Non-linear Regression Analysis with Fitter Software Application

1. Non-linear Regression Analysiswith Fitter Software Application Alexey Pomerantsev Semenov Institute of Chemical PhysicsRussian Chemometrics Society

2. Agenda • Introduction • TGA Example • NLR Basics • Multicollinearity • Prediction • Testing • Bayesian Estimation • Conclusions

3. 1. Introduction

4. Linear and Non-linear Regressions 2 Close relatives?

5. 2. Thermo Gravimetric Analysis Example Let’s see it!

6. This is Experiment! Not a Hell of Flame! TGA Experiment and Data TGA Experiment TGA Data

7. TGA Example Variables Small sizeproblem!

8. Plasticizer Evaporation Model Diffusion isnot relevant!

9. Fitter Worksheet for TGA Example

10. Service Life Prediction by TGA Data

11. 3. NLR Basics

12. Data and Errors Weight isan effectiveinstrument!

13. Rathercomplexmodel! Model f(x,a) Presentation at worksheet

14. Values Response Predictor Comment Weight Fitting Equation Parameters Data & Model Prepared for Fitter Apply Fitter!

15. Objective Function Q(a) Objective function Qis a sum of squaresand may be more… Parameter estimates Weighted variance estimate

16. Very Important Matrix A Matrix A is the cause of troubles..

17. Quality of Estimation Matrix A is the measure of quality!

18. Search by Gradient Method Matrix A is the key to search!

19. 4. Multicollinearity

20. Multicollinearity: View Multicollinearity is degradation of matrixA Objective function Q(a) N(A) = 1 2 4 5 6 7

21. Multicollinearity: Source

22. Data & Model Preprocessing ((a + b) + c) + da + (b + (c + d)) as1+10 –20 = 1

23. Example: The Arrhenius Law

24. 14+2=16 10 8 10+2=12 8+2=10 10+0=10 12+0=12 10+2=12 12+2=14 12+2=14 1) Numerical calculation of difference derivatives Derivative Calculation and Precision

25. 5. Prediction

26. Reliable Prediction Forecast shouldinclude uncertainties!

27. Nonlinearity and Simulation Non-linear models callfor special methods ofreliable prediction!

28. Prediction: Example Model ofrubber aging Accelerated aging tests Upper confidence limits

29. 6. Testing

30. Fromexperiment Fromtheory Hypotheses Testing Test statistics x is compared with critical value t(a) Test don’t prove a model! It just shows that the hypothesis is accepted or rejected!

31. Variancesbyreplicas Replica 1 Replica 2 Lack-of-Fit and Variances Tests These hypotheses are based on variances and they can’t be tested without replicas! Lack-of-Fitis a wily test!

32. Acceptabledeviation Positiveresidual Negativeresidual Outlier and Series Tests These hypotheses are based on residuals and they can be tested without replicas Series test isvery sensitive!

33. 7. Bayesian Estimation

34. Bayesian Estimation How to eat awayan elephant?Slice by slice!

35. Posterior and Prior Information. Type I The same error ineach portion of data!

36. Posterior and Prior Information. Type II Different errors in each portion of data!

37. 8. Conclusions Mysterious Nature LR Model NLR Model Thankyou!