Next: Linear regression, Up: Least-Squares Fitting [Index]

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 *< >* 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

\delta c_a = \sum_i (dc_a/dy_i) \delta y_i

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

C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>

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

C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i)

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}}*.
The correlation coefficient of the fit parameters *c_a* and *c_b*
is given by *\rho_{ab} = C_{ab} / \sqrt{C_{aa} C_{bb}}*.

Next: Linear regression, Up: Least-Squares Fitting [Index]