response surfaces max(s( )) marco lattuada swiss federal institute of technology - eth institut...
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Response SurfacesResponse Surfacesmax(S(max(S())))
Response SurfacesResponse Surfacesmax(S(max(S())))
Marco Lattuada
Swiss Federal Institute of Technology - ETHInstitut für Chemie und BioingenieurwissenschaftenETH Hönggerberg/ HCI F135 – Zürich (Switzerland)
E-mail: lattuada@chem.ethz.chhttp://www.morbidelli-group.ethz.ch/education/index
Marco Lattuada
Swiss Federal Institute of Technology - ETHInstitut für Chemie und BioingenieurwissenschaftenETH Hönggerberg/ HCI F135 – Zürich (Switzerland)
E-mail: lattuada@chem.ethz.chhttp://www.morbidelli-group.ethz.ch/education/index
Marco Lattuada– Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 2
Response SurfacesResponse 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 variables2. find the optimal conditions for the response
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 variables2. find the optimal conditions for the response
ExampleConsider 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.
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 3
"COVT" Approach"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.
"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 EngineersResponse Surfaces – Page # 4
"COVT" Approach"COVT" Approach
T
P
5060
70
80
Contour curves for the yield (Y)
Starting point
Design of experiments
Optimum ???
Optimum !!!COVT approach assumes the effect of changing one parameter per time is independent of the effect in changes of the others. This is usually NOT true.
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 5
22kk Factorial Design Factorial Design
T
P
5060
70
80
Contour curves for the yield (Y)
Design of experiments
Optimum-1
-1
+1
+1
P 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
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 6
Example: Plastic WrapExample: 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...)
Objective
Maximize the strength of the plastic wrap
Suggested starting conditions: T = 140°C P = 4.0%
Optimal conditions: T = 216°C P = 9.2%
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...)
Objective
Maximize the strength of the plastic wrap
Suggested starting conditions: T = 140°C P = 4.0%
Optimal conditions: T = 216°C P = 9.2%
2 220 0.85 1.5 0.0025 0.375 0.025Y T P T P T P
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 7
True Response SurfaceTrue Response Surface
20
30
30
40
40
50
50
50
50
50
60
6060
60
60
60
70
70
70
70
70
70
75
75
75
75
78
78
PE Additive (%)
Te
mp
era
ture
(oC
)
0 5 10 15100
120
140
160
180
200
220
240
260
280
300
Starting point
Optimum
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 8
2222 Factorial Design Factorial Design
T PCoded
t p
120 2 -1 -1
120 6 -1 +1
160 2 +1 -1
160 6 +1 +1
140
20
Tt
4
2
Pp
0 1 2Y b b p b t Initial regression model:
-1
-1
+1
+1
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 9
2222 Factorial Design Factorial Design
-1
0
1
-1
0
1
45
50
55
60
65
70
75
pt
YTrue Response Surface
Contour Curves of Y
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 10
2222 Factorial Design Factorial Design
-1.5-1
-0.50
0.51
1.5
-1
0
1
45
50
55
60
65
70
75
pt
YExperimental Responses
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 11
First Order RegressionFirst Order Regression
-1.5-1
-0.50
0.51
1.5
-1
0
1
40
50
60
70
80
pt
Y
Regressed Response
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 12
2222 Factorial Design with Center Point Factorial Design with Center Point
T PCoded
t p
120 2 -1 -1
120 6 -1 +1
160 2 +1 -1
160 6 +1 +1
140 4 0 0
140
20
Tt
4
2
Pp
0 1 2Y b b p b t Initial regression model:
-1
-1
+1
+1
Central point does not influence the regression of the
slope
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 13
2222 Factorial Design with Center Point Factorial Design with Center Point
-1.5-1
-0.50
0.51
1.5
-1.5-1
-0.50
0.51
1.540
50
60
70
80
pt
Y
True Response Surface
Contour Curves of Y
Experimental Responses
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 14
First Order RegressionFirst Order Regression
-1.5-1
-0.50
0.51
1.5
-1.5-1
-0.50
0.51
1.540
50
60
70
80
pt
YRegressed Response
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 15
CurvatureCurvature
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)
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)
2
1 1, var
2 2curv center centercenter
s t n Yn
center corner curvC E Y E Y s
C- C+
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 16
Tukey-Ancombe PlotTukey-Ancombe Plot
50 55 60 65 70 75-4
-3
-2
-1
0
1
2
3
Y Regressed
Re
sid
ua
ls
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 17
Steepest Ascent DirectionSteepest Ascent Direction
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
p
t
Contour Lines of theRegressed 1st order Surface
Steepest Ascent Direction
Steepest Ascent Direction
Experimental Points
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 18
Steepest Ascent DirectionSteepest Ascent Direction
-1.5-1
-0.50
0.51
1.5
-1
0
1
45
50
55
60
65
70
75
80
pt
Y
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 19
Monodimensional SearchMonodimensional Search
20
20
30
30
40
4050
50
50
50
50
60
60
60
6060
60
70
70
70
70
70
70
75
75
75
75
7878
P
T
0 5 10 15100
120
140
160
180
200
220
240
260
280
300
Steepest Ascent Direction
Monodimensional search
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 20
Monodimensional SearchMonodimensional Search
0 1 2 3 4 5 6 7 8 964
66
68
70
72
74
76
78
80
Step Number
Y
Experimental points
True Response along thesteepest ascent direction
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 21
2222 Factorial Design with Center Points Factorial Design with Center Points
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
p
t
Maximum from themonodimensional search
Maximum of responsesurface (unknown)
New 2k Factorial Design
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 22
2222 Factorial Design with Center Points Factorial Design with Center Points
-1.5-1
-0.50
0.51
1.5
-1.5-1
-0.50
0.51
1.570
72
74
76
78
80
pt
Experimental Points
True response surface
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 23
First Order RegressionFirst Order Regression
-1.5-1
-0.50
0.51
1.5
-1.5-1
-0.50
0.51
1.570
72
74
76
78
80
pt
Regressed Response
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 24
Central Composite DesignCentral Composite Design
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
p
t2k Factorial Design
r = 21/2
Central Composite
Design
At least three different levels are needed to estimate a second order function
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 25
Central Composite DesignCentral Composite Design
-1.5-1
-0.50
0.51
1.5
-1
0
1
70
72
74
76
78
80
pt
Y
2 20 1 2 3 4 5Y p t p t pt
Check Jacobian of the regression to verify the nature of the stationary point
Alessandro Butté – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 26
Central Composite DesignCentral Composite Design
73 74 75 76 77 78 79-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Regressed Y
Re
sid
ua
ls
Tukey-Ancombe Plot
Principal Component Analysis (PCA)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
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
Marco Lattuada – Statistical and Numerical Methods for Chemical EngineersResponse Surfaces – Page # 27
n
p
A n
k
X
PCAPCA
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
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 EngineersResponse Surfaces – Page # 28
PCA with MatlabPCA 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.
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 EngineersResponse Surfaces – Page # 29
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