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Comparison of Nakagami and Gamma distributions for a list of data

Alex Teymouri
Posted 10 months ago

Hi,

I tried to evaluate the Nakagami and Gamma distributions for monthly wind speed data.

Which distribution is better?

How to show two distributions in one plot for comparison.

I appreciate your support and time.

Alex

data = {9.9`, 9.`, 12.7`, 13.3`, 11.3`, 10.`, 9.8`, 8.8`, 8.`, 7.4`, 6.8`, 
   6.8`, 7.1`, 9.4`, 8.4`, 10.`, 15.2`, 11.2`, 9.3`, 9.7`, 8.8`, 8.4`, 6.2`, 
   7.9`, 7.4`, 8.9`, 10.5`, 9.5`, 9.3`, 11.5`, 7.1`, 8.9`, 6.5`, 5.4`, 7.1`, 
   7.6`, 8.8`, 10.2`, 14.7`, 11.9`, 9.9`, 9.8`, 10.8`, 10.2`, 8.6`, 8.9`, 
   8.5`, 8.7`, 8.3`, 10.`, 9.`, 10.5`, 10.2`, 11.5`, 9.6`, 9.9`, 7.8`, 6.8`, 
   7.3`, 5.7`, 7.6`, 5.9`, 7.4`, 8.1`, 9.2`, 10.7`, 9.9`, 9.`, 7.8`, 7.`, 5.`,
    5.7`, 6.4`, 7.1`, 9.7`, 9.9`, 7.3`, 10.7`, 10.7`, 8.2`, 8.2`, 8.8`, 4.7`, 
   5.9`, 7.4`, 6.4`, 8.`, 9.9`, 7.4`, 10.`, 9.3`, 10.8`, 9.6`, 8.6`, 8.4`, 
   8.4`, 7.`, 8.6`, 9.6`, 7.9`, 9.9`, 11.5`, 10.1`, 9.7`, 8.2`, 6.7`, 4.6`, 
   7.5`, 6.`, 7.8`, 9.9`, 10.3`, 8.4`, 10.1`, 11.6`, 12.`, 10.`, 7.9`, 6.5`, 
   5.9`, 10.6`, 9.`, 10.3`, 9.2`, 10.3`, 11.`, 11.2`, 10.4`, 10.3`, 9.5`, 
   8.3`, 10.1`, 8.1`, 9.`, 10.6`, 11.4`, 10.6`, 12.2`, 11.5`, 12.2`, 10.7`, 
   9.4`, 8.2`, 8.5`, 9.3`, 11.3`, 11.5`, 10.3`, 11.`, 12.5`, 12.2`, 12.9`, 
   10.6`, 9.2`, 7.8`, 8.8`, 9.5`, 9.9`, 11.`, 10.9`, 11.7`, 12.8`, 11.4`, 
   13.1`, 12.7`, 10.9`, 9.6`, 10.6`, 10.1`, 12.8`, 11.5`, 11.2`, 11.2`, 11.9`,
    12.2`, 11.3`, 11.2`, 10.`, 8.9`, 7.9`, 8.7`, 11.`, 11.1`, 13.4`, 13.8`, 
   12.5`, 12.1`, 12.`, 10.1`, 10.8`, 8.1`, 10.9`, 9.3`, 11.9`, 12.2`, 11.1`, 
   11.3`, 12.1`, 10.9`, 9.9`, 11.2`, 9.5`, 10.`, 9.9`, 8.6`, 9.6`, 11.3`, 
   11.2`, 11.5`, 11.3`, 10.9`, 10.9`, 10.1`, 10.8`, 8.9`, 6.9`, 9.9`, 10.3`, 
   12.1`, 11.9`, 10.7`, 11.3`, 12.5`, 11.1`, 10.3`, 9.8`, 10.`, 8.8`, 11.3`, 
   10.9`, 12.6`, 11.7`, 11.3`, 11.5`, 12.5`, 11.`, 11.4`, 9.7`, 7.2`, 8.7`, 
   9.8`, 12.6`, 13.2`, 11.8`, 12.2`, 12.4`, 11.5`, 10.2`, 10.3`, 8.1`, 9.7`, 
   7.7`, 8.4`, 10.1`, 13.`, 12.8`, 12.3`, 13.2`, 12.`, 11.6`, 11.8`, 9.`, 
   8.9`, 8.1`, 9.7`, 10.8`, 13.5`, 12.8`, 13.5`, 12.8`, 11.9`, 12.4`, 10.3`, 
   9.5`, 8.4`, 7.7`, 11.4`, 9.8`, 12.8`, 12.4`, 13.1`, 12.5`, 13.1`, 12.3`, 
   9.9`, 8.9`, 8.2`, 6.8`};

