Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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(±δx), θ±δθ) “knobs” parameters in model we can physically adjust all other we generally parameters know bounds that affect on θ results prior information (cannot control) about θ 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>,σdata)≈ P(<Y>Nexpts) ⎡ − (< Y > − < Y > true )2 ⎤ exp ⎢ ⎥ 2 2σ mean σ 2π ⎣⎢ ⎦⎥ 1 σexp.data|Nexpts <Y>Nexpts σ mean ≈ σ 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 |>> σmean means Inconsistent (akin to confidence interval: CI ≅ tα ,ν ⋅ σ 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 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. χ ≡ 2 N action < Y n > ( x n ) − Y mod el ( x n ,θ ) i σ n2 ∑ σ mean ~ 2 ~ N data for the “true” model S .D.exp N exp ts parameter refinement: χ2(θ) Å minimize χ2 by changing θ experimental 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 P( E1 ∩ E 2 ) = P( E1 ) P( E 2 | E1 ) : 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 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].