------------------------------------------------------------ Ignacio Taboada Sep 24, 2012 --------------- One of my students this year asked me specifically to cover least squares. To me, this is the absolute most basic thing you need to know about data fitting - and usually I use more advanced methods. I've uploaded my pdf and power-point to your dropbox folder: ChaosBook.org/~predrag/courses/PHYS-6124-12/taboada12/MathMethods.pptx ChaosBook.org/~predrag/courses/PHYS-6124-12/TaboadaMMeth.pdf Here are a few things that particle and astroparticle people do often for hypothesis testing: Li and Ma (1983) http://adsabs.harvard.edu/full/1983ApJ...272..317L Feldman and Cousins (1998) http://prd.aps.org/abstract/PRD/v57/i7/p3873_1 Both of these two papers are probably too complex to do in a course, but the idea of hypothesis testing can be studied in simpler cases. ------------------------------------------------------------ Peter Dimon ----------- thoughts on how to teach math methods needed by experimentlists: Probability theory Inference random walks Conditional probability Bayes rule (another look at diffusion) Machlup has a classic paper on analysing simple on-off random spectrum. Hand out to students. (no Baysians use of information that you do not have) (Peter takes a dim view) Fourier transforms power spectrum - Wiener-Kitchen for correlation function works for stationary system useless on drifting system (tail can be due to drift only) must check whether the data is stationary measure: power spectrum, work in Fourier space do this always in the lab power spectra for processes: Brownian motion, Langevin -> get Lorenzian connect to diffusion equation they need to know: need to know contour integral to get from Langevin power spectrum, to get to the correlation function why is power spectrum Lorenzian - look at the tail 1/omega^2 because the cusp at small times that gives the tails flat spectrum at origin gives long time lack of correlation position is not stationary diffusion Green's function delta fct -> Gaussian + additivity Nayquist theorem sampling up to a Nayquist theorem easy to prove Other processes: what signal you expect for a given process Fluctuation-dissipation theorem connection to response function (lots of them measure that) for Brownian motion power spectrum realted to imaginary part of response function Use Numerical Recipes (stupid on correlation functions) zillion filters (murky subject) Kalman (?) (last 3 lecturs): how to make a measurement finite time sampling rates (be intellingent about it) Peter Dimon postscript: ----------------------- Did I suggest all that? I thought I mentioned, like, three things. Anyway, looking over your previous lectures, it seems like you've covered a lot. I'm not sure I have anything to add. Did you do the diffusion equation? It's an easy example for PDEs, Green's function, etc. And it has an unphysically infinite speed of information, so you can add a wave term to make it finite. This is called the Telegraph Equation (it was originally used to describe damping in transmission lines). What about Navier-Stokes? There is a really cool exact solution (stationary) in two-dimensions called Jeffery-Hamel flow that involves elliptic functions and has a symmetry-breaking. (It's outlined in Landau and Lifshitz, Fluid Dynamics). ------------------------------------------------------------ Mike Schatz says: ----------------- 1D bare minimum: temporal signal, time series analysis discrete Fourier transform, FFT in 1 and 2D - exercises make finite set periodic Image processing: Fourier transforms, correlations, convolution, particle tracking, PDEs in 2D (Matlab): will give it to Predrag ------------------------------------------------------------ file: PHYS-6124-12/experimentMMs.txt svn: $Author: predrag $ $Date: 2012-12-02 23:45:16 -0500 (Sun, 02 Dec 2012) $