Changeset a7c6f38 in sasview for src/sas/sasgui/perspectives/fitting/media/residuals_help.rst
 Timestamp:
 Mar 4, 2018 4:07:55 PM (6 years ago)
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 master, magnetic_scatt, release4.2.2, ticket1009, ticket1094headless, ticket12422dresolution, ticket1243, ticket1249, unittestsaveload
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 84ac3f1
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 943e17a
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src/sas/sasgui/perspectives/fitting/media/residuals_help.rst
r99ded31 ra7c6f38 27 27 28 28 $\chi^2$ is a statistical parameter that quantifies the differences between 29 an observed data set and an expected dataset (or 'theory'). 30 31 When showing the a model with the data, *SasView* displays this parameter 32 normalized to the number of data points, $N_\mathrm{pts}$ such that 29 an observed data set and an expected dataset (or 'theory') calculated as 33 30 34 31 .. math:: 35 32 36 \chi^2 _N37 = \sum[(Y_i  \mathrm{theory}_i)^2 / \mathrm{error}_i^2] / N_\mathrm{pts}33 \chi^2 34 = \sum[(Y_i  \mathrm{theory}_i)^2 / \mathrm{error}_i^2] 38 35 39 When performing a fit, *SasView* instead displays the reduced $\chi^2_R$, 40 which takes into account the number of fitting parameters $N_\mathrm{par}$ 41 (to calculate the number of 'degrees of freedom'). This is computed as 36 Fitting typically minimizes the value of $\chi^2$. However, for assessing the 37 quality of the model and its "fit" this parameter is not terribly helpful on its 38 own. Thus *SasView* instead displays a normalized version of this parameter, 39 using the traditional reduced $\chi^2_R$. This is the $\chi^2$ divided by the 40 degrees of freedom (or DOF). The DOF is simply the number of data points being 41 considered reduced by the number of free (i.e. fitted) parameters. Note that 42 model parameters that are kept fixed do *not* contribute to the DOF (they are 43 not"free". This reduced value is then given as 42 44 43 45 .. math:: … … 47 49 / [N_\mathrm{pts}  N_\mathrm{par}] 48 50 49 The normalized $\chi^2_N$ and the reduced $\chi^2_R$ are very close to each 50 other when $N_\mathrm{pts} \gg N_\mathrm{par}$. 51 where $N_\mathrm{par}$ is the number of *fitted* parameters. Note that this 52 means the displayed value will vary depending on the number of parameters used 53 in the fit. In particular, when doing a calculation without a fit (e.g. 54 manually changing a parameter) the DOF will now equal $N_\mathrm{pts}$ and the 55 $\chi^2_R$ will be the smallest possible for that combination of model, data 56 set and set of parameter values. 57 58 When $N_\mathrm{pts} \gg N_\mathrm{par}$ as it should for proper fitting, the 59 value of the reduced $\chi^2_R$ will not change very much. 51 60 52 61 For a good fit, $\chi^2_R$ tends to 1.
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