Nonlinear Chi Square Fits

The terms “linear” and “nonlinear” refer to the dependence of the fitting function on the parameters and not the dependent variable $x$. Thus if we are fitting to $y = a + b \exp(-cx)$, and we do not adjust $c$, but vary only $a$ and $b$, then it is a linear fit, because of the linear dependence in $a$ and $b$. But if we are also varying $c$, it becomes a nonlinear fit. To be a linear fit, the second and higher derivatives of the fitting function with respect to all fitting parameters must all be zero.

If the parameter dependence is nonlinear we must use a more general approach. Suppose our fitting function depends nonlinearly on the parameters ${\bf a} = (a_{1},a_{2},\ldots{},a_{m})$. Then we must minimize

\chi^{2}(a_{1},a_{2},\ldots{},a_{m})= \sum_{i=1}^{N}
[y_i-\bar y_{i,model}(a_{1},a_{2},\ldots{},a_{N})]^2/\sigma_i^2.
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