# How to quantify elevated tails of a Gaussian like signal?

From some simulations I have obtained as an output a signal which roughly looks like a Gaussian with some elevated tails. Note that the input was a Gaussian.

Now I would like to quantify the deviation from the Gaussian, or, to be more specific, I would like to quantify the elevation of the tails. Any ideas how to do that?

(Maybe fitting some kind of "almost-Gaussian" type of function where the elevation of the tails is one of the fit parameter, but what would I use then as "almost-Gaussian" type of function?)

• Kurtosis is defined as basically exactly what you're asking for. – probably_someone Aug 4 '17 at 10:24
• @probably_someone hmm, that works if I have a data set, but in the case where I have "continuous" signal (i.e. the points on the measured/obtained curve are equidistant), I cannot apply the standard kurtosis functions (looking at the corresponding histogram is probably a better explanation, see here) – Alf Aug 4 '17 at 10:36
• I don't really know what you mean here. For either the discrete or the continuous case, you can simply implement the appropriate definition of kurtosis, as given by Wikipedia, Wolfram MathWorld and others. True, there may not be library functions that do this, but the coding required is pretty basic, so it shouldn't be a problem. – probably_someone Aug 4 '17 at 10:41
• @probably_someone what you are saying is basically that it does not matter that I have a continuous signal: to calculate its mean (which I need to calculate the kurtosis), I would need to use something like a weighted mean, where the height of the measured curve at a certain position corresponds to its weight...? (I would then get the mean position) – Alf Aug 4 '17 at 10:52

Besides the Gaussian or normal distribution, the Cauchy, or Lorentzian or Breit–Wigner distribution shows up frequently in physics and engineering phenomenon and measurements. While the Gaussian is its own Fourier transform, the Lorentzian is the Fourier transform of an exponential decay for t>0.

Gaussian distirbution (solid line): $exp(-x^2)$

Cauchy distribution (dotted line): $1/(1+x^2)$

Spectral line shapes often sums of Voigt distributions, which are the convolution of a natural Cauchy line shape with an instrumental plus thermal-doppler broadenging modeled as a gaussian shape.

A good goal would be to first try to better understand the physics behind the shape of your distribution then choose functional forms that are representative of what is actually happening. It is possible that you should not expect your signal to be Gaussian shaped in the first place. • it is actually as you said: depending on the simulations parameters, i.e. what happens between input and output, I cannot expect the signal at the output to be Gaussian. A Cauchy distribution seems to be what I was looking for (I'll check now), thanks! – Alf Aug 4 '17 at 11:40
• I'm happy if it helps! If you get some data and would like to post a new, and more specific question about it, and include a description, you may get more help with the choice of a fitting function. – uhoh Aug 4 '17 at 11:50
• thanks, for now, I will play around a little bit first :) (and then considering to ask a more specific question). – Alf Aug 4 '17 at 11:52