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. 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. 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 $\chi^2$ 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>