Nonlinear Regression Analysis with Fitter Software Application

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

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

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1. Introduction
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Linear and Non-linear Regressions
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Close relatives?
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2. Thermo Gravimetric Analysis Example
Lets see it!
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TGA Experiment and Data
TGA Experiment
TGA Data
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TGA Example Variables
Small sizeproblem!
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Plasticizer Evaporation Model
Diffusion isnot relevant!
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Fitter Worksheet for TGA Example
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Service Life Prediction by TGA Data
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3. NLR Basics
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Data and Errors
Weight isan effectiveinstrument!
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Model f(x,a)
Presentation at worksheet
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Data Model Prepared for Fitter
Apply Fitter!
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Objective Function Q(a)
Objective function Qis a sum of squaresand may
be more
Parameter estimates
Weighted variance estimate
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Very Important Matrix A
Matrix A is the cause of troubles..
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Quality of Estimation
Matrix A is the measure of quality!
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Search by Gradient Method
Matrix A is the key to search!
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4. Multicollinearity
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Multicollinearity View
Multicollinearity is degradation of matrix A
Objective function Q(a)
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N(A)
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Multicollinearity Source
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Data Model Preprocessing
((a b) c) d ? a (b (c d)) as
110 20 1
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Example The Arrhenius Law
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Derivative Calculation and Precision
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5. Prediction
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Reliable Prediction
Forecast shouldinclude uncertainties!
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Nonlinearity and Simulation
Non-linear models callfor special methods
ofreliable prediction!
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Prediction Example
Model ofrubber aging
Accelerated aging tests
Upper confidence limits
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6. Testing
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Hypotheses Testing
Test statistics x is compared with critical
value t (a)
Test dont prove a model! It just shows that
the hypothesis is accepted or rejected!
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Lack-of-Fit and Variances Tests
These hypotheses are based on variances and they
cant be tested without replicas!
Lack-of-Fitis a wily test!
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Outlier and Series Tests
These hypotheses are based on residuals and they
can be tested without replicas
Series test isvery sensitive!
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7. Bayesian Estimation
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Bayesian Estimation
How to eat awayan elephant?Slice by slice!
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Posterior and Prior Information. Type I
The same error ineach portion of data!
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Posterior and Prior Information. Type II
Different errors in each portion of data!
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8. Conclusions
Mysterious Nature
LR Model
NLR Model
Thankyou!