## 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

$$y(x)=a{e}^{-bx}+{a}_{1}{e}^{-{\left(\frac{x-{b}_{1}}{{c}_{1}}\right)}^{2}}+{a}_{2}{e}^{-{\left(\frac{x-{b}_{2}}{{c}_{2}}\right)}^{2}}$$

where *a _{i}* are the peak
amplitudes,

*b*are the peak centroids, and

_{i}*c*are related to the peak widths. Because unknown coefficients are part of the exponential function arguments, the equation is nonlinear.

_{i}

Load the data and open the Curve Fitter app.

`load gauss3 curveFitter`

The workspace contains two new variables:

`xpeak`

is a vector of predictor values.`ypeak`

is a vector of response values.

In the Curve Fitter app, on the

**Curve Fitter**tab, in the**Data**section, click**Select Data**. In the Select Fitting Data dialog box, select`xpeak`

as the**X data**value and`ypeak`

as the**Y data**value. Enter`Gauss2exp1`

as the**Fit name**value.On the

**Curve Fitter**tab, in the**Fit Type**section, click the arrow to open the gallery. In the fit gallery, click**Custom Equation**in the**Custom**group.In the

**Fit Options**pane, 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, constrain

*a*_{1},*a*_{2},*c*_{1}, and*c*_{2}to be greater than 0.In the

**Fit Options**pane, click**Advanced Options**.In the

**Coefficient Constraints**table, change the**Lower**bound for*a*_{1},*a*_{2},*c*_{1}, and*c*_{2}to`0`

, as the peak amplitudes and widths cannot be negative.Enter the

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

Observe the fit and residuals plots. To create a residuals plot, click

**Residuals Plot**in the**Visualization**section of the**Curve Fitter**tab.