10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 22 Introduction Models vs Data Models vs Data Engineers think of practical problems and efficient solutions from the top down Scientists use a micro view and can neglect the big picture in the bottom up analysis Models are always wrong But experiments also never match more important p k Ymodel x x knobs parameters in model we can physically adjust all other we generally parameters know bounds that affect on results prior information cannot control about diffusion limit Figure 1 Normal distribution seldom have sampling capable of making Ydata 1 x P Ydata x Ydata 2 x true distribution curve Ydata Figure 2 Example of sampling Average Value Ydata Nexpts P Ydata data P Y Nexpts Y Y true 2 exp 2 2 mean 2 1 exp data Nexpts Y Nexpts mean data N exp ts Figure 3 Normal distribution curve showing 1 standard deviation Why do we compare the model to data Is The Model Consistent With The Data Ydata Ymodel mean means Inconsistent akin to confidence interval CI t mean 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 Model Discrimination Often more than 1 model If they are consistent would like to be able to pick one closer to the data or say that either model works fine Parameter Refinement How narrow can you make the range on Experimental Design Identify which i are not determined by data A few i often control the fit Some i cannot be determined well by experiment poorly conditioned matrices Introduction to Chi Squared Analysis Assume all error is Gaussian 2 N action Y n x n Y mod el x n i n2 mean 2 N data for the true model S D exp N exp ts parameter refinement 2 minimize 2 by changing experimental design derivatives of 2 with respect to Bayesian View Prior knowledge p posterior p Ydata More knowledge after experiment Use to narrow error bars Conditional Probability P E1 E 2 P E1 P E 2 E1 probability of E2 knowing E1 happened correlation if independent P E2 model prior Pposterior Ydata P Ydata P d d P Y P P Ydata normalize P Ydata probability of observing Ydata we really observed if are true values We do not know and exactly 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 22 Page 2 of 2 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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