Plotting Gamma Distribution, log-normal and histogram

Hello all,

I have witten a piece of code to plot

Gamma Distribution

Log-normal distribution

Histogram (pdf)

from my raw data (droplet diameter).

I am not sure if I have done it correctly because the height of the bins do not match with the fittings (see image). Have I done something wrong? I have also attached the code and data.

Thanks in advance

clear all
close all

%% 

load droplet_diameter







%%gamma plot

x= 0:0.01:8;

Y= gamfit(droplet_diameter) 
G= gampdf(x,Y(1,1),Y(1,2))
 
figure
plot(x,G)

%%log

mean=mean(droplet_diameter);
variance= var(droplet_diameter);
sigma= (variance)^0.5
 
 
y_lognormal=lognpdf(x,log(mean),sigma)

figure
plot(x,y_lognormal)

hold on
 plot(x,G)
 hold on
 histogram(droplet_diameter,'Normalization','pdf')

     

[markus tripolt]

Hi

I think it is too late, but because I am curious.

Could it be because you used for your y_lognormal the log(mean), but the log of the mean (log(mean)) value is not the same like the mean of the log of the indivudial values (mean(log(xi))).

Kind regards

Markus

Answer 1

[John D'Errico]

Hmm. I recall answering exactly this same question recently. For a second or so, I wondered if I should expect a spate of such questions. (Did something change when histogramwas introduced?) But you did properly use the histogram method of 'pdf', so the scaling is correct.

That suggests the problem is lack of fit. When I look at the histogram, I'd bet that your actual distribution has a bit heavier upper tail than either a lognormal or a gamma distribution can offer. So the mismatch is in terms of a poorly fitting distribution.

Does that mean your distribution is not in fact either lognormal or gamma in reality? Well, NO real world sampled data ever truly follows any fixed distribution. There is always some amount of crap in there as noise. As well, your distribution is based on far too little data to pin down anything very well. So the true distribution might have been either of those or maybe neither one. At best, you can use a statistcal test to decide which is most probable.

Basically, get better (more) data.