C1 - method of least squares

image: TPA Hase, PX271 data analysis course

the residual of a data point is defined as:

R_{i} = y_{i} - y (x_{i})

where, $y_{i}$ is the measured value, and $y (x_{i})$ is the theoretical value

the method of least squares minimises the sum of the squares of the residuals by changing the slope and the intercept of the line of fit
the 'goodness of fit' is tested by the chi squared statistic:

χ^{2} = \sum_{i} R_{i}^{2} = \sum_{i} (y_{i} - y)^{2}

homoscedastic data refers to data with the same uncertainties
heteroscedastic data refers to data with different uncertainties, hence each point is weighted equally
normalised residuals weigh the residuals by the uncertainties such that:

R_{n o r m} = \frac{R}{α}

the weighting is inversely proportional to its error

χ^{2} = \sum_{i} \frac{R_{i}^{2}}{α_{i}^{2}} = \sum_{i} R_{i, n o r m}^{2}

for poisson data:

χ^{2} = \sum_{i} \frac{(O_{i} - E_{i})^{2}}{E_{i}}

where, $O$ and $E$ are the obtained and the expected values

the data are re-binned such that each interval contains expected counts more than 5
$χ^{2} = 1$ equates to $1 σ$ confidence limit
the ellipticity and shape of the error surface around $χ_{m i n}^{2}$ indicated correlations between the errors and the fit parameters
this needs to be incorporated into error propagation

image: TPA Hase, PX271 data analysis course