| 1 | .. residuals_help.rst |
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| 2 | |
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| 3 | .. |chi| unicode:: U+03C7 |
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| 4 | |
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| 5 | |
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| 6 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 7 | |
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| 8 | .. _Assessing_Fit_Quality: |
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| 9 | |
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| 10 | Assessing Fit Quality |
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| 11 | --------------------- |
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| 12 | |
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| 13 | When performing model-fits to some experimental data it is helpful to be able to |
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| 14 | gauge how good an individual fit is, how it compares to a fit of the *same model* |
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| 15 | *to another set of data*, or how it compares to a fit of a *different model to the* |
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| 16 | *same data*. |
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| 17 | |
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| 18 | One way is obviously to just inspect the graph of the experimental data and to |
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| 19 | see how closely (or not!) the 'theory' calculation matches it. But *SasView* |
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| 20 | also provides two other measures of the quality of a fit: |
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| 21 | |
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| 22 | * |chi|\ :sup:`2` (or 'Chi2'; pronounced 'chi-squared') |
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| 23 | * *Residuals* |
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| 24 | |
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| 25 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 26 | |
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| 27 | Chi2 |
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| 28 | ^^^^ |
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| 29 | |
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| 30 | Chi2 is a statistical parameter that quantifies the differences between an observed |
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| 31 | data set and an expected dataset (or 'theory'). |
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| 32 | |
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| 33 | *SasView* actually returns this parameter normalized to the number of data points, |
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| 34 | *Npts* such that |
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| 35 | |
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| 36 | *Chi2/Npts* = { SUM[(*Y_i* - *Y_theory_i*)^2 / (*Y_error_i*)^2] } / *Npts* |
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| 37 | |
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| 38 | This differs slightly from what is sometimes called the 'reduced chi-squared' |
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| 39 | because it does not take into account the number of fitting parameters (to |
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| 40 | calculate the number of 'degrees of freedom'), but the 'normalized chi-squared' |
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| 41 | and the 'reduced chi-squared' are very close to each other when *Npts* >> number of |
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| 42 | parameters. |
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| 43 | |
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| 44 | For a good fit, *Chi2/Npts* tends to 0. |
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| 45 | |
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| 46 | *Chi2/Npts* is sometimes referred to as the 'goodness-of-fit' parameter. |
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| 47 | |
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| 48 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 49 | |
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| 50 | Residuals |
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| 51 | ^^^^^^^^^ |
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| 52 | |
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| 53 | A residual is the difference between an observed value and an estimate of that |
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| 54 | value, such as a 'theory' calculation (whereas the difference between an observed |
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| 55 | value and its *true* value is its error). |
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| 56 | |
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| 57 | *SasView* calculates 'normalized residuals', *R_i*, for each data point in the |
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| 58 | fit: |
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| 59 | |
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| 60 | *R_i* = (*Y_i* - *Y_theory_i*) / (*Y_err_i*) |
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| 61 | |
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| 62 | For a good fit, *R_i* ~ 0. |
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| 63 | |
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| 64 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 65 | |
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| 66 | .. note:: This help document was last changed by Steve King, 08Jun2015 |
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