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EM Algorithm for Gaussian Mixture Model (EM GMM)

version 1.16 (4.33 KB) by

EM algorithm for Gaussian mixture. Works on arbitray dimensions with high speed and precision.

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This is a function tries to obtain the maximum likelihood estimation of Gaussian mixture model by expectation maximization (EM) algorithm.
It works on data set of arbitrary dimensions. Several techniques are applied to avoid the float number underflow problems that often occurs when computing probability of high dimensional data. Also the code is carefully tuned to be efficient by utilizing vertorization and matrix factorization.
This is a widely used algorithm. The detail of this algorithm can be found in many textbooks or tutorials online. Just google EM Gaussian Mixture or you can read the wiki page:
This function is robust and efficient yet the code structure is organized so that it is easy to read. Please try following code for a demo:
close all; clear;
d = 2;
k = 3;
n = 500;
[X,label] = mixGaussRnd(d,k,n);

m = floor(n/2);
X1 = X(:,1:m);
X2 = X(:,(m+1):end);
% train
[z1,model,llh] = mixGaussEm(X1,k);
% predict
z2 = mixGaussPred(X2,model);

Besides using EM to fit GMM, I highly recommend you to try another submission of mine: Variational Bayesian Inference for Gaussian Mixture Model
( which perform Bayesian inference on GMM. It has the advantage that the number of mixture components can be automatically identified by the algorithm.

Upon request, I provided a prediction function for out-of-sample inference.

This function is now a part of the PRML toolbox (

Comments and Ratings (116)

Mo Chen

Mo Chen (view profile)

@ Bernardo Noronha,
the latest update of this function use a new syntax of Matlab R2016b. That's why you get a error due to your Matlab is old. And yeah, replace the logsumexp in this release with a older one should work.

Ok I saw what the error was, the toolbox probably has an outdated version. I replaced it in my toolbox for the one available here. Other than that, looks good :)

Error using -
Matrix dimensions must agree.

Error in logsumexp (line 14)
s = a+log(sum(exp(X-a),dim)); % TODO: use log1p

Error in mixGaussEm>expectation (line 54)
T = logsumexp(R,2);

Error in mixGaussEm (line 21)
    [R, llh(iter)] = expectation(X,model);

Could you help me out with the error below? Sorry, accidentally submitted it without explaining more.
I downloaded the toolbox and included it in my MATLAB work folder.

Thanks a lot!

Jiyu Tian


 Can you help me with the EM for BMM (beta mixture model)? It's my homework,and this weekend is deadline.
Many Thanks!

I am facing a huge problem to find gamrnd function. Can you guide me with the same?


lsvih (view profile)


I am struggling to find the parameter phi in your code, so the variable for prior probability. How is it defined?
Otherwise great work!


Can we apply this algorithm on set of images ?

yy zhu

yy zhu (view profile)

Suwoong Heo

[~,x] = histc(r,[0;p/p(end)]);
this statement is showing error how should i get it balanced ....reply me pls

i'm getting this error Error: File: G:\dissertation mtech\EmGm\EmGm\mixGaussRnd.m Line: 62 Column: 3
Expression or statement is incorrect--possibly unbalanced (, {, or [. can anybody suggest something

shao shao

Anyone knows how to modify this code to consider weights to the samples. I my case, I have biased samples, so I would like to give weights to each sample in order to fit unbiased model.

 Hi ... I find this file is very usefull to clear the concepts. Thanks.... Nitisha

@@ najah G can u tell me how to use this code please.

Hie sir,
I have an image of 466616*16*1 dimension where I have decided to take my cluster size as 16 .I want to use this code can someone help me on this please.
How to run this code?
What Inputs do I need to give?


Cody (view profile)

Does anyone know how I could incorporate sample weights into the EM algorithm so that samples with higher weights more heavily influence the EM?

Ankit Singh

najah G

Can you please provide an example of initializing using a structure that has mu and sigma. The code looks for this:
if isstruct(init) % initialize with a model
R = expectation(X,init);
I wish to know how to initialize values in this init structure. Thanks so much for this implementation. It really is a blessing

najah G



Diogo (view profile)

Anyone knows how to modify this code to consider weights to the samples. I my case, I have biased samples, so I would like to give weights to each sample in order to fit unbiased model.

