The data points shown here are a synthetic sample meant to illustrate a typical spectrophotometric peak. These peaks are generally intermediate between a pure Gaussian peak and a pure Lorentzian peak.

A Gaussian peak is the first component of the model shown below; the Lorentzian peak is the second component. The overall model is a weighted average of the two.

To simulate experimental data, Gaussian noise (mean = 0, sigma = 1) was added to the computed data points. Parameter values used to synthesize the dataset are given below [in brackets]. Compare these values to the least-squares values (see below, and figure caption). Units are arbitrary.


y = C [p exp (−Log2 z^2) + (1 − p)/(z^2 + 1)] + D

where z = (x - A)/B and Log2 = log(2)


Since Regress+ can compute confidence intervals, it is interesting to consider these. In this example, the errors are Gaussian and the error variance is the same for all points. Therefore, an unweighted least-squares regression is appropriate. The optimum parameter values are as follows:
[R-squared = 0.99732, standard-error-of-estimate = 1.30564]

Least-squares Parameters

Here is the final Regress+ display showing these optimum values plus the respective 90-, 95-, and 99-percent central confidence intervals.


Numerical values for the central 95-percent confidence intervals, based on a nonparametric bootstrap (BCa method with 1,000 bootstrap samples), are as follows:

Parameter C is the most variable (thickest line in display) in units of its own, bootstrapped standard deviation. In this sense, C is the least precise parameter in the model (and A is the most precise).

This dataset is one of the examples included in the Regress+ software package.