# Segment 20. Nonlinear Least Squares Fitting

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## Contents

#### 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.)

{{#widget:Iframe |url=http://www.youtube.com/v/xtBCGPHRcb0&hd=1 |width=800 |height=625 |border=0 }}

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

Links to the slides: PDF file or PowerPoint file

### Problems

#### To Calculate

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

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

#### To Think About

1. We often rather casually assume a uniform prior $\displaystyle P(\mathbf b)= \text{constant}$ on the parameters $\displaystyle \mathbf b$ . If the prior is not uniform, then is minimizing $\displaystyle \chi^2$ 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 $\displaystyle e_i \sim \text{Cauchy}(0,\sigma_i)$ instead of $\displaystyle e_i \sim N(0,\sigma_i)$ ? How would you find MLE estimates for the parameters $\displaystyle \mathbf b$ ?

### Class Activity

Here is some data: Media:Chisqfitdata.txt

In class we will work on fitting this to some models as explained 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