Gaussian Fitting with an Exponential Background

This example fits two poorly resolved Gaussian peaks on a decaying exponential background using a general (nonlinear) custom model.

Fit the data using this equation

$y (x) = a e^{- b x} + a_{1} e^{- {(\frac{x - b_{1}}{c_{1}})}^{2}} + a_{2} e^{- {(\frac{x - b_{2}}{c_{2}})}^{2}}$

wherea_iare the peak amplitudes,b_iare the peak centroids, andc_iare related to the peak widths. Because unknown coefficients are part of the exponential function arguments, the equation is nonlinear.

Load the data and open the Curve Fitter app.
```
loadgauss3curveFitter
```
The workspace contains two new variables:
- xpeakis a vector of predictor values.
- ypeakis a vector of response values.
In the Curve Fitter app, on theCurve Fittertab, in theData部分中,点击Select Data. In the Select Fitting Data dialog box, selectxpeakas theX Datavalue andypeakas theY Datavalue. EnterGauss2exp1as theFit namevalue.
On theCurve Fittertab, in theFit Type部分中,点击the arrow to open the gallery. In the fit gallery, clickCustom Equationin theCustomgroup.
In theFit Optionspane, replace the example text in the equation edit box with these terms:
```
a*exp(-b*x) + a1*exp(-((x-b1)/c1)^2) + a2*exp(-((x-b2)/c2)^2)
```
The fit is poor (or incomplete) at this point because the starting points are randomly selected and no coefficients have bounds.

Specify reasonable coefficient starting points and constraints. Deducing the starting points is particularly easy for the current model because the Gaussian coefficients have a straightforward interpretation and the exponential background is well defined. Additionally, as the peak amplitudes and widths cannot be negative, constraina₁,a₂,c₁, andc₂to be greater than 0.

In theFit Optionspane, clickAdvanced Options.
In theCoefficient Constraintstable, change theLowerbound fora₁,a₂,c₁, andc₂to0, as the peak amplitudes and widths cannot be negative.

Enter theStartPointvalues as shown for the specified coefficients.

Coefficients	Start Point
`a`	100
`a1`	100
`a2`	80
`b`	0.1
`b1`	110
`b2`	140
`c1`	20
`c2`	20

Advanced Options section with specified start points and lower bounds for the coefficients

As you change the fit options, the Curve Fitter app updates the fit.

Observe the fit and residuals plots. To create a residuals plot, clickResiduals Plotin theVisualizationsection of theCurve Fittertab.