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Peterson, Peter authoredPeterson, Peter authored
Description
Calculates a measure of the goodness of a fit in four ways.
The ChiSquared property returns the sum of squares of differences between the calculated and measured values:
\chi_{1}^{2} = \sum_{i} (y_i - f_i)^2
where y_i and f_i are the measured and calculated values at i-th point.
The ChiSquaredDividedByDOF is ChiSquared divided by the number of degrees of freedom (DOF):
\chi_{2}^{2} = \frac{1}{DOF}\sum_{i} (y_i - f_i)^2
DOF = N_d - N_p where N_d is the number of data points used in fitting and N_p is the number of free (not fixed or tied) parameters of the function.
The ChiSquaredWeighted property sums the squares of the differences divided by the data errors:
\chi_{3}^{2} = \sum_{i} \left(\frac{y_i - f_i}{\sigma_i}\right)^2
Finally, ChiSquaredWeightedDividedByDOF is
\chi_{4}^{2} = \chi_{3}^{2} / DOF
Parameter errors
Setting the Output property to a non-empty string makes the algorithm explore the surface of the \chi^{2} around its minimum and estimate the standard deviations for the parameters. The value of the property is a base name for two output table workspaces: '<Output>_errors' and '<Output>_pdf'. The former workspace contains parameter error estimates and the latter shows \chi^{2}'s 1d slices along each parameter (keeping all other fixed).
The error table has the following columns:
| Column | Description |
|---|---|
| Parameter | Parameter name |
| Value | Parameter value passed with the Function property |
| Value at Min | The minimum point of the 1d slice of the \chi^{2}. If the Function is at the minimum then Value at Min should be equal to Value. |
| Left Error | The negative deviation from the minimum point equivalent to 1\sigma. Estimated from analisys of the surface. |
| Right Error | The positive deviation from the minimum point equivalent to 1\sigma. Estimated from analisys of the surface. |
| Quadratic Error | 1\sigma standard deviation in the quadratic approximation of the surface. |
| Chi2 Min | The value of \chi^{2} at the minimum relative to the test point. |
The pdf table contains slices of the \chi^{2} along each parameter. It has 3 column per parameter. The first column of the 3 is the parameter values, the second has the \chi^{2} and the third is the probability density function normalised to have 1 at the maximum.
Usage
Example 1
Output:
Example 2