posterior
Posterior probability of Gaussian mixture component
Description
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 j
th 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 j
th
Gaussian mixture component given observation i
, Probability(component
j
| observation i
).
Version History
Introduced in R2007b
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