From Computational Statistics (CSE383M and CS395T)
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Here are the stages that each group should get to:

1. Read in the data and plot the <math>(x_i,y_i)</math> data points, including error bars or some other graphical indication of the <math>\sigma_i</math>'s.

2. Hmm. They look kind of like a raised parabola, don't they? Try fitting a model of the form <math>y = b_0 + b_1 x^2</math>. What are the best fitting values for <math>b_0, b_1</math>? Plot the best fit curve on the same plot as you produced in stage 1. Does it look like a good fit? What is your value of <math>\chi^2_{min}</math>?

At this stage you might want to automate your process so that you can quickly plug in the following models and get best-fit parameters, <math>\chi^2_{min}</math>, and a graphical plot.

3. Do a linear fit to see how bad it is: <math>y = b_0 + b_1 x</math>

4. Try an exponential: <math>y = b_0 \exp(b_1 x)</math>

5. Try adding a linear term to the parabola to get a general quadratic: <math>y = b_0 + b_1 x + b_2 x^2</math>

6. Does the ordering of values <math>\chi^2_{min}</math> seem to match your intuitive impression of which curves fits best?

7. Calculate standard errors for your fitted parameters using the Hessian matrix (as described in the segment). Is the value of <math>b_1</math> in stage 5 different enough from zero so that you are sure it isn't zero? (That is, are you justified in adding the extra parameter to the original stage 2 parabola?)

8. Can you answer the same question by looking at the <math>\chi^2_{min}</math> values? (We'll learn more about this later in the course.)