posterior

Posterior probability of Gaussian mixture component

Syntax

P = posterior(gm,X)

[P,nlogL] = posterior(gm,X)

Description

P = posterior(gm,X) returns the posterior probability of each Gaussian mixture component in gm given each observation in X.

[P,nlogL] = posterior(gm,X) also returns the negative loglikelihood of the Gaussian mixture model gm given the data X.

Examples

Compute Posterior Probabilities

Generate random variates that follow a mixture of two bivariate Gaussian distributions by using the mvnrnd function. Fit a Gaussian mixture model (GMM) to the generated data by using the fitgmdist function, and then compute the posterior probabilities of the mixture components.

Define the distribution parameters (means and covariances) of two bivariate Gaussian mixture components.

mu1 = [2 2];          % Mean of the 1st component
sigma1 = [2 0; 0 1];  % Covariance of the 1st component
mu2 = [-2 -1];        % Mean of the 2nd component
sigma2 = [1 0; 0 1];  % Covariance of the 2nd component

Generate an equal number of random variates from each component, and combine the two sets of random variates.

rng('default') % For reproducibility
r1 = mvnrnd(mu1,sigma1,1000);
r2 = mvnrnd(mu2,sigma2,1000);
X = [r1; r2];

The combined data set X contains random variates following a mixture of two bivariate Gaussian distributions.

Fit a two-component GMM to X.

gm = fitgmdist(X,2)

gm = 

Gaussian mixture distribution with 2 components in 2 dimensions
Component 1:
Mixing proportion: 0.500765
Mean:   -1.9675   -0.9654

Component 2:
Mixing proportion: 0.499235
Mean:    1.9657    2.0342

Plot X by using scatter. Visualize the fitted model gm by using pdf and fcontour.

figure
scatter(X(:,1),X(:,2),10,'.') % Scatter plot with points of size 10
hold on
gmPDF = @(x,y) arrayfun(@(x0,y0) pdf(gm,[x0 y0]),x,y);
fcontour(gmPDF,[-6 8 -4 6])
c1 = colorbar;
ylabel(c1,'Probability Density Function')

Compute the posterior probabilities of the components.

P = posterior(gm,X);

P(i,j) is the posterior probability of the jth Gaussian mixture component given observation i.

Plot the posterior probabilities of Component 1 by using the scatter function. Use the circle colors to visualize the posterior probability values.

figure
scatter(X(:,1),X(:,2),10,P(:,1))
c2 = colorbar;
ylabel(c2,'Posterior Probability of Component 1')

Plot the posterior probabilities of Component 2.

figure
scatter(X(:,1),X(:,2),10,P(:,2))
c3 = colorbar;
ylabel(c3,'Posterior Probability of Component 2')

Input Arguments

gm — Gaussian mixture distribution

gmdistribution object

Gaussian mixture distribution, also called Gaussian mixture model (GMM), specified as a gmdistribution object.

You can create a gmdistribution object using gmdistribution or fitgmdist. Use the gmdistribution function to create a gmdistribution object by specifying the distribution parameters. Use the fitgmdist function to fit a gmdistribution model to data given a fixed number of components.

X — Data

n-by-m numeric matrix

Data, specified as an n-by-m numeric matrix, where n is the number of observations and m is the number of variables in each observation.

If a row of X contains NaNs, then posterior excludes the row from the computation. The corresponding value in P is NaN.

Data Types: single | double

Output Arguments

P — Posterior probability

n-by-k numeric vector

Posterior probability of each Gaussian mixture component in gm given each observation in X, returned as an n-by-k numeric vector, where n is the number of observations in X and k is the number of mixture components in gm.

P(i,j) is the posterior probability of the jth Gaussian mixture component given observation i, Probability(component j | observation i).

nlogL — Negative loglikelihood

numeric value

Negative loglikelihood value of the Gaussian mixture model gm given the data X, returned as a numeric value.

Version History

Introduced in R2007b