Fit Gaussian mixture distribution to data
returns
a Gaussian mixture distribution model with additional options specified
by one or more GMModel
= fitgmdist(X
,k
,Name,Value
)Name,Value
pair arguments.
For example, you can specify a regularization value or the covariance type.
Generate data from a mixture of two bivariate Gaussian distributions.
mu1 = [1 2];
Sigma1 = [2 0; 0 0.5];
mu2 = [3 5];
Sigma2 = [1 0;0 1];
rng(1); % For reproducibility
X = [mvnrnd(mu1,Sigma1,1000);mvnrnd(mu2,Sigma2,1000)];
Fit a Gaussian mixture model. Specify that there are two components.
GMModel = fitgmdist(X,2);
Plot the data over the fitted Gaussian mixture model contours.
figure y = [zeros(1000,1);ones(1000,1)]; h = gscatter(X(:,1),X(:,2),y); hold on ezcontour(@(x1,x2)pdf(GMModel,[x1 x2]),get(gca,{'XLim','YLim'})) title('{\bf Scatter Plot and Fitted Gaussian Mixture Contours}') legend(h,'Model 0','Model1') hold off
Generate data from a mixture of two bivariate Gaussian distributions. Create a third predictor that is the sum of the first and second predictors.
mu1 = [1 2];
Sigma1 = [1 0; 0 1];
mu2 = [3 4];
Sigma2 = [0.5 0; 0 0.5];
rng(1); % For reproducibility
X1 = [mvnrnd(mu1,Sigma1,100);mvnrnd(mu2,Sigma2,100)];
X = [X1,X1(:,1)+X1(:,2)];
The columns of X
are linearly dependent. This can cause illconditioned covariance estimates.
Fit a Gaussian mixture model to the data. You can use try
/ catch
statements to help manage error messages.
rng(1); % Reset seed for common start values try GMModel = fitgmdist(X,2) catch exception disp('There was an error fitting the Gaussian mixture model') error = exception.message end
There was an error fitting the Gaussian mixture model error = Illconditioned covariance created at iteration 2.
The covariance estimates are illconditioned. Subsequently, optimization stops and an error appears.
Fit a Gaussian mixture model again, but use regularization.
rng(1); % Reset seed for common start values GMModel = fitgmdist(X,2,'RegularizationValue',0.1)
GMModel = Gaussian mixture distribution with 2 components in 3 dimensions Component 1: Mixing proportion: 0.507057 Mean: 0.9767 2.0130 2.9897 Component 2: Mixing proportion: 0.492943 Mean: 3.1030 3.9544 7.0574
In this case, the algorithm converges to a solution due to regularization.
Gaussian mixture models require that you specify a number of components before being fit to data. For many applications, it might be diffcult to know the appropriate number of components. This example shows how to explore the data, and try to get an initial guess at the number of components using principal component analysis.
Load Fisher's iris data set.
load fisheriris
classes = unique(species)
classes = 'setosa' 'versicolor' 'virginica'
The data set contains three classes of iris species. The analysis proceeds as if this is unknown.
Use principal component analysis to reduce the dimension of the data to two dimensions for visualization.
[~,score] = pca(meas,'NumComponents',2);
Fit three Gaussian mixture models to the data by specifying 1, 2, and 3 components. Increase the number of optimization iterations to 1000
. Use dot notation to store the final parameter estimates. By default, the software fits full and different covariances for each component.
GMModels = cell(3,1); % Preallocation options = statset('MaxIter',1000); rng(1); % For reproducibility for j = 1:3 GMModels{j} = fitgmdist(score,j,'Options',options); fprintf('\n GM Mean for %i Component(s)\n',j) Mu = GMModels{j}.mu end
GM Mean for 1 Component(s) Mu = 1.0e14 * 0.2861 0.0871 GM Mean for 2 Component(s) Mu = 1.3212 0.0954 2.6424 0.1909 GM Mean for 3 Component(s) Mu = 1.9642 0.0062 2.6424 0.1909 0.4750 0.2292
GMModels
is a cell array containing three, fitted gmdistribution
models. The means in the three component models are different, suggesting that the model distinguishes among the three iris species.
Plot the scores over the fitted Gaussian mixture model contours. Since the data set includes labels, use gscatter
to distinguish between the true number of components.
figure for j = 1:3 subplot(2,2,j) gscatter(score(:,1),score(:,2),species) h = gca; hold on ezcontour(@(x1,x2)pdf(GMModels{j},[x1 x2]),... [h.XLim h.YLim],100) title(sprintf('GM Model  %i Component(s)',j)); xlabel('1st principal component'); ylabel('2nd principal component'); if(j ~= 3) legend off; end hold off end g = legend; g.Position = [0.7 0.25 0.1 0.1]; %set(g,'Position',[0.7,0.25,0.1,0.1])
The threecomponent Gaussian mixture model, in conjunction with PCA, looks like it distinguishes between the three iris species.
