lec25_11082006 - 10.34 Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #25: Conclude Models vs. Data Parameter Estimation 1) Model definition/Formulation: choosing θ 2) Compile/Assess what you already knew before adjusting θ a. estimate parameters, error bars b. initial guess θ 3) Adjust θ a. Determine if Model is Consistent with Data: if inconsistent, you have learned something important b. θ bestfit at θ localminima of χ 2 ( θ ) 4) Refine p( θ ): narrow range for the parameters a. summarize what we have learned 4-STEP PROCESS IS ALSO CALLED: “LEAST-SQUARES FITTING” 1) Repeat measurement “i” N replicates times Y i (j) j = 1,N replicates ( ) = = = = = = data rep rep N i i i data i rep N j data i j i i rep N j j i data i Y Y N Y Y N Y Y 1 2 model 2 1 2 ) ( 1 ) ( ) ( 1 σ θ χ σ Quantitative Definition of “Consistent” calc χ 2 ( θ ; Y data ) Prob(measure Y data with χ 2 χ 2 expt ) {if small, unlikely that our model is right} Î Γ
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