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sasview/src/sas/sasgui/perspectives/fitting/media/residuals_help.rst
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Assessing Fit Quality
When performing model-fits to some experimental data it is helpful to be able to gauge how good an individual fit is, how it compares to a fit of the same model to another set of data, or how it compares to a fit of a different model to the same data.
One way is obviously to just inspect the graph of the experimental data and to see how closely (or not!) the 'theory' calculation matches it. But SasView also provides two other measures of the quality of a fit:
- $chi^2$ (or 'Chi2'; pronounced 'chi-squared')
- Residuals
Chi2
$chi^2$ is a statistical parameter that quantifies the differences between an observed data set and an expected dataset (or 'theory').
When showing the a model with the data, SasView displays this parameter normalized to the number of data points, $N_mathrm{pts}$ such that
When performing a fit, SasView instead displays the reduced $chi^2_R$, which takes into account the number of fitting parameters $N_mathrm{par}$ (to calculate the number of 'degrees of freedom'). This is computed as
The normalized $chi^2_N$ and the reduced $chi^2_R$ are very close to each other when $N_mathrm{pts} gg N_mathrm{par}$.
For a good fit, $chi^2_R$ tends to 1.
$chi^2_R$ is sometimes referred to as the 'goodness-of-fit' parameter.
Residuals
A residual is the difference between an observed value and an estimate of that value, such as a 'theory' calculation (whereas the difference between an observed value and its true value is its error).
SasView calculates 'normalized residuals', $R_i$, for each data point in the fit:
Think of each normalized residual as the number of standard deviations between the measured value and the theory. For a good fit, 68% of $R_i$ will be within one standard deviation, which will show up in the Residuals plot as $R_i$ values between $-1$ and $+1$. Almost all the values should be between $-3$ and $+3$.
Residuals values larger than $pm 3$ indicate that the model is not fit correctly, the wrong model was chosen (e.g., because there is more than one phase in your system), or there are problems in the data reduction. Since the goodness of fit is calculated from the sum-squared residuals, these extreme values will drive the choice of fit parameters. Any uncertainties calculated for the fitting parameters will be meaningless.
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