Matt Cheng

Thx Chen, I add some code for 1-D data by following your code:

Change the code below otherwise in switch structure as
for i = 1:c
            idc = label==i;
            plot(X(1,label==i),['.' color(i)],'MarkerSize',15,'LineWidth',1.2);
            hold on
Hope it is helpful!

Easy to be used and efficient Implementation Thx

If I had a feature vector with 6 features,the data matrix would be like 6x100 for a 10x10 block image. When trying to use spread,it can't take such dimensional data. Any way to visualize the clusters ?


Does this script work for 1-D cases?


Sachin (view profile)

Very easy to implement. Once you get the vector set in d x n, and specify init, algorithm runs flawlessly. Thank you! I expected the output to be better than k-means but for some reason it is not and it may very well be my data set or I may not be using the emgm as it should.


shawin (view profile)

its very nice code

may you give flowchart of this program?
because when i read the code, this is not the same as any algorithm that I read.


Zhixin (view profile)


Romain (view profile)


Anton (view profile)


Is there an exact R version of this implementation? I have found many R implementations of EM for GMM but none of them are as fast as this one.



Fizi (view profile)

Can you please provide an example of initializing using a structure that has mu and sigma. The code looks for this:
if isstruct(init) % initialize with a model
    R = expectation(X,init);
I wish to know how to initialize values in this init structure. Thanks so much for this implementation. It really is a blessing

thanks.but I dont understand some code.
can you answer this What is bellow codes' mathematical mean ?
y = loggausspdf(X, mu, Sigma)
[U,p]= chol(Sigma);
Q = U'\X;
q = dot(Q,Q,1);
c = d*log(2*pi)+2*sum(log(diag(U)));


Rahul (view profile)


Judy (view profile)

thanks a lot. after finding many materials , finally i find it.

Quan Wang

Quan Wang (view profile)

It helps me a lot!

It would be better if you include a "compute_pdf_from_GMM" file, which I have to write myself.


mutah (view profile)

EM for Gaussian mixture: running ...
??? Input argument "X" is undefined.

Error in ==> emalgorithm at 8
R =initialization(X,init);


zjut (view profile)

Hi,chen,can I define the number of cluster by myself?


vxxx (view profile)

hi chen
how to see the plot of pdf for this function

Mo Chen

Mo Chen (view profile)

Hi, cjain, you have function call mu in path. It is your problem to solve, not mine.


cjain (view profile)

hi, i m finding following error:1.Error: File: emgm.m Line: 77 Column: 33
 "mu" previously appeared to be used as a function or command,
 conflicting with its use here as the name of a variable.
 A possible cause of this error is that you forgot to initialize the
 variable, or you have initialized it implicitly using load or eval.
2.Input argument "X" is undefined.

Error in ==> emgm_1 at 8
R = initialization(X,init);
plz plz resolve it


cjain (view profile)


Hi, thanks for the code; well written.
Can you help me out with a simple query? When we specify the number of Gaussians to (say 2), can we find the weight of each Gaussian component, (i.e weight of all samples that have label=1 and weight of all samples that have label=2)?

Fantastic code.In fact i am getting following error when i execute in MATLAB 2009a.

??? Error: File: emgm.m Line: 9 Column: 3
Expression or statement is incorrect--possibly unbalanced (, {, or

Please tell me any one how to correct it.


siyam (view profile)

hi is there anyway to set the covariance matrix to diagonal in this code?


chen (view profile)


I wanna ask what does this eye(d)*1e-6.
You said this is for numerical stability.
Could you explain a little bit?



hgyfgh (view profile)


bubbas (view profile)

Yang Liu

Tom Hall

Fantastic. Does a much better job at fitting than the built-in Signal Processing gaussian mixtures function, which commonly fits an obviously bimodal dist with a unimodal function.

Nikolay S.

