Histogram Fit: Scaling and offset

I have one dimensional data (~12500 entries) with values reaching from ~135 to ~1150, yielding 3 peaks (see attachment).

Now I want to create a histogram showing the data distribution, as well as a fitting curve and a goodness of fit (chi squared) test.

Thus far I got the following:

    load Data.mat   

    bins = round(sqrt(length(Data)));           % Number of bins
    [f, x] = hist(Data,bins);                   % Calculate histogram
    
    pd = fitdist(x','Kernel');                  % Calculate fit
    y = pdf(pd,f);                              % Calculate pdf
    
    
    figure(1)
    dx = diff(x(1:2));
    bar(x, f/sum(f*dx));                        % Normalizing and plotting
    hold on
    
    plot(x,y,'Linewidth',2)                     % Plot fit
    hold off  
    
    [h,p] = chi2gof(x,'CDF',pd,'Alpha',0.05);   % Chi squared test

While my chi2gof test yields expected results (h=0 ; p = 0.9983) my plot doesn't look to well:

The scale of the fitting curve sems to be way off for all 3 peaks. Additionally I'd expect the curve to get a lot closer to 0 for very low and very high values.

Thanks in advance for any suggestions on how to improve/fix my code!

[Jeff Miller]

Regarding the scaling problem, what is sum(y*dx)?

Regarding the above-0 tails of the estimated pdf, do they drop off when you compute the density over a wider range, e.g. -500 to 2000? If so, the problem may be that the kernel bandwidth is not optimal. The default is to choose a good bandwidth to estimate a normal distribution. This looks much more like a mixture of three different distributions, so MATLAB's bandwidth guess may be pretty far from optimal.

[Marcel Dorer]

@Scaling problem

sum(y*dx) = 0.3353

I also attached the data file to the OP, in case it's needed :)

@ 0-tails

Thanks, I managed to achieve better tails by adjusting the bandwidth of the kernel!

pd = fitdist(x','Kernel','Width',75);

Answer 1

[Jeff Miller]

I think there are a couple of problems. Try this:

load Data.mat
bins = round(sqrt(length(Data)));           % Number of bins
[f, x] = hist(Data,bins);                   % Calculate histogram
pd = fitdist(Data,'Kernel','Width',5);      % Calculate fit
y = pdf(pd,x);                              % Calculate pdf of bin values

figure(1)
dx = diff(x(1:2));
bar(x, f/sum(f*dx));                        % Normalizing and plotting
hold on

y = y / sum(y*dx);

plot(x,y,'Linewidth',2)                     % Plot fit
hold off

[h,p] = chi2gof(x,'CDF',pd,'Alpha',0.05);   % Chi squared test

[Marcel Dorer]

Works perfectly! Thanks a lot!

I didn't work a lot with fitdist and pdf yet, I'm glad you were able to point out my mistakes :)