/k 2DS00 Statistics 1 for Chemical Engineering lecture 5

/k Week schedule Week 1: Measurement and statistics Week 2: Error propagation Week 3: Simple linear regression analysis Week 4: Multiple linear regression analysis Week 5: Non-linear regression analysis

/k Detailed contents of week 5 intrinsically linear models well-known non-linear models non-linear regression –model choice –start values –Marquardt and Gauss-Newton algorithm –confidence intervals –hypothesis testing –residual plots –overfitting

/k Approaches to non-linear models 1.transformation to linear model 2.approximation of non-linear model by linear model (linearization through Taylor approximation) 3.non-linear regression analysis (numerically find parameters for which sum of squares is minimal) Remark: although 2) is often applied in the chemical literature, we strongly recommend against this procedure because there is no guarantee that it yields accurate results.

/k Intrinsically linear models Some non-linear models may be transformed into linear models transformed model must fulfil usual assumptions!

/k Examples of non-linear models exponential growth model Mitscherlich model inverse polynomial model logistic growth model Gompertz growth model Von Bertalanffy model Michaelis-Menten model

/k Exponential growth model non-linear additive error term intrinsically linear multiplicative error term

/k Mitscherlich model horizontal asymptote determines speed of growth if monomolecular model

/k Inverse polynomial model Slow convergence to asymptote 1/  1

/k Logistic (autoclytic) growth model  0 is horizontal asymptote

/k Gompertz growth model Logarithm of Gompertz curve is monomolecular curve horizontal asymptote determines growth speed

/k Von Bertalanffy model Special cases of this general model are: m=0 en : monomolecular model m=2 en : logistic model m  1: Gompertz model

/k Michaelis-Menten model This model is often used to describe diffusion kinetics Watch out for overfitting in model with many parameters.

/k Marquardt algorithm Non-linear regression requires numerical search for parameter values that minimise error sum of squares. Most important algorithms: 1. Gauss-Newton algorithm (uses 1 st -order approximation; may overshoot minimum) 2. steepest descent algorithm (searches for direction with largest downhill slope; may be slow) 3. Marquardt algorithm (switches according to situation between above mentioned algorithm)

/k Gauss algorithm

/k Marquardt algorithm Choice between both methods is determined by Marquardt parameter :  0  algorithm approaches to Gauss-Newton    algorithm approaches to steepest descent The Marquardt algorithm is (deservedly) the most used method in practice.

/k Numerical search for minimum of error sum of squares local minimum true minimum Where should we start the numerical search?

/k Choice of start values inspect data and use interpretation of parameters in model –parameter is related to value of asymptote –model value at certain setting use linear regression to obtain approximations to parameter values –transform model to linear model –approximate model by linear model

/k Possible causes for non-convergence model does not match data badly determined numerical derivatives overfitting: –model has too many parameters –some model parameters have almost same function

/k Important issues in non-linear regression analysis carefully consider choice of model choose starting values that relate to the model at hand experiment with different starting values to prevent convergence to local minimum watch out for overfitting

/k Fritz and Schluender equation: start values for a and b For C2=0, this reduces to Use first 10 measurements (i.e., those with C2=0) to obtain start values for a and b.

/k Fritz and Schluender equation: other initial values

/k Examination bring your notebook Monday October 6, – in Paviljoen J17 and L10 (not Auditorium) clean copy of Statistisch Compendium is allowed contents: –one exercise on error propagation – three statistical analyses to be performed on your notebook