R2 between histogram and (weibull) distribution

Looking for the following:

R2 between histogram of parameter 'Weibull.Wh_Wp' and weibull distribution of parameter 'Weibull.Wh_Wp'.

[parmHat, parmCI]=wblfit(Weibull.Wh_Wp);
X=linspace(min(Weibull.Wh_Wp),max(Weibull.Wh_Wp));

plot(X,wblpdf(X,parmHat(1),parmHat(2)),'Color','k', 'LineWidth',1.3)
hold on
histogram(Weibull.Wh_Wp,'BinWidth',10,'FaceColor',[0.4660 0.6740 0.1880], 'EdgeColor', [0.45 0.45 0.45],'Normalization','pdf')

Below the figure of the parameter 'Weibull.Wh_Wp' with its histogram and fitted weibull distribution.

Answer 1

[Star Strider]

Try something like this —

wblfcn = @(p,x) wblpdf(x,p(1),p(2));

Weibull.Wh_Wp = wblrnd(150, 3, 1, 100);
X=linspace(min(Weibull.Wh_Wp),max(Weibull.Wh_Wp));

[parmHat,parmCI] = wblfit(Weibull.Wh_Wp);

plot(X,wblpdf(X,parmHat(1),parmHat(2)),'Color','k', 'LineWidth',1.3)
hold on
h1 = histogram(Weibull.Wh_Wp,'BinWidth',10,'FaceColor',[0.4660 0.6740 0.1880], 'EdgeColor', [0.45 0.45 0.45],'Normalization','pdf');

ctrs = h1.BinEdges(1:end-1) + diff(h1.BinEdges(1:2))/2;
y = h1.Values;

wblmdl = fitnlm(ctrs, h1.Values, wblfcn, parmHat)

wblmdl = 
Nonlinear regression model:
    y ~ wblpdf(x,p1,p2)

Estimated Coefficients:
          Estimate      SE       tStat       pValue  
          ________    _______    ______    __________

    p1     151.42      9.9346    15.241    1.7925e-12
    p2     2.6423     0.36199    7.2994    4.6607e-07


Number of observations: 22, Error degrees of freedom: 20
Root Mean Squared Error: 0.00251
R-Squared: 0.373,  Adjusted R-Squared 0.341
F-statistic vs. zero model: 41.9, p-value = 7.05e-08

I’m not certain that an value on a distribution fit using maximum likelihood is statistically correct, however this will provide it if desired. See the documentation for fitnlm for details on it.

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[Dora de Jong]

Thank you so much for your quick response and the script.

Now that I think about it more, I think you're right that the using a R2 value statistically not correct.

Do you have an idea how I can prove that a certain distribution fits well?

[Star Strider]

My pleasure.

The problem is that the independent variable value here is artificial (bin locations), and the dependent variable data are histogram counts.

Probably the most reliable fit information here is ‘parmCI’ since that tells how well the parameters are estimated. If the confidence limits for each parameter are of the same sign, the parameter is significant and the fit is appropriate. If they are opposite signs for any parameter, that means that the confidence limits include zero, and the parameter is not required in the estimation (so the fit is likely not good).

So I would use (or cite) the parameter confidence limits as the only necessary measure of the goodness-of-fit here.

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[Dora de Jong]

Thank you so much for explaining and taking the time. It has helped a lot.

[Star Strider]

As always, my pleasure!

Thank you!

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