15.3.5.2 Global Fitting with Parameter Sharing

See more related video:Global Fitting

Global fitting in Origin involves fitting multiple datasets with the same fitting function. Parameters in the fitting function can optionally be shared amongst all datasets. If a parameter is shared, the fitting procedure will yield the same value for that parameter for all datasets. If a parameter is not shared, the fitting procedure will yield a unique value for that parameter for each dataset.

Key points of Global Fitting:

Multiple datasets are fitted with one model simultaneously.
Fit parameters are optionally shared between the datasets.
- For each shared (global) parameter, one best-fit value is estimated from all of the datasets that are fitted.
- For each non-shared (local) parameter, a unique best-fit value is generated for individual dataset that is fitted.
Other options such as constraints and weighting are also available when you choose to perform Global Fitting.

To do global fitting with parameter sharing:

Select multiple datasets when you open the NLFit dialog.
Select a fitting function.
In the Data Mode drop-down list of Data Selection settings, select Global Fit.

In the Parameters tab, select the checkbox in the Share column which corresponds to the parameter you want to set as shared.
Click the Fit or OK button to perform the fitting.

Related Algorithm

The fitting report for global fit will output the Parameters, Statistics and ANOVA tables for each dataset and a global Statistics and ANOVA table for all of the datasets. When global fitting is performed, the Chi-square for n datasets is computed as:

$\chi ^2=\sum_{i=1}^m[\frac{Y1_i-f(x1_i^{\prime };\hat \theta 1)}{\sigma 1_i}]^2+\sum_{i=1}^m[\frac{Y2_i-f(x2_i^{\prime };\hat \theta 2)}{\sigma 2_i}]^2+\ldots +\sum_{i=1}^m[\frac{Yn_i-f(xn_i^{\prime };\hat \theta n)}{\sigma n_i}]^2$

and

$reduced X^2=\frac {X^2}{dof}=\frac {X^2}{n-p}$

The global ANOVA table is:

	df	Sum of Squares	Mean Square	F Value	Prob > F
Model	p-1	$SS_{reg} = SYY- RSS$	$MS_{reg} = SS_{reg} /p-1$	$MS_{reg} / MSE$	p-value
Error	-p	RSS	MSE = RSS /(n-p)
Total	n-1	SYY

In the above formula, n is the total number of data points, and p is the total number of parameters. Note that when parameters are shared, it will reduce the number of parameters, p. For example, to do a global fit for two datasets with simple linear function, y = a + bx, with the parameter a shared, the number of parameters becomes three because we have reduced one parameter. Therefore, p = 3.