(*Gamma Distribution*)

gam = EstimatedDistribution[data, GammaDistribution[\[Alpha], \[Beta]]];

Show[
 Histogram[data, 15, "PDF"],
 Plot[PDF[GammaDistribution[23.81, 0.41], x], {x, Min[data], Max[data]}, 
  Filling -> Axis, Exclusions -> None]]

(*Nakagami Distribution*)

nak = EstimatedDistribution[data, NakagamiDistribution[\[Mu], \[Omega]]];

Show[
 Histogram[data, 15, "PDF"],
 Plot[PDF[NakagamiDistribution[6.39, 102.37], x], {x, Min[data], Max[data]}, 
  Filling -> Axis, Exclusions -> None]]
POSTED BY: Alex Teymouri
6 Replies
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Jim Baldwin
Jim Baldwin
Posted 10 months ago

Which is best depends on what you mean by "best", your subject matter, and your objective.

Because you appear to be selecting between a Gamma and a Nakagami distribution, I would assume that there is no theoretical reason for either. As Henrik Schachner demonstrated even a Normal distribution has a smaller AIC value (which is likely involved in FindDistribution.

If the objective is just to summarize the distribution of this particular dataset, then a nonparametric density estimate (i.e., a SmoothHistogram) is what I would recommend. (It's the 21st century so histograms should not be the first thing to come to mind.)

If there is some appropriate subject matter knowledge or more specific objective is given, then better and more specific advice can be obtained.
POSTED BY: Jim Baldwin
Alex Teymouri
Alex Teymouri
Posted 10 months ago

Dear Rohit and Jim,

Thank you so much.

Is it possible to show that SmoothHistogram is better than the Nakagami and Gamma distributions by calculating the AIC values?

With best Regards
POSTED BY: Alex Teymouri
Jim Baldwin
Jim Baldwin
Posted 10 months ago

@RohitNamjoshi and @AlexTeyMouri: Don't you both want to have 2 * 2 (i.e., 2 * number of estimated parameters) rather than 2 * 1?

Also, is there a real need for a parametric representation of the distribution of the observed data? In other words, why not consider the following?

Show[Histogram[data, "FreedmanDiaconis", "PDF"], SmoothHistogram[data]]

Histogram and nonparametric density function
POSTED BY: Jim Baldwin
Rohit Namjoshi
Rohit Namjoshi
Posted 10 months ago

Hi Alex,

There are a couple of syntax errors. Try

aicgam = -2 LogLikelihood[GammaDistribution[\[Alpha], \[Beta]] /. mlegam, data] + 2*1

aicnak = -2 LogLikelihood[NakagamiDistribution[\[Mu], \[Omega]] /. mlenak, data] + 2*1
POSTED BY: Rohit Namjoshi
Alex Teymouri
Alex Teymouri
Posted 10 months ago

Thank you, Henrik.

As you know, we can evaluate these statistics distributions using the AIC.

Where I did I mistake?

(*Maximum Likelihood for Gamma*)

mlegam = 
 FindDistributionParameters[data, GammaDistribution[\[Alpha], \[Beta]]]

(*AIC for Gamma*)

aicgam = -2 LogLikelihood[
    GammaDistribution[{\[Alpha], \[Beta]} /. mlegam], data] + 2*1

(*Maximum Likelihood for Nakagami*)

mlenak = 
 FindDistributionParameters[data, 
  NakagamiDistribution[\[Mu], \[Omega]]]

(*AIC for Nakagami*)

aicnak = -2 LogLikelihood[
    NakagamiDistribution[{\[Mu], \[Omega]} /. mlenak], data] + 2*1
POSTED BY: Alex Teymouri

Hi Alex,

as it seems, Nakagami is slightly better. But if you let the system decide, the NormalDistribution winns:

gam = EstimatedDistribution[data, GammaDistribution[\[Alpha], \[Beta]]];
nak = EstimatedDistribution[data, NakagamiDistribution[\[Mu], \[Omega]]];
best = FindDistribution[data];

plot = Plot[{PDF[gam, x], PDF[nak, x], PDF[best, x]}, {x, Min[data], Max[data]}, PlotStyle -> {Red, Green, {Dashed, Blue}}, PlotLabels -> {"Gamma", "Nakagami", "best:Normal"}];

Show[Histogram[data, Automatic, "PDF"], plot, PlotRange -> {{4, 17.7}, Automatic}, ImageSize -> Large]

enter image description here

Does that help? Regards -- Henrik

POSTED BY: Henrik Schachner
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