pdf

Probability density function for Gaussian mixture distribution

Syntax

y = pdf(gm,X)

Description

y = pdf(gm,X) returns the probability density function (pdf) of the Gaussian mixture distribution gm, evaluated at the values in X.

Examples

Compute pdf Values

Create a gmdistribution object and compute its pdf values.

Define the distribution parameters (means and covariances) of a two-component bivariate Gaussian mixture distribution.

mu = [1 2;-3 -5];
sigma = [1 1]; % shared diagonal covariance matrix

Create a gmdistribution object by using the gmdistribution function. By default, the function creates an equal proportion mixture.

gm = gmdistribution(mu,sigma)

gm = 

Gaussian mixture distribution with 2 components in 2 dimensions
Component 1:
Mixing proportion: 0.500000
Mean:     1     2

Component 2:
Mixing proportion: 0.500000
Mean:    -3    -5

Compute the pdf values of gm.

X = [0 0;1 2;3 3;5 3];
pdf(gm,X)

ans = 4×1

    0.0065
    0.0796
    0.0065
    0.0000

Plot pdf

Create a gmdistribution object and plot its pdf.

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

p = [0.4 0.6];               % Mixing proportions     
mu = [1 2;-3 -5];            % Means
sigma = cat(3,[2 .5],[1 1])  % Covariances 1-by-2-by-2 array

sigma = 
sigma(:,:,1) =

    2.0000    0.5000


sigma(:,:,2) =

     1     1

The cat function concatenates the covariances along the third array dimension. The defined covariance matrices are diagonal matrices. sigma(1,:,i) contains the diagonal elements of the covariance matrix of component i.

Create a gmdistribution object by using the gmdistribution function.

gm = gmdistribution(mu,sigma)

gm = 

Gaussian mixture distribution with 2 components in 2 dimensions
Component 1:
Mixing proportion: 0.500000
Mean:     1     2

Component 2:
Mixing proportion: 0.500000
Mean:    -3    -5

Plot the pdf of the Gaussian mixture distribution by using fsurf.

gmPDF = @(x,y) arrayfun(@(x0,y0) pdf(gm,[x0 y0]),x,y);
fsurf(gmPDF,[-10 10])

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 — Values at which to evaluate pdf

n-by-m numeric matrix

Values at which to evaluate the pdf, 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.

Data Types: single | double

Output Arguments

y — pdf values

n-by-1 numeric vector

pdf values of the Gaussian mixture distribution gm, evaluated at X, returned as an n-by-1 numeric vector, where n is the number of observations in X.

The pdf function computes the pdf values by using the likelihood of each component given each observation and the component probabilities.

$$y(i)={\displaystyle \sum _{j=1}^{k}L(C_j|O_i)\text{P}({\text{C}}_j)},$$

where L(C_j|O_j) is the likelihood of component j given observation i, and P(C_j) is the probability of component j. The pdf function computes the likelihood term by using the multivariate normal pdf of the jth Gaussian mixture component evaluated at observation i. The component probabilities are the mixing proportions of mixture components, the ComponentProportion property of gm.

Version History

Introduced in R2007b