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EM algorithm for Gaussian mixture model


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EM algorithm for Gaussian mixture model



23 Dec 2009 (Updated )

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 on 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.

load data;
label = emgm(x,3);

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.

For all the question regarding to use the code for image segmentation, you have to orgnize the image into a matrix, where each column is the feature vector of one pixel of the image. For example, if RGB value is used, for a 10x10 image the data matrix is a 3x100 matrix where each column is a vector of RGB value of a pixel.


Variational Bayesian Inference For Gaussian Mixture Model inspired this file.

This file inspired Em Algorithm For Gaussian Mixture Model With Background Noise.

MATLAB release MATLAB 7.9 (R2009b)
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Comments and Ratings (85)
21 Aug 2014 Putu Ananta Wijaya

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

23 Jul 2014 Zhixin  
09 Apr 2014 Romain  
09 Apr 2014 Anton  
26 Jan 2014 mitilma

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.

25 Jan 2014 mitilma  
05 Dec 2013 Fizi

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

08 Nov 2013 nurullah ates

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)));

31 Oct 2013 Rahul  
24 Sep 2013 Judy

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

15 Sep 2013 Quan Wang

It helps me a lot!

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

27 Aug 2013 mutah

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

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

01 Jul 2013 zjut

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

11 Jun 2013 vxxx

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

11 Mar 2013 Mo Chen

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

04 Mar 2013 cjain

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

04 Mar 2013 cjain  
28 Feb 2013 Praveen

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)?

19 Dec 2012 Chandra Shekhar

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.

30 Nov 2012 siyam

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

14 Nov 2012 chen


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


24 Oct 2012 hgyfgh  
08 Oct 2012 bubbas  
08 Oct 2012 Yang Liu  
27 Aug 2012 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.

25 Jul 2012 Bayarbadrakh Baramsai  
12 Jul 2012 Nikolay S.

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...

03 Jul 2012 Alex  
26 Jun 2012 zalayeto  
18 May 2012 marouane ayech

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

30 Apr 2012 Bogdan Dzyubak

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

16 Apr 2012 Cong

Excellent Work! Thank you !

16 Apr 2012 fateme

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

18 Mar 2012 keerthi

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 :

14 Mar 2012 shamla

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.

14 Mar 2012 shamla  
14 Mar 2012 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)

22 Feb 2012 Mo Chen

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)

20 Feb 2012 Adili neila

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

14 Feb 2012 Prasath

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

08 Feb 2012 Mo Chen

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

08 Feb 2012 Andreas

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.

07 Feb 2012 Mo Chen

I dont see problem

07 Feb 2012 Mo Chen

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?

13 Jan 2012 Mark

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:

23 Dec 2011 Nicolas

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


06 Aug 2011 Michael

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.

27 Jul 2011 Amish

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

25 Jul 2011 Amish

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?

25 Jul 2011 Amish  
05 Jul 2011 HONGJING  
29 Jun 2011 Ting

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

23 Jun 2011 dattatray

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..

12 Jun 2011 minni sharma

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

thanking you

10 Jun 2011 Mo Chen

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.

10 Jun 2011 Mo Chen

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.

20 May 2011 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.

15 May 2011 Anathea Pepperl  
04 May 2011 Brian

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.

27 Apr 2011 Neha

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);

27 Apr 2011 Enrique Martí  
13 Mar 2011 Mo Chen

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

07 Mar 2011 Silvina

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!

04 Mar 2011 Michael Davis

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.

09 Feb 2011 Daniel Zoran  
23 Dec 2010 Nevine

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

23 Dec 2010 Nevine

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.


04 Nov 2010 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.

03 Nov 2010 Mo Chen

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

03 Nov 2010 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.

03 Nov 2010 Mo Chen

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.

02 Nov 2010 Giang Le

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

28 Oct 2010 dattatray

Thank you Very much sir..!!

27 Oct 2010 Mo Chen

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

27 Oct 2010 Mo Chen

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

27 Oct 2010 Mo Chen

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

27 Oct 2010 dattatray

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...

19 Oct 2010 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);


22 Sep 2010 Patrick

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

24 Jun 2010 Tianfan XUE  
16 Feb 2010 Mo Chen

Can you send me your data via email?

16 Feb 2010 Daniil Kocharov

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?

09 Feb 2010 Mo Chen

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

09 Feb 2010 Gayathri

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 [.

25 Dec 2009

fix missing files

28 Dec 2009

add missing files

24 Jan 2010

update description

27 Jan 2010

Fixed missing file

04 Mar 2010

fix bug for 1d data

04 Mar 2010

fix bug for 1d data

30 Sep 2010

reorganize and clean the code a bit

03 Nov 2010

update loggausspdf due to api change of matlab

13 Mar 2011

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

04 Feb 2012


29 Feb 2012

Update description

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