Nikolay S. (view profile)

Those missing the Statistics Toolbox and getting an error:
"??? Undefined function or method 'randsample' for input arguments of type 'double'." can use the following code as a substitute for randsample function.

function y = randsample(n, k)
y=round(1+ (n-1)*rand(k, 1) );

Mo (Michael), thanks for the submission, but a few comments I have:
1) You should have mentioned that Statistics Toolbox is needed.
2) when applied following command: label = emgm(x, 10);
where size(x)= 2 84480 , it did not converge in 500 interations, (which took about 2 minutes), as opposed to k_means by Yi Cao, which worked juts fine...
label = emgm(x, 3); worked fine btw...


Alex (view profile)


Hi Michael,
I want to know wheither there is a theoretical proof for the technique you have used in logsumexp to avoid numerical underflow ?

Simple to use, fast, and doesn't crash.


Cong (view profile)

Excellent Work! Thank you !


fateme (view profile)

hello,I want to apply emgm on adult dataset,which it's attributes are both categorical and numerical,I tried to apply clustering on data saved in dataset and in cell array,but this data types are not defined for emgm. can emgm be used for string array?pleas help me.tnx


hello sir, we are using em algorithm for detecting resampling (tampering of images). for this we need to get the fourier transform of the probability map. how can we modify this code for the above purpose. kindly help.
my mail id :


shamla (view profile)

i need to apply gmm to iris dataset and obtain 3 clusters.i need to display the (index of datapoints)datapoints in each cluster.please help me.


shamla (view profile)

zheng zhou

Hi Michael,
I have a 65*100matrix,can I use this code to get the two-dimension GMM,in which the mu sigma and weight are two dimension.(z=f(x,y),f is the function for GMM)

Mo Chen

Mo Chen (view profile)

For all the questions about how to use is for image segmentation:
You have to organize the image into a matrix where each column is the feature of a pixel(say rgb)

Adili neila

Hi Michael,
how can I use your code for images?


we doing project on statistical pixel intensity segmentation of clsm images..
we need coding for gaussian mixture, normal distribution, poisson...
plss mail coding to tis mail id

Mo Chen

Mo Chen (view profile)

Hi Andreas, that function is in statistics toolbox. It random sample k integers between 1 to n.


Hi Michael,

A small question: the randsample function called at line 44 seemingly does not exist, as I get an error. I am running R2011b. Are other, non-native, files required to run emgm? Thanks.

Mo Chen

Mo Chen (view profile)

I dont see problem

Mo Chen

Mo Chen (view profile)

For any one having question about changing result:

Please read wiki page. EM is only finding local minimal, which means the result depend on initialization.

Thank You for this Excellent Work,
is there any paper that may help to understand the program?


Mark (view profile)

For people getting different results on each run, this is due to the use of psuedorandom number generators in initialization. Try setting the psuedorandom number seed:


Hi, I try with 1 D array, and I have this problem

>> label = emgm(a,1);
EM for Gaussian mixture: running ...
Converged in 2 steps.
>> spread(a,label);
??? Error using ==> spread at 33
ERROR: only support data of 2D or 3D.

How I can solve it



Very easy to use and fast, but like some of the above posters, I am getting different results every time I run it on the same data.


Amish (view profile)

Fixed seed for random generator and got the same plot. This is a very useful utility. Many Thanks.


Amish (view profile)

Same question as Ting:
"converging steps are changing for the same data"
Must have to do with the latest Matlab release. I am using R2011b.
Michael, can you confirm?


Amish (view profile)



Ting (view profile)

Dear Chen,
When I using EM to analysis my data, the result is always changing, and converge step is changing too, is there any way to make it stable?
Thank you


I have a small Question,
suppose i have modeled a data.
it has 100 vectors each having 36 features.
so my input is 36 x 100
now i have given 5 clusters,model is trained now.
now i have set of 5 mu(36 x 1) and 5 sigma(36 x 36 x 5).
lets us say i have a unknown vector x of size(36 x 1)
now how to find out,in which cluster this particular vector fits..?
for each cluster we have only mu,and sigma,and for a single vector matlab gives sigle value of mean,and cov()
can you help me in this..?
if it would have been k-means,its easy to calculate euclidean distance with cluster centroid,which ever is minimum,that is answer..

minni sharma

To chen
can u send me a code for image fusion using EM algorithm please.

thanking you

Mo Chen

Mo Chen (view profile)

To Venkat R,

This code uses general form ofthe multivariable Gaussian distribution, not the one in your comment, which is simply the 1d special case.

You cannot arbitrayly add a parameter there. You have to ensure the density function is actually a valid density function (means it has to integrate to 1). Otherwise, EM does not work.

Mo Chen

Mo Chen (view profile)

To Brian,
"Furthermore since you draw the centers from the points themselves, there will always be at least 1 point in each cluster, making even the intended code pointless."
If the initialized k centers are very close, after one iteration of the EM, there will be only one cluster there.
This piece of code simply prevent this from happening. It ensure that there is no more than two initialized centers belong to one cluster.

Venkat R

Dear Chen,
Very good and fast implementation.
I guess the normal distribution you are using is exp( -(x-mu)^2/2*sigma^2 )/sqrt(2*pi*sigma^2)

In that case, if I were to slightly modify the sigma by w*sigma(or mu by w*mu), where 'w' is another design parameter, Can you help me which functions I need to change to utilize your code.

Thanking you very much.


Brian (view profile)

Found this pointless piece of code in the initialization:

while k ~= unique(label)
        idx = randsample(n,k);
        m = X(:,idx);
        [~,label] = max(bsxfun(@minus,m'*X,sum(m.^2,1)'/2),[],1);

Unless I am missing something, I'm assuming you were trying to make sure at least one point is assigned to each cluster? Well, this just checks if at least 1 point is assigned to the kth cluster. E.g. try:
if k ~= unique([5;5;5;5])
it will say that label assignment is OK.

Furthermore since you draw the centers from the points themselves, there will always be at least 1 point in each cluster, making even the intended code pointless.

You may want to use another strategy to ensure centers are chosen that take more than a single point for instance.

Another common initialization strategy is to partition the points randomly into k clusters.


Neha (view profile)

k = 1;
X = imread('image.png');
label = emgm(X,3);

Please tell me how to fix the errors listed below:

EM for Gaussian mixture: running ...
??? Error using ==> mtimes
Integer data types are not fully supported for this operation.
At least one operand must be a scalar.

Error in ==> emgm>initialization at 46
    [dum,label] = max(bsxfun(@minus,m'*X,sum(m.^2,1)'/2),[],1);

Error in ==> emgm at 8
R = initialization(X,init);

Error in ==> Untitled at 2
label = emgm(X,3);

Mo Chen

Mo Chen (view profile)

Hi, I will upload a new version. Please try it and tell me if it still happens.


Dear Michael,
I'm trying to use your code on images (using reshape to give them a vector structure) and I'm getting the following error message:
Error using ==> loggausspdf at 10
ERROR: sigma is not SPD.

Interestingly, after calling the command many times, the function eventually works.

Any feedback on this issue will be greatly appreciated!

A couple of minor bugs:

1. I came across the same problem as Nofil Barlas above when the size of the input vector is [ N 1 ]. Reshape to [ 1 N ] and it works.

2. If you tell it to find only 1 mixture, it keeps going until it runs out of memory. The code should either disallow an init parameter of 1, or else have a short function to handle this trivial case.

Otherwise, great, very useful! Thanks.

Daniel Zoran


Nevine (view profile)

The email address in the file bounced. Please send me your address so that I can email you the data file.


Nevine (view profile)

I am getting the error:
??? Error using ==> loggausspdf at 10
ERROR: sigma is not SPD.
I am using matlab Release R2010a.
The input data X is 24x57600 with 2 clusters.

labels = emgm(X, 2);

I will send you the data via email.


Giang Le

i see now, I have tried with 2009a and earlier version and it gave me error when i increased number of clusters. Work fine with 2009b although it is not converge.
I am very thankful for your reply.

Mo Chen

Mo Chen (view profile)

Not happened here. which version of matlab are your using?

Giang Le

Hi Michael,
Thanks for a quick reply. Here is the problem, I am try to clustering 11208 samples to 128 with dim is 14.
> x = rand(14,11208);
>[est_label,model] = emgm(x,128);
EM for Gaussian mixture: running ... ??? Error using ==> loggausspdf at 7
ERROR: sigma is not SPD.

Error in ==> emgm>expectation at 65
    R(:,i) = loggausspdf(X,mu(:,i),Sigma(:,:,i));

Error in ==> emgm at 16
    [R, llh(t)] = expectation(X,model);
I have tried to increase the sigma0 but the problem is still there.

Mo Chen

Mo Chen (view profile)

Giang Le,
How does it happen? The function can hardly produce a non positive definite sigma.
However, if it does, you may try to change the sigma0 in line 76 to be a larger value.

Giang Le

Can you please let me know how to fix the ERROR: sigma is not SPD?


Thank you Very much sir..!!

Mo Chen

Mo Chen (view profile)

Hi, Nofil Barlas,
Maybe you forget clear your memory before load the data.

Mo Chen

Mo Chen (view profile)

Hi dattatray,
Take a look at the comment in the code of
You may get the idea.

Mo Chen

Mo Chen (view profile)

Hi, Patrick
Sigma(:,:,1) is the covariance matrix of the first gaussian mixture component.


Can You please tell me..about initialization which you have made.
generating random values is fine..
but i havent understood

use of maxVal=bsxfun(@minus,m'*X, sum(m.^2,1)'/2 )
[dum,label] = max(bsxfun(@minus,m'*X,sum(m.^2,1)'/2));
can you please tell me that...?
code is really wonderful,
but if i could get any theory material regarding functions you have written,especially for expectation,maximization,and log gaussian pdf..
please mail me on
thank you in advance...

Nofil Barlas

Quick question. I ran the code and what the error:

>> load data;
label = emgm(x,3);
EM for Gaussian mixture: running ... ??? Error using ==> randsample at 117
K must be less than or equal to N for sampling without replacement.

Error in ==> emgm>initialization at 36
    idx = randsample(n,k);

Error in ==> emgm at 9
R = initialization(X,init);



Apologize for the following simple question. What exactly does the sigma data mean from the example given? The first Sigma (1,1) is the sigma for the first estimated cluster and the second sigma (2,2) is for the first estimated cluster on the second row ??

Could you please clarify? Thanks.

>> model.Sigma

ans(:,:,1) =

    0.7227 0.8439
    0.8439 1.8252

ans(:,:,2) =

    0.2629 -0.1116
   -0.1116 0.2411

ans(:,:,3) =

    0.4209 -0.0600
   -0.0600 0.0967

Tianfan XUE

Mo Chen

Mo Chen (view profile)

Can you send me your data via email?

Sorry for asking such a silly question:
I got an error trying to use 1d data.
Error using ==> loggausspdf at 7 ERROR: sigma is not SPD.
Error in ==> emgmm>expectation at 68
    R(i,:) = loggausspdf(X,mu(:,i),Sigma(:,:,i));
I think that's because Sigma is:
Sigma(:,:,1) = NaN
Sigma(:,:,2) = NaN
Sigma(:,:,3) = 7.0826e-005
But why is it NaN I cannot understand, or is there anything else wrong?

Mo Chen

Mo Chen (view profile)

Before you give any bad rating, you should really notice that this function require MATLAB 7.9 (2009b).
It use a new feature of matlab.
upgrade your matlab, or you can modify all


Produces the following error with the above steps.

label = emgmm(x,3);
??? Error: File: emgmm.m Line: 21 Column: 7
Expression or statement is incorrect--possibly unbalanced (, {, or [.



update title, description, and tag


tweak and add prediction function


Update description




Fix several minor bugs and reorganize the code structure a bit.


update loggausspdf due to api change of matlab


reorganize and clean the code a bit


fix bug for 1d data


fix bug for 1d data


Fixed missing file


update description


add missing files


fix missing files

MATLAB Release
MATLAB 7.9 (R2009b)

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