Tutorial 9. Series Functions - Part II
In this tutorial, we will learn how to perform nonlinear curve fit. It is
recommended that you read the topic Nonlinear Fit
before continue.
In nonlinear curve fit, if a pre-defined model function (i.e. Exponential,
Gaussian, Lorentzian, Polynomial, or Voigt) is selected by the user, DataScene
will try to estimate a reasonable set of initial parameters for use in the fit;
if a user defined model function is selected, the user will need to supply the
initial values of the parameters. We will demonstrate this in the following
sections.
Using Pre-defined Model Functions
Start the DataScene program and create a new data table. Set column formulas
"Row" and "Exp[ -0.5 * (Cell("Table1", "Column
1", Row) - 22)^2 / 36 ] + 0.35 * Exp[ - 0.5 * (Cell("Table1",
"Column 1", Row) - 39)^2 / 9 ] + Rnd() / 25" for rows 1 to 50 of Column
1 and Column 2, respectively. Plot a Line
series, line1, in a graph document, Graph 1, with Column 1
and Column 2 as the Position and Value components,
respectively. The series line1 represents two Gaussian peaks with small
random noises.
With Graph 1 as the active document, select the Graph: Series
Function: Nonlinear Fit menu command. Switch to the Nonlinear Fit
tab-page of the Series Function dialog
box. Select Gaussian in the Function Type combo-box, and
DataScene switches to the Fit Gaussian tab-page. Increment the value in
the Number of Peaks spin box to 2 because we have two Gaussian peaks in
the source series. Check the Show individual peaks check box to enable
DataScene to draw the two fitted Gaussian peaks along with the source and
destination series. Move the Initial Position track bar so that the left
dotted line in Graph 1 moves to the center of the first peak. Leave the
value in the Initial Width spin box unchanged. By now, we have set the
initial position and width for the first peak. Increment the value in the Peak
Number spin box to 2. Move the Initial Position track bar so that the
right dotted line in Graph 1 moves to the center of the second peak.
Again, we accept the estimated width in the Initial Width spin box for
the second peak.
Fig. 1. Choosing the initial peak position for the second Gaussian peak.
Click the Apply button of the Series
Function dialog box to perform the fit. After the fit is finished,
DataScene automatically prints the fitting information including the fitted
parameters in the Messages window. DataScene
also automatically switches to the Nonlinear Fit tab-page where the the
fitted parameter are displayed in the Parameter List grid. For a simple
fit like this, DataScene should be able to converge within the maximum number of
iterations - which is by default 30 as shown in the Max. Iterations spin
box on the Nonlinear Fit tab-page. If this is not the case, you may need
to press the Apply button again to start another fit session using the
current parameters. After you finish fitting, close the Series
Function dialog box. In addition to the source and destination series,
two Line series are created to represent the two fitted Gaussian peaks. You may
want to change the colors of the series to make them more visible.
Fig. 2. Two-peak Gaussian fit of the line1 series.
Using User Defined Model Functions
Add a new data table, Table2, to the project. Set column formulas
"Row" and "Exp[-Cell("Table1", "Column 1",
Row) / 20] * Sin[Cell("Table1", "Column 1", Row) / 3] + Rnd()
/ 10" for rows 1 to 50 of Column 1 and Column 2 of Table2,
respectively. Plot a Line series, line1, in a
graph document, Graph 2, with Column 1 and Column 2 of Table2
as the Position and Value components, respectively.
Select the Graph: Series Function: Nonlinear Fit menu command. Switch
to the Nonlinear Fit tab-page of the Series
Function dialog box. With User Defined selected in the Function
Type combo-box, type "y0 + exp(-x/a0) * sin(x/b0)" in the Function
text box as the model function for the fit. In the Parameter List grid,
enter the three parameters: y0, a0, and b0 in the Name column; enter 0,
1, 1 in the Value column as the initial values of the three parameters,
respectively. The variable x in the model function is a built-in variable
representing the independent variable. You can verify this by opening the Formula
Editor (by clicking the Editor button) and switching to the Variable
& Parameters library mode to see the descriptions. Click the Apply
button to perform the fit. Although the initial values we provided are far away
for the optimized values, the nonlinear fit converged successfully for this
simple fit.
Fig. 3. Source line1 series and the fitted destination line2
series in Graph 2.
Close the project without saving it.
See Also
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