View source for Segment 20. Nonlinear Least Squares Fitting

====Watch this segment====
(Don't worry, what you see statically below is not the beginning of the segment.  Press the play button to start at the beginning.)

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The direct YouTube link is [http://youtu.be/xtBCGPHRcb0 http://youtu.be/xtBCGPHRcb0]

Links to the slides: [http://wpressutexas.net/coursefiles/20.NonlinearLeastSquares.pdf PDF file] or [http://wpressutexas.net/coursefiles/20.NonlinearLeastSquares.ppt PowerPoint file]

===Problems===
====To Calculate====
1. (See lecture slide 3.)  For one-dimensional <math>x</math>, the model <math>y(x | \mathbf b)</math> is called "linear" if <math>y(x | \mathbf b) = \sum_k b_k X_k(x)</math>, where <math>X_k(x)</math> are arbitrary known functions of <math>x</math>.  Show that minimizing <math>\chi^2</math> produces a set of linear equations (called the "normal equations") for the parameters <math>b_k</math>.

2. A simple example of a linear model is <math>y(x | \mathbf b) = b_0 + b_1 x</math>, which corresponds to fitting a straight line to data.  What are the MLE estimates of <math>b_0</math> and <math>b_1</math> in terms of the data: <math>x_i</math>'s, <math>y_i</math>'s, and <math>\sigma_i</math>'s?

====To Think About====
1.  We often rather casually assume a uniform prior <math>P(\mathbf b)= \text{constant}</math> on the parameters <math>\mathbf b</math>.  If the prior is not uniform, then is minimizing <math>\chi^2</math> the right thing to do?  If not, then what should you do instead?  Can you think of a situation where the difference would be important?

2.  What if, in lecture slide 2, the measurement errors were <math>e_i \sim \text{Cauchy}(0,\sigma_i)</math> instead of <math>e_i \sim N(0,\sigma_i)</math>?  How would you find MLE estimates for the parameters <math>\mathbf b</math>?

===Class Activity===
Here is some data: [[Media:Chisqfitdata.txt]]

In class we will work on fitting this to some models as explained [[ClassActivity20130325|here]].

Here are Bill's numerical answers, so that you can see whether you are on the right track (or whether Bill got it wrong!): [[Media:Chisqfitanswers.txt]]