Response Surfaces max(S( )) Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg

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: [email protected]://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: [email protected]://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.


"COVT" Approach"COVT" Approach

T

P

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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.


22kk Factorial Design Factorial Design

T

P

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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


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


True Response SurfaceTrue Response Surface

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Starting point

Optimum


2222 Factorial Design Factorial Design

T PCoded

t p

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Pp

0 1 2Y b b p b t Initial regression model:

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YTrue Response Surface

Contour Curves of Y



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YExperimental Responses


First Order RegressionFirst Order Regression

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Regressed Response


2222 Factorial Design with Center Point Factorial Design with Center Point

T PCoded

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Pp

0 1 2Y b b p b t Initial regression model:

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Central point does not influence the regression of the

slope


2222 Factorial Design with Center Point Factorial Design with Center Point

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True Response Surface

Contour Curves of Y

Experimental Responses



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YRegressed Response


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+


Tukey-Ancombe PlotTukey-Ancombe Plot

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sid

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ls


Steepest Ascent DirectionSteepest Ascent Direction

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Contour Lines of theRegressed 1st order Surface

Steepest Ascent Direction


Experimental Points


Steepest Ascent DirectionSteepest Ascent Direction

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Y


Monodimensional SearchMonodimensional Search

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Monodimensional search


Monodimensional SearchMonodimensional Search

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Step Number

Y

Experimental points

True Response along thesteepest ascent direction


2222 Factorial Design with Center Points Factorial Design with Center Points

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Maximum from themonodimensional search

Maximum of responsesurface (unknown)

New 2k Factorial Design


2222 Factorial Design with Center Points Factorial Design with Center Points

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Experimental Points

True response surface



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Regressed Response


Central Composite DesignCentral Composite Design

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

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p

t2k Factorial Design

r = 21/2

Central Composite

Design

At least three different levels are needed to estimate a second order function



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Check Jacobian of the regression to verify the nature of the stationary point



73 74 75 76 77 78 79-2.5

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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


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.


Documents

Response Surfaces max(S( )) Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg