1. Response Surfaces max(S(  )) 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 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

2. Marco Lattuada– Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 2 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 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.

3. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 3 "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.

4. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 4 "COVT" Approach T P 50 60 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.

5. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 5 2 k Factorial Design T P 50 60 70 80 Contour curves for the yield (Y) Design of experiments Optimum +1 PTY 40 +178 +159 +1 58 Initial investigation starts with a first order approximation of the response surface

6. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 6 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...) Objective Maximize the strength of the plastic wrap Suggested starting conditions:T = 140°CP = 4.0% Optimal conditions:T = 216°CP = 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°CP = 4.0% Optimal conditions:T = 216°CP = 9.2%

7. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 7 True Response Surface Starting point Optimum

8. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 8 2 2 Factorial Design TP Coded tp 1202 1206+1 1602+1 1606+1 Initial regression model: +1

9. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 9 2 2 Factorial Design True Response Surface Contour Curves of Y

10. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 10 2 2 Factorial Design Experimental Responses

11. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 11 First Order Regression Regressed Response

12. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 12 2 2 Factorial Design with Center Point TP Coded tp 1202 1206+1 1602+1 1606+1 140400 Initial regression model: +1 Central point does not influence the regression of the slope

13. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 13 2 2 Factorial Design with Center Point True Response Surface Contour Curves of Y Experimental Responses

14. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 14 First Order Regression Regressed Response

15. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 15 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) 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) C - C +

16. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 16 Tukey-Ancombe Plot

17. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 17 Steepest Ascent Direction p t Contour Lines of the Regressed 1st order Surface Steepest Ascent Direction Experimental Points

18. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 18 Steepest Ascent Direction

19. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 19 Monodimensional Search Steepest Ascent Direction Monodimensional search

20. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 20 Monodimensional Search Experimental points True Response along the steepest ascent direction

21. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 21 2 2 Factorial Design with Center Points Maximum from the monodimensional search Maximum of response surface (unknown) New 2 k Factorial Design

22. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 22 2 2 Factorial Design with Center Points Experimental Points True response surface

23. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 23 First Order Regression Regressed Response

24. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 24 Central Composite Design 2 k Factorial Design r = 2 1/2 Central Composite Design At least three different levels are needed to estimate a second order function

25. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 25 Central Composite Design Check Jacobian of the regression to verify the nature of the stationary point

26. Alessandro Butté – Statistical and Numerical Methods for Chemical Engineers Response Surfaces – Page # 26 Central Composite Design Tukey-Ancombe Plot

27. 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 Engineers Response Surfaces – Page # 27 n p A n k X

28. 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 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

29. 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. 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