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I have simulated a Gaussian curve in 50 bins of data. I have then repeated this many times, drawing the amplitude of the Gaussian from a log-normal distribution. Here are a 10 realizations:

(IMAGE 1) enter image description here

I actually create 10,000 realizations, I take the sum of the 50 bins (that you see above) for each of the 10,000 realizations and then I plot a histogram of the result:

(IMAGE 2) enter image description here

This histogram is a log-normal distribution, which I think makes sense because I am drawing the amplitudes (of my Gaussian curves) from a log-normal distribution.

So when I plot a log-normal curve over the top, it fits well:

(IMAGE 3) enter image description here

Now to make the situation more realistic, I am adding lots of noise to my Gaussian signals, so now they look more like:

(IMAGE 4) enter image description here

Now when I plot my histogram, the log-normal curve no longer matches the data. This is because of the noise that I have added of course:

(IMAGE 5) enter image description here

To find the function that best describes my histogram now, I believe that I have to take the convolution of my log-normal curve and a Gaussian curve that describes the distribution of the noise that I added.

Here is the histogram of the noise I added along with a Gaussian curve fitting the data (mean -0.07 standard deviation 42.89):

(IMAGE 6) enter image description here

So I believe that to describe the shape of the histogram in IMAGE 5, I need to convolve the dashed line in IMAGE 5 with the dashed line in IMAGE 6:

(IMAGE 7) enter image description here

So I try to do that convolution and I get the blue curve:

(IMAGE 8) enter image description here

So then to check if this convolution does indeed describe my histogram from IMAGE 5 I plot everything together:

(IMAGE 9) enter image description here

As you can see the convolution still doesn't quite fit the histogram. The curve is a little too wide.

Can anyone see what I am doing wrong here? Am I correct to think that to get the distribution of the sum of my 50 noisy data bins I need to convolve the log-normal distribution in IMAGE 2 with the Gaussian distribution in IMAGE 6?

Thank you!

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  • $\begingroup$ I'm not sure I fully understand this either, but you'd get the convolution if you add Gaussian noise to lognormally distributed data, i.e., the amplitudes. But you're saying that you add the noise to the "Gaussian signals" rather than directly to the lognormally distributed amplitudes, if I understand things correctly (of which I'm not sure). $\endgroup$ Jun 22, 2023 at 21:45
  • $\begingroup$ For your image 1 it might be better to plot dots along with the curves, since you are talking about bins. The use of curves is confusing (although defensible because it makes it easier to compare them) and make it appear as if they are functions of a continuous variable instead of discrete variable. $\endgroup$ Jun 22, 2023 at 21:54

2 Answers 2

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There is a discrepancy between the lognormal from image 3 and image 5. Check the peak height and position. (Not just the histogram, but also the red curve)

It seems like you have refitted a log-normal distribution to the data with noise and used that log-normal distribution for the convolution, instead of the original log-normal distribution.

example of difference

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As Sextus Empiricus already pointed out, the discrepancy is quite likely due to the wrong density for the amplitudes. I don't believe this stems from refitting to the data with noise, but suspect it could come from a minor typo or glitch in the parameter $\sigma$ of the lognormal.

The figure below illustrates how the outcome of the convolution chances when you replace the true distribution for the amplitudes by one with a larger standard deviation parameter. Here: if lognorm(4.8, 0.4) is used instead of lognorm(4.8, 0.2):

densities of signal, noise, and convolution

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