How Do We Know if Model and Data are really Consistent?
χ2(θ) = minθ ||(yexpt-ymodel(θ))/σ||2
θ are the parameters we are adjusting to fit the data
• Uncertainty σ = σmodel + σexpt
• If minθ χ2 > tolerance it is very likely that the model is inconsistent with the data.
• USUALLY χ2 HAS MANY LOCAL MINIMA
• Need globally optimal choice of adjustable parameters θ to be 100% sure model & data are inconsistent. It just
became possible to guarantee global minimum in χ2(p) for nonlinear ODE kinetic models:
Singer et al., J. Phys. Chem. A (2006).
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Example 1: Local optimization suggested model is wrong, but global optimization finds good fit
Optical Density @316 nm
Time (μs)
Best-possible least-squares fit
Fits
found using
conventional least-squares
Experiment: laser-initated oxidation of cyclohexadiene; most of the signal comes from cyclohexadienyl radical (C6H7) in partial equilibrium with
two isomers of C6H7OO. For details see Taylor et al. J.Phys.Chem.A (2004), Singer et al. J.Phys.Chem.A (2006)
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Example 2: Global Optimization Proves Model & Data Inconsistent
Data Global optimum Local optimum (χ2 = 500)
Optical Density
Time (μs)
Simple least-squares fit: close but not quite
right (only 5% chance of being consistent with the data)
The best possible choice of parameters
for this particular chemistry model: not much better than the local fit (only 16% chance that data
would be this far off if the model were correct).
Singer et al.,
J. Phys. Chem. A (2006) Conclusion: this chemistry model is not correct. (Could strengthen this conclusion by tightening the error bars…)
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10.34 Numerical Methods Applied to Chemical Engineering
Fall 2015
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