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Gaussian Mixture Models

Introduction to Gaussian Mixture Models

Gaussian mixture models are formed by combining multivariate normal density components.

In Statistics Toolbox™ software, use the gmdistribution class to fit data using an expectation maximization (EM) algorithm, which assigns posterior probabilities to each component density with respect to each observation.

Gaussian mixture models are often used for data clustering. Clusters are assigned by selecting the component that maximizes the posterior probability. Like k-means clustering, Gaussian mixture modeling uses an iterative algorithm that converges to a local optimum. Gaussian mixture modeling may be more appropriate than k-means clustering when clusters have different sizes and correlation within them. Clustering using Gaussian mixture models is sometimes considered a soft clustering method. The posterior probabilities for each point indicate that each data point has some probability of belonging to each cluster.

Cluster with Gaussian Mixtures

Gaussian mixture distributions can be used for clustering data, by realizing that the multivariate normal components of the fitted model can represent clusters.

  1. To demonstrate the process, first generate some simulated data from a mixture of two bivariate Gaussian distributions using the mvnrnd function.

    rng('default')  % For reproducibility
    mu1 = [1 2];
    sigma1 = [3 .2; .2 2];
    mu2 = [-1 -2];
    sigma2 = [2 0; 0 1];
    X = [mvnrnd(mu1,sigma1,200);mvnrnd(mu2,sigma2,100)];
    
    scatter(X(:,1),X(:,2),10,'ko')
    

  2. Fit a two-component Gaussian mixture distribution. Here, you know the correct number of components to use. In practice, with real data, this decision would require comparing models with different numbers of components.

    options = statset('Display','final');
    gm = fitgmdist(X,2,'Options',options);
    
    33 iterations, log-likelihood = -1210.59
    
  3. Plot the estimated probability density contours for the two-component mixture distribution. The two bivariate normal components overlap, but their peaks are distinct. This suggests that the data could reasonably be divided into two clusters.

    hold on
    ezcontour(@(x,y)pdf(gm,[x y]),[-8 6],[-8 6]);
    hold off
    

  4. Partition the data into clusters using the cluster method for the fitted mixture distribution. The cluster method assigns each point to one of the two components in the mixture distribution.

    idx = cluster(gm,X);
    cluster1 = (idx == 1);
    cluster2 = (idx == 2);
    
    scatter(X(cluster1,1),X(cluster1,2),10,'r+');
    hold on
    scatter(X(cluster2,1),X(cluster2,2),10,'bo');
    hold off
    legend('Cluster 1','Cluster 2','Location','NW')
    

    Each cluster corresponds to one of the bivariate normal components in the mixture distribution. cluster assigns points to clusters based on the estimated posterior probability that a point came from a component; each point is assigned to the cluster corresponding to the highest posterior probability. The posterior method returns those posterior probabilities. For example, plot the posterior probability of the first component for each point.

    P = posterior(gm,X);
    
    scatter(X(cluster1,1),X(cluster1,2),10,P(cluster1,1),'+')
    hold on
    scatter(X(cluster2,1),X(cluster2,2),10,P(cluster2,1),'o')
    hold off
    legend('Cluster 1','Cluster 2','Location','NW')
    clrmap = jet(80); colormap(clrmap(9:72,:))
    ylabel(colorbar,'Component 1 Posterior Probability')
    

Soft Clustering Using Gaussian Mixture Distributions

An alternative to the previous example is to use the posterior probabilities for "soft clustering". Each point is assigned a membership score to each cluster. Membership scores are simply the posterior probabilities, and describe how similar each point is to each cluster's archetype, i.e., the mean of the corresponding component. The points can be ranked by their membership score in a given cluster.

[~,order] = sort(P(:,1));
plot(1:size(X,1),P(order,1),'r-',1:size(X,1),P(order,2),'b-');
legend({'Cluster 1 Score' 'Cluster 2 Score'},'location','NW');
ylabel('Cluster Membership Score');
xlabel('Point Ranking');

Although a clear separation of the data is hard to see in a scatter plot of the data, plotting the membership scores indicates that the fitted distribution does a good job of separating the data into groups. Very few points have scores close to 0.5.

Soft clustering using a Gaussian mixture distribution is similar to fuzzy k-means clustering, which also assigns each point to each cluster with a membership score. The fuzzy k-means algorithm assumes that clusters are roughly spherical in shape, and all of roughly equal size. This is comparable to a Gaussian mixture distribution with a single covariance matrix that is shared across all components, and is a multiple of the identity matrix. In contrast, gmdistribution allows you to specify different covariance options. The default is to estimate a separate, unconstrained covariance matrix for each component. A more restricted option, closer to k-means, would be to estimate a shared, diagonal covariance matrix.

gm2 = fitgmdist(X,2,'CovType','Diagonal',...
  'SharedCov',true);

This covariance option is similar to fuzzy k-means clustering, but provides more flexibility by allowing unequal variances for different variables.

You can compute the soft cluster membership scores without computing hard cluster assignments, using posterior, or as part of hard clustering, as the third output from cluster.

P2 = posterior(gm2,X); % equivalently [idx,~,P2] = cluster(gm2,X)
[~,order] = sort(P2(:,1));
plot(1:size(X,1),P2(order,1),'r-',1:size(X,1),P2(order,2),'b-');
legend({'Cluster 1 Score' 'Cluster 2 Score'},'location','NW');
ylabel('Cluster Membership Score');
xlabel('Point Ranking');

Assign New Data to Clusters

In the previous example, fitting the mixture distribution to data using fitgmdist, and clustering those data using cluster, are separate steps. However, the same data are used in both steps. You can also use the cluster method to assign new data points to the clusters (mixture components) found in the original data.

  1. Given a data set X, first fit a Gaussian mixture distribution. The previous code has already done that.

    gm
    
    gm = 
    
    Gaussian mixture distribution with 2 components in 2 dimensions
    Component 1:
    Mixing proportion: 0.629379
    Mean:    1.0758    2.0426
    
    Component 2:
    Mixing proportion: 0.370621
    Mean:   -0.8292   -1.8482
    
    
    
    
  2. You can then use cluster to assign each point in a new data set, Y, to one of the clusters defined for the original data.

    Y = [mvnrnd(mu1,sigma1,50);mvnrnd(mu2,sigma2,25)];
    
    idx = cluster(gm,Y);
    cluster1 = (idx == 1);
    cluster2 = (idx == 2);
    
    scatter(Y(cluster1,1),Y(cluster1,2),10,'r+');
    hold on
    scatter(Y(cluster2,1),Y(cluster2,2),10,'bo');
    hold off
    legend('Class 1','Class 2','Location','NW')
    

    As with the previous example, the posterior probabilities for each point can be treated as membership scores rather than determining "hard" cluster assignments.

For cluster to provide meaningful results with new data, Y should come from the same population as X, the original data used to create the mixture distribution. In particular, the estimated mixing probabilities for the Gaussian mixture distribution fitted to X are used when computing the posterior probabilities for Y.

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