Parameter Estimation Quantitative Definition of “Consistent”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. θbestfit at θlocalminima of χ2(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” Nreplicates times Yi(j) j = 1,Nreplicates()∑∑∑===⎟⎟⎠⎞⎜⎜⎝⎛−=−−==datareprepNiiidatairepNjdataijiirepNjjidataiYYNYYNYY12model212)(1)()(1σθχσ Quantitative Definition of “Consistent” calc χ2(θ; Ydata) Prob(measure Ydata with χ2 ≥ χ2expt) {if small, unlikely that our model is right} Î ⎟⎠⎞⎜⎝⎛Γ⎟⎟⎠⎞⎜⎜⎝⎛Γ=⎟⎠⎞⎜⎝⎛Γ∫∞−−22,2222expt2N2122exptvxvdtvetxtv ν = Ndata - Nparams_adjustedGAMMA FUNCTION Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].If ⎟⎠⎞⎜⎝⎛Γ⎟⎟⎠⎞⎜⎜⎝⎛Γ22,22vvχ < 0.01 Æ very unlikely model is consistent with data χ2(θ) < χ2max for consistency If you have more data, you have more confidence. Need lots more data than number of θ’s. 10.34, Numerical Methods Applied to Chemical Engineering Lecture 25 Prof. William Green Page 2 of 3 Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. P(χ2) Ndata χ2expt [χ2] If = : consistent If >> : inconsistent MatLab: chi2cdf.m neglect χ2(θ) = χ2(θbestfit) + ½(θ – θbest)T*H(θbest)*(θ – θbest) + O(∆θ3) H ≈ JTJ Jin = bestniYθθ∂∂model ∆H ∆H1 x k k1 ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ΔdataTHk,112expχ θthat is inconsistent with data consistent inconsistent Figure 1. Chi-squared distribution tests.Figure 2. An example of two parameter fitting.∆H k χ2(θ) = χ2m=x δ χ2 = χ2best+ 2χ2 = χ2best+ 1 \ Figure 3. Contours around best fit. People want these contours to be circles Range of parameters that are acceptable: ∆H ± δ(∆H) k = kbest + δk Covariance Matrix H kinetics experiments give the ellipses P(θ) ∆Hdist. on ∆k Equilibrium Concentrati10.34, Numerical Methods Applied to Chemical Engineering Lecture 25 Prof. William Green Page 3 of 3 Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Figure 4. Each experiment tells you about different “cuts” or ellipses and where they all intersect is the answer. Experiment gives the values of the horizontal cuts. BAYESIAN: store p(θ) STORE ALL THE DATA: Thermochemistry Active Tables,
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