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EM Algorithm for Gaussian Mixture Model (EM GMM)

version 1.16.0.1 (4.71 KB) by Mo Chen

Mo Chen (view profile)

EM algorithm for Gaussian mixture.

Updated 05 Dec 2018

This package fits Gaussian mixture model (GMM) by expectation maximization (EM) algorithm.It works on data set of arbitrary dimensions.
Several techniques are applied to improve numerical stability, such as computing probability in logarithm domain to avoid float number underflow which often occurs when computing probability of high dimensional data.
The code is also carefully tuned to be efficient by utilizing vertorization and matrix factorization.

This algorithm is widely used. The detail can be found in the great textbook "Pattern Recognition and Machine Learning" or the wiki page
http://en.wikipedia.org/wiki/Expectation-maximization_algorithm

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);
plotClass(X,label);

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

Besides using EM to fit GMM, I highly recommend you to try another submission of mine: Variational Bayesian Inference for Gaussian Mixture Model
(http://www.mathworks.com/matlabcentral/fileexchange/35362-variational-bayesian-inference-for-gaussian-mixture-model) which performs Bayesian inference on GMM. It has the advantage that the number of mixture components can be automatically identified by the algorithm.

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

This function is now a part of the PRML toolbox (http://www.mathworks.com/matlabcentral/fileexchange/55826-pattern-recognition-and-machine-learning-toolbox)
For anyone who wonders how to finish his homework, DONT send email to me.

Cite As

Mo Chen (2019). EM Algorithm for Gaussian Mixture Model (EM GMM) (https://www.mathworks.com/matlabcentral/fileexchange/26184-em-algorithm-for-gaussian-mixture-model-em-gmm), MATLAB Central File Exchange. Retrieved .

liangku guo

WELL

Wenwen Zhou

Chao

crixus

crixus (view profile)

may I know how is the mu generated?

mu(:,i) = gaussRnd(mu0,beta0*Sigma(:,:,i))

is it randomly generated to create some sample data?

polo sepian

YI-CHEN CHANG

YI-CHEN CHANG (view profile)

hi, I have a 240*320 matrix, I want to know how to input it?

Md Mehedi Hassan

Md Mehedi Hassan (view profile)

hi! i have 20x2 matrix data. how do i utilize this code and stop generating random data?

James Too

Dhruv Shah

Bahman

chengjuan gong

Yi Wang

Yi Wang (view profile)

Thanks for your good work. It is very useful for me.

xiaoli liu

xiaoli liu (view profile)

Does anyone know that why the loglikelihood is llh and the function of T?
T = logsumexp(R,2);
llh = sum(T)/n; % loglikelihood
R = exp(bsxfun(@minus,R,T));
Thank you so much~

xiaoli liu

xiaoli liu (view profile)

Sorry! My interface showed 5 stars yesterday. My network was wrong yesterday.

xiaoli liu

Matt J

Matt J (view profile)

xiaoli liu, why are you giving such a poor, 1-star rating to code that you are thankful for???

xiaoli liu

xiaoli liu (view profile)

To Mo Chen
Thanks for your great work. While there are some codes I don't understand.
T = logsumexp(R,2);
llh = sum(T)/n; % loglikelihood
R = exp(bsxfun(@minus,R,T));
Especially the loglikelihood and the function of T.
Thank you so much !

zhuang yu

zhuang yu (view profile)

Great work Mo chen. It saves me lots of time

Wei-Jie Chen

Wei-Jie Chen (view profile)

Good work thanks！
I have a question, what is the following function? Is there some reference? I could not understand why to do this.
function y = loggausspdf(X, mu, Sigma)
d = size(X,1);
X = bsxfun(@minus,X,mu);
[U,p]= chol(Sigma);
if p ~= 0
error('ERROR: Sigma is not PD.');
end
Q = U'\X;
q = dot(Q,Q,1); % quadratic term (M distance)
c = d*log(2*pi)+2*sum(log(diag(U))); % normalization constant
y = -(c+q)/2;

ERIC LI

Gurmeet Singh

Gurmeet Singh (view profile)

To Mo Chen
How can we use this code for training GMM from image patches?

Faris Nafiah

rania bensaied

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.

Bernardo Noronha

Bernardo Noronha (view profile)

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 :)

Bernardo Noronha

Bernardo Noronha (view profile)

Error:
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);

Bernardo Noronha

Bernardo Noronha (view profile)

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

Jiyu Tian (view profile)

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

Manjushree Aithal

Manjushree Aithal (view profile)

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

lsvih

Jennifer

Jennifer (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!

