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 2 Compile Assess what you already knew before adjusting a estimate parameters error bars b initial guess 3 Adjust a Determine if Model is Consistent with Data if inconsistent you have learned something important b bestfit at localminima of 2 4 Refine p 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 N rep Yi data 2 Yi j 1 N rep N data i 1 N rep j Y data Yi model i i i Yi j Yi data 2 j 1 N rep 1 2 Quantitative Definition of Consistent calc 2 Ydata Prob measure Ydata with 2 2expt 2 xexpt 2 v xexpt 2 2 t e dt N v v 2 2 2 2 v t 1 2 2 if small unlikely that our model is right Ndata Nparams adjusted GAMMA 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 v 2 2 2 If v 2 0 01 very unlikely model is consistent with data 2 2max for consistency If you have more data you have more confidence Need lots more data than number of s If inconsistent If consistent 2 P 2expt Ndata 2 Figure 1 Chi squared distribution tests MatLab chi2cdf m inconsistent H1 k1 Tdata H 1 x 2 exp H consistent k1 k that is inconsistent with data neglect Figure 2 An example of two parameter fitting 2 2 bestfit best T H best best O 3 H JTJ Jin Yi model n best 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 25 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 2 2best 2 2 2best 1 H k 2 2m x Figure 3 Contours around best fit People want these contours to be circles Range of parameters that are acceptable H H k kbest k Covariance Matrix Equilibrium Concentration kinetics experiments give the ellipses Hdist P Experiment H gives the values of k the horizontal Figure 4 Each experiment tells you about different cuts or ellipses and cuts where they all intersect is the answer BAYESIAN store p STORE ALL THE DATA Thermochemistry Active Tables PrIMe 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 25 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
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