Non Linear Regression Y i = f(  x i ) +  i Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg/ HCI F135 – Zürich (Switzerland) Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg/ HCI F135 – Zürich (Switzerland)

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 2 Puromycin Description: Puromycin is an antibiotic used by scientists in bio-research to select cells modified by genetic engineering. Mechanism of action: This is described by the Michaelis-Menten model for enzyme kinetics, which relates the initial velocity on an enzymatic reaction to the substrate concentration x trough the equation:

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 3 Puromycin Kinetics The model:

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 4 Model Linearization Puromycin Kinetics: Model Rearrangement: Linearized Model: Puromycin Kinetics: Model Rearrangement: Linearized Model:

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 5 Model Linearization Regression Line  1 =  2 =

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 6 Model Linearization Regression from linearized model  1 =  2 =

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 7 Puromycin Kinetics The model:   Linearized model is needed to estimate  2

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 8 Nonlinear Regression Object To minimize the objective function where n is the number of observations, y i the responses, x i is the vector of the observations,  the vector of the parameters and f(x i,  ) the nonlinear model function. It is possible to plot the objective function S(  ) as a function of the parameter values, in order to reveal the presence of a minimum. Object To minimize the objective function where n is the number of observations, y i the responses, x i is the vector of the observations,  the vector of the parameters and f(x i,  ) the nonlinear model function. It is possible to plot the objective function S(  ) as a function of the parameter values, in order to reveal the presence of a minimum.

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 9 Objective Function S(  ) Contour plot of S(q) Minimum Estimated value of  from linearization

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 10 Minimization of S(  ) Model linearization: where: so the residuals are: Search for minimum with Gauss-Newton method: Model linearization: where: so the residuals are: Search for minimum with Gauss-Newton method: J 0 = Jacobian

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 11 Gauss-Newton Method Applied to S(  ) Convergence path of Gauss- Newton Method (  1 ) opt = (  2 ) opt =

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 12 Nonlinear Regression Regression from linearized model

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 13 Ellipsoidal Confidence Region The ellipsoidal confidence region can be evaluate from the linearized model around the point, which is the vector of the parameters for which the objective function has a minimum. In practice, every vector of the parameters  which satisfies the following condition: is within the confidence interval, where n is the number of observations, p the number of parameters and s the standard deviation: The ellipsoidal confidence region can be evaluate from the linearized model around the point, which is the vector of the parameters for which the objective function has a minimum. In practice, every vector of the parameters  which satisfies the following condition: is within the confidence interval, where n is the number of observations, p the number of parameters and s the standard deviation:

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 14 Ellipsoidal Confidence Region 

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 15 True Confidence Region for Parameters The real confidence region can be estimated by plotting the region of space for which:

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 16 Matlab Nonlinear Regression Routine First, create a function providing the residuals for the n observation as a function of the parameter values: Then, use the routine 'nlinfit'; First, create a function providing the residuals for the n observation as a function of the parameter values: Then, use the routine 'nlinfit';

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 17 Tukey-Ancombe Plot

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 18 Normal Plot >> normplot(r)

Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Nonlinear Regressions – Page # 19 Matlab Estimation of Parameter CI Parameter confidence interval can be estimated by Matlab as follows: The confidence interval can be estimated using the following Matlab GUI: Parameter confidence interval can be estimated by Matlab as follows: The confidence interval can be estimated using the following Matlab GUI: