## 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*_{i} are the peak
centroids, and *c*_{i} are 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 Fitting app:

The workspace contains two new variables:

In the Curve Fitting app, select `xpeak`

for **X
data** and `ypeak`

for **Y data**.

Enter `Gauss2exp1`

for the **Fit
name**.

Select `Custom Equation`

for
the model type.

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.

Click **Fit Options**.

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 start points as shown for the unknown coefficients.

Unknowns | Start
Point |

`a` | 100 |

`a1` | 100 |

`a2` | 80 |

`b` | 0.1 |

`b1` | 110 |

`b2` | 140 |

`c1` | 20 |

`c2` | 20 |

As you change fit options, the Curve Fitting app refits. Press **Enter** or
close the Fit Options dialog box to ensure your last change is applied
to the fit.

Following are the fit and residuals.