Lorenzian and gaussian pdf-function fitting with Matlabs nlinfit, confidence interval

[deccard]

I have data that I want to fit to both Gaussian and Lorentzian (Cauchy) distribution. I have been using Matlab's nlinfit as follows:

gaus = @(p,xdata) (p(1)/(sqrt(2*pi*p(2)))*exp(-(xdata-p(3)).^2/(2*p(2)))+min).*weights;
[g_pfit,g_residual,g_J]=nlinfit(data(:,1), data(:,2).*weights, gaus, [4030 2 -5]);
g_ci=nlparci(g_pfit,g_residual,'jacobian',J,'alpha',0.317);

loren = @(p,xdata) p(1)./(pi*p(2)*(1+((xdata-p(3))./p(2)).^2))+p(4);
[l_pfit,l_residual,l_J]=nlinfit(data(:,1), data(:,2), loren, [10030 0.5 -5 4]);
l_ci=nlparci(l_pfit,l_residual,'jacobian',l_J,'alpha',0.317);

The strange thing, however, is that my data is more like Gaussian-shaped and Gaussian curve, is by eye way more better fit. Still I get smaller errors for the width of Lorentzian fit than Gaussian (using the nlparci function).

Why would I get smaller errors for the width of the Lorentzian curve than for the width of the Gaussian curve, which is a better fit?

deccard

 

[Aaron Bex]

1: That's an interesting question! It could be that the curve fitting algorithms used by Matlab's nlinfit function are better for Lorentzian distributions than for Gaussian distributions, so it's possible that you're getting more accurate results with the Lorentzian fit. It could also be that the data is more complex than either of the two distributions can capture, so the algorithm is getting a better result with the Lorentzian fit. You might want to try out some other curve fitting algorithms to see if you can get different results.

 

[oswaler]

,

Thank you for sharing your method of fitting data to both Gaussian and Lorenzian distributions using Matlab's nlinfit function. It is not uncommon to use this approach when dealing with data that may have a mixture of both distributions present. However, it is important to understand the limitations of this approach and the potential reasons for the differences in the errors for the width of the two curves.

Firstly, it is important to note that the Gaussian and Lorenzian distributions have different shapes and therefore may not be equally suitable for fitting a given dataset. The Gaussian distribution is typically used for data that follows a normal distribution, while the Lorenzian distribution is often used for data that has a heavy-tailed or skewed distribution. Therefore, it is possible that your data may not be a perfect fit for either distribution, which could explain why the Gaussian curve appears to fit better by eye.

Secondly, the nlparci function calculates the confidence interval for the parameters based on the assumption that the model is linear in the parameters. This means that the confidence interval may not accurately reflect the uncertainty in the parameters for non-linear models such as the Gaussian and Lorenzian distributions. Therefore, the smaller errors for the width of the Lorenzian curve may not necessarily indicate a better fit, but rather a limitation of the method used to calculate the confidence interval.

In summary, it is important to carefully consider the shape of your data and the assumptions made when fitting it to a particular distribution. It is also worth exploring alternative methods for calculating the confidence interval for non-linear models to ensure more accurate results.