There are other options you can use to help select the appropriate number of components for a Gaussian mixture model. For example,
Compare multiple models with varying numbers of components using information criteria, e.g., AIC or BIC.
Estimate the number of clusters using evalclusters
, which supports, the CalinskiHarabasz criterion and the gap statistic, or other criteria.
Gaussian mixture models require that you specify a number of components before being fit to data. For many applications, it might be diffcult to know the appropriate number of components. This example uses the AIC fit statistic to help you choose the best fitting Gaussian mixture model over varying numbers of components.
Generate data from a mixture of two bivariate Gaussian distributions.
mu1 = [1 1]; Sigma1 = [0.5 0; 0 0.5]; mu2 = [2 4]; Sigma2 = [0.2 0; 0 0.2]; rng(1); X = [mvnrnd(mu1,Sigma1,1000);mvnrnd(mu2,Sigma2,1000)]; plot(X(:,1),X(:,2),'ko') title('Scatter Plot') xlim([min(X(:)) max(X(:))]) % Make axes have the same scale ylim([min(X(:)) max(X(:))])
Supposing that you do not know the underlying parameter values, the scatter plots suggests:
There are two components.
The variances between the clusters are different.
The variance within the clusters is the same.
There is no covariance within the clusters.
Fit a twocomponent Gaussian mixture model. Based on the scatter plot inspection, specify that the covariance matrices are diagonal. Print the final iteration and loglikelihood statistic to the Command Window by passing a statset
structure as the value of the Options
namevalue pair argument.
options = statset('Display','final'); GMModel = fitgmdist(X,2,'CovarianceType','diagonal','Options',options);
10 iterations, loglikelihood = 4787.38
GMModel
is a fitted gmdistribution
model.
Examine the AIC over varying numbers of components.
AIC = zeros(1,4); GMModels = cell(1,4); options = statset('MaxIter',500); for k = 1:4 GMModels{k} = fitgmdist(X,k,'Options',options,'CovarianceType','diagonal'); AIC(k)= GMModels{k}.AIC; end [minAIC,numComponents] = min(AIC); numComponents BestModel = GMModels{numComponents}
numComponents = 2 BestModel = Gaussian mixture distribution with 2 components in 2 dimensions Component 1: Mixing proportion: 0.501736 Mean: 1.9824 4.0013 Component 2: Mixing proportion: 0.498264 Mean: 0.9879 1.0511
The smallest AIC occurs when the software fits the twocomponent Gaussian mixture model.
Gaussian mixture model parameter estimates might vary with different initial values. This example shows how to control initial values when you fit Gaussian mixture models using fitgmdist
.
Load Fisher's iris data set. Use the petal lengths and widths as predictors.
load fisheriris
X = meas(:,3:4);
Fit a Gaussian mixture model to the data using default initial values. There are three iris species, so specify k = 3 components.
rng(10); % For reproducibility
GMModel1 = fitgmdist(X,3);
By default, the software:
Randomly chooses k = 3 data points
Treats the chosen data points as initial means for each component
Sets the initial covariance matrices as diagonal, where element (j
, j
) is the variance of X(:,j)
Treats the initial mixing proportions as uniform
Fit a Gaussian mixture model by connecting each observation to its label.
y = ones(size(X,1),1); y(strcmp(species,'setosa')) = 2; y(strcmp(species,'virginica')) = 3; GMModel2 = fitgmdist(X,3,'Start',y);
Fit a Gaussian mixture model by explicitly specifying the initial means, covariance matrices, and mixing proportions.
Mu = [1 1; 2 2; 3 3]; Sigma(:,:,1) = [1 1; 1 2]; Sigma(:,:,2) = 2*[1 1; 1 2]; Sigma(:,:,3) = 3*[1 1; 1 2]; PComponents = [1/2,1/4,1/4]; S = struct('mu',Mu,'Sigma',Sigma,'ComponentProportion',PComponents); GMModel3 = fitgmdist(X,3,'Start',S);
Use gscatter
to plot a scatter diagram that distinguishes between the iris species. For each model, plot the fitted Gaussian mixture model contours.
