Models vs. Data Why do we compare the model to data? Introduction to Chi-Squared Analysis Bayesian View Conditional Probability10.34, 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. p(k)diffusion limit more important Ymodel(x(±δx), θ±δθ) “knobs” all other we generally parameters parameters know bounds in model we can that affect on θ physically adjust results prior information (cannot control) about θ Figure 1. Normal distribution. Ydata(1)(x) Ydata seldom have sampling capable of making Ydata(2)(x) true distribution curve P Average Value Figure 2. Example of sampling.<Ydata>Nexpts P P(<Y>Nexpts) <Y>Nexpts(<Ydata>,σdata)≈ σexp.data|Nexpts ()tsdatameanmeantrueNYYexp222exp21σσσπσ≈⎥⎥⎦⎤⎢⎢⎣⎡><−><− Why do we compare the model to data? • Is The Model Consistent With The Data? |<Ydata> - Ymodel |>> σmean means Inconsistent (akin to confidence interval: meantCIσνα⋅≅,) (Ydata(x))Figure 3. Normal distribution curve showing 1 standard deviation.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 θ? • Experimental Design Identify which {θi} are not determined by data. A few θi often control the fit. Some θi cannot be determined well by experiment (poorly conditioned matrices). Introduction to Chi-Squared Analysis Assume all error is Gaussian. dataNinnelnnNxYxYaction~),()(22mod2∑−><≡σθχ for the “true” model 10.34, Numerical Methods Applied to Chemical Engineering Lecture 22 Prof. William Green 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]. tsmeanNDSexpexp..~σ parameter refinement: χ2(θ) Å minimize χ2 by changing θexperimental design: derivatives of χ2 with respect to θ. Bayesian View Prior knowledge p(θ, σ) Æ posterior p(θ, σ; Ydata) More knowledge after experiment. Use to narrow error bars. Conditional Probability )|()()(12121EEPEPEEP =∩: probability of E2 knowing E1 happened (correlation) if independent: P(E2) model prior Pposterior(θ,σ|Ydata) = P(Ydata|θ,σ)P(θ,σ) ∫∫dθ dσ P(Y|θ,σ)P(θ,σ) ≈ P(Ydata) normalize P(Ydata|θ,σ): probability of observing Ydata we really observed if θ, σ are true values. We do not know θ and σ
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