Analysis 2: Difference between revisions
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[[2012 | Back to 2012 Main Page]] | [[2012 | Back to 2012 Main Page]] | ||
==A Model for Data== | |||
From the [[Analysis 1| 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. | From the [[Analysis 1| 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. | ||
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<math> d_i = a t_i^2 + b t_i + c </math> | <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 a number of times and computed the sample variance of the trials.) We construct the $\chi^2$ of the model, given the data like so | 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== | |||
We construct the $\chi^2$ 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> | <math> \chi^2 = \sum_i^N \frac{(d_i - (a t_i^2 + b t_i + c))^2}{s_i^2} </math> | ||
It's worht | |||
Revision as of 14:52, 9 February 2012
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
We construct the $\chi^2$ 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>
It's worht