| [ad2ce4e] | 1 | .. residuals_help.rst |
|---|
| 2 | |
|---|
| 3 | |
|---|
| 4 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
|---|
| 5 | |
|---|
| 6 | .. _Assessing_Fit_Quality: |
|---|
| 7 | |
|---|
| 8 | Assessing Fit Quality |
|---|
| 9 | --------------------- |
|---|
| 10 | |
|---|
| 11 | When performing model-fits to some experimental data it is helpful to be able to |
|---|
| 12 | gauge how good an individual fit is, how it compares to a fit of the *same model* |
|---|
| 13 | *to another set of data*, or how it compares to a fit of a *different model to the* |
|---|
| 14 | *same data*. |
|---|
| 15 | |
|---|
| 16 | One way is obviously to just inspect the graph of the experimental data and to |
|---|
| 17 | see how closely (or not!) the 'theory' calculation matches it. But *SasView* |
|---|
| 18 | also provides two other measures of the quality of a fit: |
|---|
| 19 | |
|---|
| 20 | * |chi|\ :sup:`2` (or 'Chi2'; pronounced 'chi-squared') |
|---|
| 21 | * *Residuals* |
|---|
| 22 | |
|---|
| 23 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
|---|
| 24 | |
|---|
| 25 | Chi2 |
|---|
| 26 | ^^^^ |
|---|
| 27 | |
|---|
| 28 | Chi2 is a statistical parameter that quantifies the differences between an observed |
|---|
| 29 | data set and an expected dataset (or 'theory'). |
|---|
| 30 | |
|---|
| 31 | *SasView* actually returns this parameter normalized to the number of data points, |
|---|
| 32 | *Npts* such that |
|---|
| 33 | |
|---|
| 34 | *Chi2/Npts* = { SUM[(*Y_i* - *Y_theory_i*)^2 / (*Y_error_i*)^2] } / *Npts* |
|---|
| 35 | |
|---|
| 36 | This differs slightly from what is sometimes called the 'reduced chi-squared' |
|---|
| 37 | because it does not take into account the number of fitting parameters (to |
|---|
| 38 | calculate the number of 'degrees of freedom'), but the 'normalized chi-squared' |
|---|
| 39 | and the 'reduced chi-squared' are very close to each other when *Npts* >> number of |
|---|
| 40 | parameters. |
|---|
| 41 | |
|---|
| [ff42a26] | 42 | For a good fit, *Chi2/Npts* tends to 0. |
|---|
| [ad2ce4e] | 43 | |
|---|
| [68f5fbb] | 44 | *Chi2/Npts* is sometimes referred to as the 'goodness-of-fit' parameter. |
|---|
| [ad2ce4e] | 45 | |
|---|
| 46 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
|---|
| 47 | |
|---|
| 48 | Residuals |
|---|
| 49 | ^^^^^^^^^ |
|---|
| 50 | |
|---|
| 51 | A residual is the difference between an observed value and an estimate of that |
|---|
| 52 | value, such as a 'theory' calculation (whereas the difference between an observed |
|---|
| 53 | value and its *true* value is its error). |
|---|
| 54 | |
|---|
| 55 | *SasView* calculates 'normalized residuals', *R_i*, for each data point in the |
|---|
| 56 | fit: |
|---|
| 57 | |
|---|
| 58 | *R_i* = (*Y_i* - *Y_theory_i*) / (*Y_err_i*) |
|---|
| 59 | |
|---|
| 60 | For a good fit, *R_i* ~ 0. |
|---|
| 61 | |
|---|
| 62 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
|---|
| 63 | |
|---|
| 64 | .. note:: This help document was last changed by Steve King, 08Jun2015 |
|---|