figure subplot(2,2,1) h = gscatter(X(:,1),X(:,2),species,[],'o',4); haxis = gca; xlim = haxis.XLim; ylim = haxis.YLim; d = (max([xlim ylim])min([xlim ylim]))/1000; [X1Grid,X2Grid] = meshgrid(xlim(1):d:xlim(2),ylim(1):d:ylim(2)); hold on contour(X1Grid,X2Grid,reshape(pdf(GMModel1,[X1Grid(:) X2Grid(:)]),... size(X1Grid,1),size(X1Grid,2)),20) uistack(h,'top') title('{\bf Random Initial Values}'); xlabel('Sepal length'); ylabel('Sepal width'); legend off; hold off subplot(2,2,2) h = gscatter(X(:,1),X(:,2),species,[],'o',4); hold on contour(X1Grid,X2Grid,reshape(pdf(GMModel2,[X1Grid(:) X2Grid(:)]),... size(X1Grid,1),size(X1Grid,2)),20) uistack(h,'top') title('{\bf Initial Values from Labels}'); xlabel('Sepal length'); ylabel('Sepal width'); legend off hold off subplot(2,2,3) h = gscatter(X(:,1),X(:,2),species,[],'o',4); hold on contour(X1Grid,X2Grid,reshape(pdf(GMModel3,[X1Grid(:) X2Grid(:)]),... size(X1Grid,1),size(X1Grid,2)),20) uistack(h,'top') title('{\bf Initial Values from the Structure}'); xlabel('Sepal length'); ylabel('Sepal width'); legend('Location',[0.7,0.25,0.1,0.1]); hold off
According to the countors, GMModel2
seems to suggest a slight trimodality, while the others suggest bimodel distributions.
Display the estimated component means.
table(GMModel1.mu,GMModel2.mu,GMModel3.mu,'VariableNames',... {'Model1','Model2','Model3'})
ans = Model1 Model2 Model3 _________________ ________________ ________________ 5.0241 1.8658 4.2857 1.3339 1.4604 0.2429 4.7471 1.4616 1.462 0.246 4.7509 1.4629 1.4605 0.24306 5.5507 2.0316 5.0158 1.8592
GMModel2
seems to distinguish between the iris species the best.
X
— Datanumeric matrixData to which the Gaussian mixture model is fit, specified as a numeric matrix.
The rows of X
correspond to observations,
and columns correspond to variables.
NaN
s indicate missing values. The software
removes rows of X
containing at least one NaN
before
fitting, which decreases the effective sample size.
Data Types: double
k
— Number of componentspositive integerNumber of components to use when fitting Gaussian mixture model,
specified as a positive integer. For example, if you specify k
= 3
, then the software fits a Gaussian mixture model with
three distinct means, covariances matrices, and component proportions
to the data (X
).
Specify optional commaseparated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.
'RegularizationValue',0.1,'CovarianceType','diagonal'
specifies
a regularization parameter value of 0.1 and to fit diagonal covariance
matrices.'CovarianceType'
— Type of covariance matrix'full'
(default)  'diagonal'
Type of covariance matrix to fit to the data, specified as the
commaseparated pair consisting of 'CovarianceType'
and
either 'diagonal'
or 'full'
.
If you set 'diagonal'
, then the software
fits diagonal covariance matrices. In this case, the software estimates k*d
covariance
parameters, where d
is the number of columns in X
(i.e., d
= size(X,2)
).
Otherwise, the software fits full covariance matrices. In this
case, the software estimates k*d*(d+1)/2
covariance
parameters.
Example: 'CovarianceType','diagonal'
Data Types: char
'Options'
— Iterative EM algorithm optimization optionsstatset
options structureIterative EM algorithm optimization options, specified as the
commaseparated pair consisting of 'Options'
and
a statset
options structure.
This table describes the available namevalue pair arguments.
Name  Value 

'Display' 

'MaxIter'  Positive integer indicating the maximum number of iterations
allowed. The default is 100 
'TolFun'  Positive scalar indicating the termination tolerance for the
loglikelihood function value. The default is 1e6 . 
Example: 'Options',statset('Display','final','MaxIter',1500,'TolFun',1e5)
'RegularizationValue'
— Regularization parameter value0
(default)  nonnegative scalarRegularization parameter value, specified as the commaseparated
pair consisting of 'RegularizationValue'
and a
nonnegative scalar.
Set RegularizationValue
to a small positive
scalar to ensure that the estimated covariance matrices are positive
definite.
Example: 'RegularizationValue',0.01
Data Types: double
'Replicates'
— Number of times to repeat EM algorithm1
(default)  positive integerNumber of times to repeat the EM algorithm using a new set of
initial values, specified as the commaseparated pair consisting of 'Replicates'
and
a positive integer.
If Replicates
is greater than 1
,
then:
The namevalue pair argument Start
must
be randSample
(which is the default value) or plus
.
GMModel
is the fit with the largest
loglikelihood.
