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Approximated solution for non-linear fits with normal errors

Linearity implies that the arguments of the exponential of the integrand in Eq. (28) contains only first and second powers of $\mu_{x_i}$, and then the integrals has a closed solution. Though this is not true in general, the linear case teaches us how to get an approximated solution of the problem. We can take first order expansions of $\mu_{y}(\mu_{x},{\mbox{\boldmath$\theta$}})$ around each $x_i$
$\displaystyle \mu_y(\mu_{x_i};{\mbox{\boldmath$\theta$}})$ $\textstyle \approx$ $\displaystyle \mu_y(x_i;{\mbox{\boldmath$\theta$}}) \, + \,
\mu_y^{\,\prime}(x_i;{\mbox{\boldmath$\theta$}})\cdot
(\mu_{x_i}-x_i)\,.$ (31)

The difference $y_i-m\,\mu_{x_i}-c$ in Eq. (28), that was indeed equal to $y_i-\mu_y(\mu_{x_i};{\mbox{\boldmath$\theta$}})$ in the general case, using the linear approximation becomes

\begin{displaymath}y_i - \mu_y(x_i;{\mbox{\boldmath $\theta$}}) -
\mu_y^{\,\pri...
..._y^{\,\prime}(x_i;{\mbox{\boldmath $\theta$}})\cdot x_i\,]
\,, \end{displaymath}

i.e. we have the following replacements in Eqs. (28)-(30):
$\displaystyle m$ $\textstyle \rightarrow$ $\displaystyle \mu_y^{\,\prime}(x_i;{\mbox{\boldmath$\theta$}})$ (32)
$\displaystyle c$ $\textstyle \rightarrow$ $\displaystyle \mu_y(x_i;{\mbox{\boldmath$\theta$}}) - \mu_y^{\,\prime}(x_i;{\mbox{\boldmath$\theta$}})\cdot x_i\,.$ (33)

The approximated equivalent of Eq. (30) is then
$\displaystyle f({\mbox{\boldmath$\theta$}}\,\vert\,{\mbox{\boldmath$x$}},{\mbox{\boldmath$y$}},I)$ $\textstyle \propto\approx$ $\displaystyle \prod_i
\frac{1}{\sqrt{\sigma_{y_i}^2+{\mu_y^{\,\prime}}^2
(x_i;{...
...math$\theta$}})\cdot\sigma_{x_i}^2] }
\right]}\, f(\theta\,\vert\,I)\,,\ \ \ \ $ (34)

where the unusual symbol ` $\propto\approx$' stands for `approximately proportional to'.


next up previous
Next: Extra variability of the Up: Fits, and especially linear Previous: Linear fit with normal
Giulio D'Agostini 2005-11-21