Generation of large data sets (10^4) for log Pearson type III distribution and finding its paramters

Dear All, I have a data set which is log pearson type III distributed fitted. Using that data how can I find the parameters associated with that distribution. further using that paramters I have to generate more number of samples. That samples which I have generated should overlap twith the data given below. Can anyone please help me with this I am working on this from a long time.

Q PDF(LP3)

14.162562 0.000011

28.325123 0.000723

31.403941 0.001391

33.866995 0.002169

36.945813 0.002904

39.408867 0.003638

41.871921 0.004439

45.566502 0.005262

49.261084 0.006063

53.571429 0.006875

59.729064 0.007632

70.197044 0.008288

89.901478 0.007654

100.369458 0.006875

113.300493 0.005996

125.000000 0.005173

137.315271 0.004350

150.862069 0.003571

167.487685 0.002815

192.118227 0.002025

225.985222 0.001246

275.246305 0.000679

326.970443 0.000345

370.073892 0.000211

418.103448 0.000133

459.359606 0.000089

500 5.56242E-05

[Torsten]

Dear All, I have a data set which is log pearson type III distributed fitted. Using that data how can I find the parameters associated with that distribution.

So you assume that your data follow the log pearson type III distribution and you want to find the parameters of this distribution ?

Maybe you could include a link on how the distribution with its parameters looks like.

[Raj Arora]

No, actually this data surely fits log pearson type 3 distribution. I have digitized this data from a research article. If you can plot it Q vs PDF it will show you.

[Raj Arora]

https://ascelibrary.org/doi/full/10.1061/%28ASCE%291084-0699%282007%2912%3A5%28482%29

this article mentions the pdf and the paramter associated with it.

[Torsten]

Too much to read and understand - sorry.

Answer 1

[Jeff Miller]

If the distribution you have in mind is the same one described here then this is a good problem for Cupid (see especially the file DemoLogPearsonIII.m).

% Initialize distribution information
Q = [14.162562, 28.325123, 31.403941, 33.866995, 36.945813, 39.408867, 41.871921, 45.566502, 49.261084, 53.571429, 59.729064, 70.197044, 89.901478, 100.369458, 113.300493, 125.000000, 137.315271, 150.862069, 167.487685, 192.118227, 225.985222, 275.246305, 326.970443, 370.073892, 418.103448, 459.359606, 500];
PDF = [0.000011,   0.000723,   0.001391,   0.002169,   0.002904,   0.003638,   0.004439,   0.005262,   0.00606,   0.006875,   0.007632,   0.008288,   0.007654,   0.006875,   0.005996,   0.005173,   0.004350,   0.003571,   0.002815,   0.002025,   0.001246,   0.000679,   0.000345,   0.000211,   0.000133,   0.000089,  5.56242E-05];

% Create a starting LogPearsonIII distribution with initial guesses
% for the parameters (which will be estimated later):
GammaShape = 5;
GammaRate = 5;
ShiftConst = 2;

% RNGamma, AddTrans, and ExpTrans are all parts of Cupid
startGamma = RNGamma(GammaShape,GammaRate);

% Shift the gamma by adding in a constant. The resulting distribution
% is a Pearson III distribution:
PearsonIII = AddTrans(startGamma,ShiftConst);

% The Log-Pearson III is a distribution whose log is Pearson III.
% That means that the Log-Pearson III distribution is an
% exponential transformation of the original PearsonIII.
% So, we can make the desired Log-Pearson III by using the Exp transform:

LogPearsonIII = ExpTrans(PearsonIII);
% LogPearsonIII is now a distribution object with the starting guesses for the parameter estimates.

% Next estimate the distribution's parameters by trying to match the PDF values at the Q points:
LogPearsonIII.EstDensity(Q,PDF);

% Print the estimated parameter values:
LogPearsonIII.ParmValues
% ans =
%       26.803       8.8801       1.6729

% Display the PDF and CDF of the estimated distribution
LogPearsonIII.PlotDens;

% Generate 1000 random numbers from the fitted distribution:
randlp = LogPearsonIII.Random(1000,1);

% Compare the histogram of the random numbers with the PDF originally specified.
figure;
histogram(randlp,'normalization','pdf');
hold on
plot(Q,PDF);

% You can also generate the random numbers like this:
randlp2 = exp( gamrnd(26.803,1/8.8801,1000,1)+1.6729 );

[Raj Arora]

Thanks Jeff Miller for you valuable comment. Yes this is the same distribution that you attached here and it is used for prediction of floods. Earlier I was having problem with RNGamma. Than I downloaded cupid file and installed it in matlab. Now an error is coming while running this code in matlab can you suggest something how I can reduce this error and run this code. So that I can have a look what is the final repsonse of this code.

Unrecognized method, property, or field 'EstDensity' for class 'ExpTrans'.

Error in Jeffmilerresponse (line 21)

LogPearsonIII.EstDensity(Q,PDF);

[Jeff Miller]

Hi Raj Arora,

I guess you installed Cupid with the Cupid.mltbx file -- sorry, I have not kept mltbx completely up to date.

Here is one shortcut that might work: go back to GitHub, download the newest file dGeneric.m, and use it to replace the older version of the file that was installed from mltbx. Then try the script again.

If that doesn't work you will probably have to download the whole project from GitHub instead of using the mltbx file, or just wait for me to update mltbx tomorrow.

[Raj Arora]

Yes Jeff, I have downloaded the Cupid.mltbx file. Ok let me try with this dGeneric.m I will let you know whether I am able to proceed or not.

[Raj Arora]

Thankyou very much Jeff for sharing this valuable information and time I was stuck on this problem from a long time. I downloaded the entire project form Github and now I pasted your code in that folder and your code is running smooothly. Finallly I am able to see the final aspect which seems to me totally correct.

Thankyou Jeff once again.

[Jeff Miller]

You are welcome, Raj. I am glad that Cupid was useful to you.