Physics307L F09:Schedule/Week 8 agenda/Linear fit theory

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

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

[math]\displaystyle{ \ y=A+Bx }[/math]

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

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

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

[math]\displaystyle{ Prob \propto \frac{1}{\sigma_y^N}e^{-\chi^2/2} }[/math]

[math]\displaystyle{ 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:

(1) [math]\displaystyle{ AN + B \sum x_i = \sum y_i }[/math]

(2) [math]\displaystyle{ A \sum x_i + B \sum x_i^2 = \sum x_i y_i }[/math]

[math]\displaystyle{ A=\frac{\sum x_i^2 \sum y_i^2 - \sum x_i \sum x_i y_i}{\Delta} }[/math]

[math]\displaystyle{ B=\frac{N\sum x_i y_i - \sum x_i \sum y_i}{\Delta} }[/math]

[math]\displaystyle{ \Delta=N \sum x_i^2 - \left ( \sum x_i \right )^2 }[/math]