Introduction toIntroduction toparameter optimizationparameter optimization

Sabine Beulke, Central Science Laboratory, York, UK

Kinetic Evaluation according to Recommendations by the FOCUS Work Group on Degradation Kinetics

Washington, January 2006

Curve fittingCurve fitting

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OptimizationOptimization

Least squares method:

Minimizes the sum of squared residuals (RSS)

Calculated line

Residual = deviation between calculated and measured data

Measured datapoint

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OptimizationOptimization

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Calculatecurve

Initial guess(starting value)

CalculateRSS

Modifyparameter 0

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Automatic optimizationAutomatic optimization

Stops when:

Convergence criteria are metComparison between RSS for actual and previous runs. Convergence reached if difference is smaller than user-specified difference

Termination criteria are metFor example, when maximum number of runs has been carried out (user-specified)

Good fit not guaranteed!

Non-uniquenessNon-uniqueness

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measured FOMC

M0 92.48 DT50 7.2alpha 957.220 DT90 24.1beta 10004.3

139.277 Residual Sum of Squares

M0 92.47 DT50 7.2alpha 6696.536 DT90 24.1beta 70030.3

139.120 Residual Sum of Squares

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Non-uniquenessNon-uniqueness

Parameter correlationParameters strongly related

Effects on RSS of changes in one parameter can be compensated by changes in another parameter

Inadequate modelFor example, selection of bi-phasic model not warranted if data follow SFO

Global versus local minimumGlobal versus local minimum

The optimisation may find a local “valley” in the RSS surface, but not the absolute, global minimum.

Different parameter combinations may be returned for different starting values.

Good fit not guaranteed!

From: http://www.ssg-surfer.com/

RSS as a function ofchanges in 2 parameters

FOCUS recommendationsFOCUS recommendations

Always evaluate the visual fit

Avoid over-parameterisation

Aim at finding reasonable starting values

Always use different starting values

Constrain parameter ranges if appropriate

Plausibility checks for parameters and endpoints

Stepwise fitting where necessary

Be aware of differences between software packages

Residual plot

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Goodness of fit - visual assessmentGoodness of fit - visual assessment

Concentration vs. time plot

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SFO

Goodness of fit - statistical criteriaGoodness of fit - statistical criteria

2 test

whereC = calculated valueO = observed value = mean of all observed valueserr = measurement error percentage

If calculated 2 > tabulated 2 then the model is not appropriate at the chosen level of significance

Error percentage unknown Calculate error level at which 2 test is passed

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1100err

Confidence in parameter estimates

Calculate e.g. from ModelMaker output

A parameter is significantly different from zero if p (t) < alpha

Others (e.g. model efficiency, F-test)

iparameteroferrordardtans

iparameterofestimateat

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Goodness of fit - statistical criteriaGoodness of fit - statistical criteria

FOCUS optimization procedureFOCUS optimization procedure

Initial guess(starting values)

Enter measureddata

Evaluate:Visual fitStatistics

ParametersEndpoints

Optimize

Select kinetic model& parameters

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