Physics307L: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 \ A and \ B to the data

\ y=A+Bx

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

Principle of maximum likelihood

For a given \ A and \ B, the probability for each \ y_i is:

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

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

Prob = Prob(y_1) \cdot Prob(y_2) \cdot ... \cdot Prob(y_N)

Each term has the same σy, so can be simplified as:

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

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

(1) AN + B \sum x_i = \sum y_i

(2) A \sum x_i + B \sum x_i^2 = \sum x_i y_i

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

Can also derive formulas for weighting each point individually

Also, formulas for calculating uncertainty in fit parameters

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