36.1 Overview

Least-squares fits are found by minimizing \chi^2 (chi-squared), the weighted sum of squared residuals over n experimental datapoints (x_i, y_i) for the model Y(c,x),

     \chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2

The p parameters of the model are c = {c_0, c_1, ...}. The weight factors w_i are given by w_i = 1/\sigma_i^2, where \sigma_i is the experimental error on the data-point y_i. The errors are assumed to be gaussian and uncorrelated. For unweighted data the chi-squared sum is computed without any weight factors.

The fitting routines return the best-fit parameters c and their p \times p covariance matrix. The covariance matrix measures the statistical errors on the best-fit parameters resulting from the errors on the data \sigma_i, and is defined as C_{ab} = <\delta c_a \delta c_b> where \langle \, \rangle denotes an average over the gaussian error distributions of the underlying datapoints.

The covariance matrix is calculated by error propagation from the data errors \sigma_i. The change in a fitted parameter \delta c_a caused by a small change in the data \delta y_i is given by

allowing the covariance matrix to be written in terms of the errors on the data,

For uncorrelated data the fluctuations of the underlying datapoints satisfy <\delta y_i \delta y_j> = \sigma_i^2 \delta_{ij}, giving a corresponding parameter covariance matrix of

When computing the covariance matrix for unweighted data, i.e. data with unknown errors, the weight factors w_i in this sum are replaced by the single estimate w = 1/\sigma^2, where \sigma^2 is the computed variance of the residuals about the best-fit model, \sigma^2 = \sum (y_i - Y(c,x_i))^2 / (n-p). This is referred to as the variance-covariance matrix. The standard deviations of the best-fit parameters are given by the square root of the corresponding diagonal elements of the covariance matrix, \sigma_{c_a} = \sqrt{C_{aa}}.