (PCA)
Consider a large sets of data (e.g., many spectra (n) of a chemical
reaction as a function of the wavelength (p))
Objective:
Data reduction: find a smaller set of (k) derived (composite)
variables that retain as much information as possible
p
n
A
Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 27
k
n
X
PCA
PCA takes a data matrix of n objects by p variables,
which may be correlated, and summarizes it by
uncorrelated axes (principal components or principal
axes) that are linear combinations of the original p
variables
New axes= new coordinate system.
Construct the Covariance Matrix of the data (which
need to be first centered), and find its eigenvalues and
eigenvectors
Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 28
PCA with Matlab
There are two possibilities to perform PCA with Matlab:
1) Use Singular Value Decomposition:
[U,S,V]=svd(data);
where U contains the scores, V the eigenvectors of the covariance
matrix, or loading vectors. SVD does not require the statistics toolbox.
2) Command [COEFF,Scores]=princomp(data), is a specialized command to
perform principal value decomposition. It requires the statistics toolbox.
Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers
Response Surfaces – Page # 29