Analysis 2
A Model for Data
From the previous analysis tutorial, you have an idea of how to estimate errors on data, including the mean of data. And you know some basic error propagation. In this tutorial, you'll learn how to fit a model to the data and propagate errors to the model parameters.
So let's say you have measured $N$ data <math>d_i</math> along with some estimates of the error/standard deviation <math>s_i</math>. You took these data with respect to some well determined independent variable <math>t_i</math>. As an example, you could have measured the altitude of a projectile (<math>d_i</math>) as a function of time could be <math>t_i</math>. In this example you would expect the altitude of the data to be modeled by
<math> d_i = a t_i^2 + b t_i + c </math>
where <math>a, b, c</math> are model parameters corresponding to the acceleration, initial velocity, and initial displacement of the projectile. (Q: How would you have measured the error for the data? A: You could have run the experiment -- that is, tossed the projectile -- a number of times and computed the sample variance of the trials.)
<math>\chi^2</math> and Goodness of Fit
For a given set of parameters <math>a, b, c</math>, we need to evaluate how well the model fits the data -- the goodness of fit. For this purpose, you can use the <math>\chi^2</math>. You construct the <math>\chi^2</math> of the model, given the data, like so
<math> \chi^2 = \sum_i^N \frac{(d_i - (a t_i^2 + b t_i + c))^2}{s_i^2} </math>,
where we have continued to use the example model from the previous section. The numerator is the difference between our data and the model, squared and normalized by our estimate of the variance. Finding the "best fit" model in practice involves minimizing <math>chi^2</math>
Now let's suppose that the model does indeed apply to the phenomenon measured by the experiment and that we have chosen the right model parameters $a,b,c$. In this case <math>a t_i^2 + b t_i + c</math> is the expectation value for <math>d_i</math>. If this is the case then the expectation of the numerator of the <math>\chi^2</math> expression is
<math> \langle (d_i - (a t_i^2 + b t_i + c))^2 \rangle = \langle (d_i - \mu_{d_i})^2 \rangle = \sigma_{d_i}^2 \approx s_i^2 </math>,
where we have used the definition of variance <math> \sigma_{d_i}^2 </math> from the previous tutorial. Given this result, the expectation value for our goodness of fit becomes <math>\langle \chi^2 \rangle = N</math>. This turns out not to be the whole story, because we don't know nature's choice of $a, b, c$ and need to fit the model to the data -- more on how to do this below. If we find the best fit $a, b, c$ to the data, then our expectation value for the goodness of fit is <math>\langle \chi^2 \rangle = N - 3</math>, where 3 corresponds to the number of parameters in our model, in this case $a, b, c$. More generally, if we have $M$ parameters, the expectation for <math>\chi^2</math> associated with a good "best fit" model is called the number of degrees of freedom of the fit: <math>N-M</math>. To understand this, let's take the case where we have taken only three data points (<math>N=3</math>) of our trajectory. Because three points define a quadratic function, these three points determine exactly a best fit parabolic trajectory and the numerator of <math>\chi^2</math> is identically zero and therefore <math>\chi^2 = 0 = N-3</math>. Here is an example of
