Chi-square normalization

If the user's function value F is supposed to be a chisquare, it must of course be properly normalized. That is, the ``weights'' must in fact correspond to the one-standard-deviation errors on the observations. The most general expression for the chi-square $\chi$ is of the form (see [5], p.163):

$\chi 2=\; \sum$_i,j(x_i- y_i(a)) V_ij(x_j- y_j(a))

where x is the vector of observations, $y(a)$ is the vector of fitted values (or theoretical expressions for them) containing the variable fit parameters a, and V is the inverse of the error matrix of the   observations x, also known as the covariance matrix of the observations.

Fortunately, in most real cases the observations x are statistically independent of each other (e.g., the contents of the bins of a histogram, or measurements of points on a trajectory), so the matrix V is diagonal only. The expression for $\chi 2$ then simplifies to the more familiar form:

$\chi 2=\; \sum$_i{(x_i- y_i(a))²e_i²}

where $e2$ is the inverse of the diagonal element of V, the square of the error on the corresponding observation x. In the case where the x are integer numbers of events in an unweighted histogram, for example, the $e2$ are just equal to the x (or to the y, see [5], pp.170-171).

The minimization of $\chi 2$ above is sometimes called weighted least     squares in which case the inverse quantities $1/e2$ are called the weights. Clearly this is simply a different word for the same thing, but in practice the use of these words sometimes means that the interpretation of $e2$ as variances or squared errors is not straightforward. The word weight often implies that only the relative weights are known (``point two is twice as important as point one'') in which case there is apparently an unknown overall normalization factor. Unfortunately the parameter errors coming out of such a fit will be proportional to this factor, and the user must be aware of this in the formulation of his problem.

The $e2$ may also be functions of the fit parameters a (see [5], pp.170-171). Normally this results in somewhat slower convergence of the fit since it usually increases the nonlinearity of the fit. (In the simplest case it turns a linear problem into a non-linear one.) However, the effect on the fitted parameter values and errors should be small.

If the user's chi-square function is correctly normalized, he should   use UP=1.0 (the default value) to get the usual one standard-deviation errors for the parameters one by one. To get two-standard-dev.eviation errors, use [SET ERRordef]ERROR DEF 4.0, etc., since the chisquare dependance on parameters is quadratic. For more general confidence regions involving more than one parameter, see section .