This repository contains MATLAB codes for Inverse-Gamma Normal mixture model.
Other codes for mixture model application for supercritical fluids (Lognormal-Nakagami mixture, Probabilistic classification) will be uploaded soon.
If you use these codes for the publication, please cite one of the following articles.
[1] Yoon, T. J., Ha, M. Y., Lee, W. B., & Lee, Y.-W. (2017),The Journal of Supercritical Fluids, 119, 36-43.
[2] Yoon, T. J., Ha, M. Y., Lee, W. B., & Lee, Y.-W. (2017), The Journal of Supercritical Fluids, 130, 364-372.
If you have any questions, please contact me with my e-mail or researchgate account. e-mail: oheim124@gmail.com
The specific information of these codes is as follows.
This code calculates the likelihood of the Inverse Gamma distribution. Please refer to Wikipedia page for the description of Inverse Gamma distribution.Population parameters of the Inverse Gamma distribution are originally shape parameter and scale parameter, but this code is written with arithmetic mean and standrad deviation as the input parameters because of the practical purposes. You can modify the code based on the Wikipedia webpage.
This codeblock draws an Inverse Gamma distribution whose mean is 5 and standard deviation is 2.
x = 0.01:0.01:20;
y = invgampdf(x,5,2);
plot(x,y)This code estimates the population parameters of the Inverse Gamma distribution based on the sample data.
data = gamrnd(10,20,[10000,1]); % random number generation sampled from the Gamma distribution
data = data.^(-1);
m1 = mean(data);
s1 = std(data);
% So far, we generated the samples which follow the Inverse Gamma distribution whose arithmetic
% mean and standard deviation is m1 and s1.
[mu,sigma]=invgamfit(data); % MLE for the Inverse Gamma distribution.
% You can check if the algorithm worked by comparing (m1, s1) with (mu and sigma)
% Let us check if the algorithm worked.
histogram(data,'Normalization','pdf')
hold on
x = 1E-4:1E-4:0.02;
y = invgampdf(x,mu,sigma);
plot(x,y);Output: arithmetic means (mu), standard deviations(sigma), and mixture weights(p) of the Inverse Gamma and the Normal distributions.
This code calculates the population parameters of the Inverse Gamma-Normal mixture model using the Method of Moments.
If the convergence rate is too slow because of a large number of population data, they can be randomly sampled.
In this example, we generate the synthetic data which consist of the random numbers sampled from the Inverse Gamma distribution and the Normal distribution.
data1 = gamrnd(10,20,[100000,1]); % random numbers sampled from the Gamma distribution
data1 = data1.^(-1); % Mean and standard deviation of these random numbers are 0.0055 and 0.0020, respectively.
data2 = normrnd(0.012,0.002,[100000,1]); % random numbers sampled from the Normal distribution
data = [data1;data2];
histogram(data,'Normalization','pdf')
hold onWe apply the Method of Moments to the data.
[mu,sigma,p,counter]=ignmom(data,1E-7,100000);
x = 1E-4:1E-4:.02;
y1 = p(1)*invgampdf(x,mu(1),sigma(1));
y2 = p(2)*normpdf(x,mu(2),sigma(2));
plot(x,y1); plot(x,y2); plot(x,y1+y2);By comparing the obtained parameters and probability distribution functions with the original data, you can know that the mixture model was well fitted to the synthetic data.
Output: arithmetic means (mu), standard deviations(sigma), and mixture weights(p) of the Inverse Gamma and the Normal distributions, loglikelihood (loglikelihood)
This code calculates the population parameters of the Inverse Gamma-Normal mixture model using the Maximum Likliehood Estimation (MLE). If the convergence rate is too slow because of a large number of population data, they can be randomly sampled.
The synthetic data obtained in the previous example is used.
We apply the Maximum Likelihood Estimation (MLE) to the data.
data1 = gamrnd(10,20,[100000,1]); % random numbers sampled from the Gamma distribution
data1 = data1.^(-1); % Mean and standard deviation of these random numbers are 0.0055 and 0.0020, respectively.
data2 = normrnd(0.012,0.002,[100000,1]); % random numbers sampled from the Normal distribution
data = [data1;data2];
histogram(data,'Normalization','pdf')
hold on
[mu, sigma, p, counter, loglikelihood]=ignmle(data,1E-7,10000);
x=1E-4:1E-4:.02;
y1=p(1)*invgampdf(x,mu(1),sigma(1));
y2=p(2)*normpdf(x,mu(2),sigma(2));
plot(x,y1)
hold on
plot(x,y2)
plot(x,y1+y2)By comparing the obtained parameters and probability distribution functions with the original data, you can know that the mixture model was well fitted to the synthetic data.