Physics307L F08:Schedule/Week 8 agenda/Linear fit theory

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Following John R. Taylor, "An Introduction to Error Analysis," 2nd edition, Chapter 8:

We have a relation as follows, and want to fit [math]\ A[/math] and [math]\ B[/math] to the data

[math]\ y=A+Bx[/math]

Assume same gaussian distribution for random error in each [math]\ y_i[/math] (same [math]\ \sigma[/math] for all). Not necessary, but simplifies derivation and results

Principle of maximum likelihood

For a given [math]\ A[/math] and [math]\ B[/math], the probability for each [math]\ y_i[/math] is:

[math]Prob(y_i) \propto \frac{1}{\sigma_y}e^{-(y_i-A-Bx_i)^2/2\sigma_y^2}[/math]

And we can call the probability of getting all of the data points as:

[math]Prob = Prob(y_1) \cdot Prob(y_2) \cdot ... \cdot Prob(y_N)[/math]

Each term has the same [math]\sigma_y[/math], so can be simplified as:

[math]Prob \propto \frac{1}{\sigma_y^N}e^{-\chi^2/2}[/math]
[math]chi-squared, \chi^2 = \sum_{i=1}^N \frac{\left (y_i - A - Bx_i \right )^2}{\sigma_y^2}[/math]

To maximize the probability, minimize the chi-squared sum ... take derivatives, solve system of equations, obtain:

[math]A=\frac{\sum x_i^2 \sum y_i^2 - \sum x_i \sum x_i y_i}{\Delta}[/math]
[math]B=\frac{N\sum x_i y_i - \sum x_i \sum y_i}{\Delta}[/math]
[math]\Delta=N \sum x_i^2 - \left ( \sum x_i \right )^2[/math]

Can also derive formulas for weighting each point individually

Also, formulas for calculating uncertainty in fit parameters