Example: 'Replicates',10
Data Types: double
'SharedCovariance'
— Flag indicating whether all covariance matrices are identicallogical false
(default)  logical true
Flag indicating whether all covariance matrices are identical
(i.e., fit a pooled estimate), specified as the commaseparated pair
consisting of 'SharedCovariance'
and either logical
value false
or true
.
If SharedCovariance
is true
,
then all k
covariance matrices are equal, and the
number of covariance parameters is scaled down by a factor of k
.
'Start'
— Initial value setting method'randSample'
(default)  'plus'
 vector of integers  structure arrayInitial value setting method, specified as the commaseparated
pair consisting of 'Start'
and 'randSample'
, 'plus'
,
a vector of integers, or a structure array.
The value of Start
determines the initial
values required by the optimization routine for each Gaussian component
parameter — mean, covariance, and mixing proportion. This table
summarizes the available options.
Value  Description 

'randSample'  The software selects k observations from X at
random as initial component means. The mixing proportions are uniform.
The initial covariance matrices for all components are diagonal, where
the element j on the diagonal is the variance of X(:,j) . 
'plus'  The software selects k observations from X using
the kmeans++ algorithm.
The initial mixing proportions are uniform. The initial covariance
matrices for all components are diagonal, where the element j on
the diagonal is the variance of X(:,j) . 
Vector of integers  A vector of length n (the number of observations)
containing an initial guess of the component index for each point.
That is, each element is an integer from 1 to k ,
which corresponds to a component. The software collects all observations
corresponding to the same component, computes means, covariances,
and mixing proportions for each, and sets the initial values to these
statistics. 
Structure array  Suppose that there are d variables (i.e., d
= size(X,2) ). The structure array, e.g., S ,
must have three fields:

Example: 'Start',ones(n,1)
Data Types: char
 double
 struct
GMModel
— Fitted Gaussian mixture modelgmdistribution
modelFitted Gaussian mixture model, returned as a gmdistribution
model.
Access properties of GMModel
using dot notation.
For example, display the AIC by entering GMModel.AIC
.
gmdistribution
might:
Converge to a solution where one or more of the components has an illconditioned or singular covariance matrix.
The following issues might result in an illconditioned covariance matrix:
The number of dimensions of your data is relatively high and there are not enough observations.
Some of the predictors (variables) of your data are highly correlated.
Some or all the features are discrete.
You tried to fit the data to too many components.
In general, you can avoid getting illconditioned covariance matrices by using one of the following precautions:
Preprocess your data to remove correlated features.
Set 'SharedCovariance'
to true
to
use an equal covariance matrix for every component.
Set 'CovarianceType'
to 'diagonal'
.
Use 'RegularizationValue'
to add
a very small positive number to the diagonal of every covariance matrix.
Try another set of initial values.
Pass through an intermediate step where one or more of the components has an illconditioned covariance matrix. Try another set of initial values to avoid this issue without altering your data or model.
The software optimizes the Gaussian mixture model likelihood using the iterative ExpectationMaximization (EM) algorithm.
The kmeans++ algorithm
uses an heuristic to find centroid seeds for kmeans
clustering. fitgmdist
can apply the same principle
to initialize the EM algorithm by using the kmeans++ algorithm to
select the initial parameter values for a fitted Gaussian mixture
model.
The kmeans++ algorithm assumes the number of clusters is k and chooses the initial parameter values as follows.
Select the component mixture probability to be the uniform probability $${p}_{i}=\frac{1}{k}$$, where i = 1, ..., k.
Select the covariance matrices to be diagonal and identical, where $${\sigma}_{i}=diag\left({a}_{1},{a}_{2},\dots ,{a}_{k}\right)$$ and $${a}_{j}=\mathrm{var}\left({X}_{j}\right)$$.
Select the first initial component center μ_{1} uniformly from all data points in X.
To choose center j:
Compute the Mahalanobis distances from each observation to each centroid, and assign each observation to its closest centroid.
For m = 1,...,n and p = 1,...,j – 1, select centroid j at random from X with probability
$$\frac{{d}^{2}\left({x}_{m},{\mu}_{p}\right)}{{\displaystyle \sum _{h;{x}_{h}\in {{\rm M}}_{p}}{d}^{2}\left({x}_{h},{\mu}_{p}\right)}}$$
where $$d\left({x}_{m},{\mu}_{p}\right)$$ is the distance between observation m and μ_{p}, and M_{p} is the set of all observations closest to centroid μ_{p} and x_{m} belongs to M_{p}.
That is, select each subsequent center with a probability proportional to the distance from itself to the closest center that you already chose.
Repeat step 4 until k centroids are chosen.
[1] McLachlan, G., and D. Peel. Finite Mixture Models. Hoboken, NJ: John Wiley & Sons, Inc., 2000.
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