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I am trying to fit some data that looks like a bell-curve: we reach a maximum at some value close to the mean, then the graph falls towards zero as we get further away from it. I am not the "owner" of the data so I cannot share it with you here, but I think the idea is clear with the "fake data" below

I would like to find a non-linear model that can fit that type of data, but my search did not give me much information. What are your suggestions?

The data looks something like this

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  • $\begingroup$ Have you tried a regular linear model already? How were the residuals? Also, is your real data also bounded in 0 and 1000? $\endgroup$ – user2974951 Aug 30 '19 at 9:39
  • $\begingroup$ @user2974951 The data is in principle not bounded, but we would get all zeros if we go too much beyond. A regular linear model does not work here as there is no upward/downward trend $\endgroup$ – David Aug 30 '19 at 9:49
  • $\begingroup$ @David If you have other variables try a regular linear model first and check the residuals. If this is all the data that you have then, as mkt suggested, try a GAM model. $\endgroup$ – user2974951 Aug 30 '19 at 10:01
  • $\begingroup$ @user2974951 There are no other predictors. All the data is plotted there $\endgroup$ – David Aug 30 '19 at 10:03
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    $\begingroup$ Related: How to fit intensity peaks from a image? $\endgroup$ – gung - Reinstate Monica Aug 30 '19 at 12:05
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If your goal is to just describe the pattern, you could try a GAMM i.e. generalized additive mixed model. Choose the residual distribution to reflect the zero bound and any other data properties you may be aware of.

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  • $\begingroup$ Why should I use an additive model when I have only one predictor? Also, what residual distribution should I use? $\endgroup$ – David Aug 30 '19 at 11:50
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    $\begingroup$ @David You may not need the additivity, but GAM(M)s are a convenient way to fit smooth responses. Splines, etc do the same, but GAMMs offer you the added benefit of allowing you to pick your distribution. I can't propose one without a more detailed description of your data and its properties (which you can edit into the question, if you want to). $\endgroup$ – mkt - Reinstate Monica Aug 30 '19 at 11:53
  • $\begingroup$ What would you do if, for instance you knew $y=f(x)$ where $f$ follows the shape of a normal-distribution density function? $\endgroup$ – David Aug 30 '19 at 11:56
  • $\begingroup$ @David If you have a good reason to expect that the response shape is captured by a specific equation/functional form, just fit that. But if you're just eyeballing it and deciding that it looks normal-ish, you don't lose much by using a GAM. It's your call and depends on your reasoning, goals and the nature of the data. FWIW, calling this a normal distribution doesn't obviously make sense to me here. $\endgroup$ – mkt - Reinstate Monica Aug 30 '19 at 11:58

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