Azam Afzaal Tahir

Azam Afzaal Tahir (view profile)

Hello,

Can we apply this algorithm on set of images ?

yy zhu

Suwoong Heo

STUTI DEVARSHI

STUTI DEVARSHI (view profile)

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

STUTI DEVARSHI

STUTI DEVARSHI (view profile)

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

shao shao (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.

Nitisha Rajgure

Nitisha Rajgure (view profile)

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

Pallawi Pallawi

Pallawi Pallawi

Pallawi Pallawi (view profile)

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

Pallawi Pallawi

Pallawi Pallawi (view profile)

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

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

najah G (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

najah G

excellent

Diogo

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

Matt Cheng (view profile)

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

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

Easy to be used and efficient Implementation Thx

Arthur Curry

Arthur Curry (view profile)

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 ?

CKanellas

CKanellas (view profile)

Does this script work for 1-D cases?

Sachin

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

shawin (view profile)

its very nice code

Putu Ananta Wijaya

Putu Ananta Wijaya (view profile)

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

Zhixin

Romain

Anton

mitilma

mitilma (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.

mitilma

Fizi

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

nurullah ates

nurullah ates (view profile)

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

Judy

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

mutah (view profile)

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

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

zjut

zjut (view profile)

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

vxxx

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

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

PChoppala

PChoppala (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)?

Chandra Shekhar

Chandra Shekhar (view profile)

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

siyam (view profile)

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

chen

chen (view profile)

Hi,

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

cyklucifer

hgyfgh

bubbas

Yang Liu

Tom Hall

Tom Hall (view profile)

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

zalayeto

marouane ayech

marouane ayech (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 ?

Bogdan Dzyubak

Bogdan Dzyubak (view profile)

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

Cong

Cong (view profile)

Excellent Work! Thank you !

fateme

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

keerthi

keerthi (view profile)

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 : keerthi2412@gmail.com

shamla

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

zheng zhou

zheng zhou (view profile)

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)

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

Prasath

Prasath (view profile)

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 vinodhinybtech@gmail.com

Mo Chen

Mo Chen (view profile)

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

Andreas

Andreas (view profile)

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)

Nicolas:
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.

AISSA BOULMERKA

AISSA BOULMERKA (view profile)

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

Mark

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:
http://www.mathworks.com/help/techdoc/ref/rng.html

Nicolas

Nicolas (view profile)

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.
??? Error using ==> spread at 33
ERROR: only support data of 2D or 3D.

How I can solve it

thanks

Michael

Michael (view profile)

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

Amish (view profile)

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

Amish

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

HONGJING

Ting

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

dattatray

dattatray (view profile)

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

minni sharma (view profile)

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

Venkat R (view profile)

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.

Anathea Pepperl

Brian

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);
end

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:
k=5;
if k ~= unique([5;5;5;5])
else
disp('OK');
end
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

Neha (view profile)

k = 1;
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);

Enrique Martí

Mo Chen

Mo Chen (view profile)

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

Silvina

Silvina (view profile)

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!

Michael Davis

Michael Davis (view profile)

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

Michael,
Thanks.

Nevine

Nevine (view profile)

Michael,
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.

Thanks.

Giang Le

Giang Le (view profile)

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.

Mo Chen

Mo Chen (view profile)

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

Giang Le

Giang Le (view profile)

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

Giang Le (view profile)

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

dattatray

dattatray (view profile)

Thank you Very much sir..!!

Mo Chen

Mo Chen (view profile)

Hi, Nofil Barlas,

Mo Chen

Mo Chen (view profile)

Hi dattatray,
Take a look at the comment in the code of
http://www.mathworks.com/matlabcentral/fileexchange/24616-kmeans-clustering
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.

dattatray

dattatray (view profile)

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..

Nofil Barlas

Nofil Barlas (view profile)

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

label = emgm(x,3);
scatterd(x,label);
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);

Thanks.

Patrick

Patrick (view profile)

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 ??

>> 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?

Daniil Kocharov

Daniil Kocharov (view profile)

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?
Thanks.

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.
[~,x]=fun();
to
[dum,x]=fun();

Gayathri

Gayathri (view profile)

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 [.

 5 Dec 2018 1.16.0.1 improve description 13 Mar 2016 1.16.0.0 update title, description, and tag 7 Mar 2016 1.16.0.0 tweak and add prediction function 29 Feb 2012 1.16.0.0 Update description 4 Feb 2012 1.15.0.0 tuning 13 Mar 2011 1.13.0.0 Fix several minor bugs and reorganize the code structure a bit. 3 Nov 2010 1.12.0.0 update loggausspdf due to api change of matlab 30 Sep 2010 1.11.0.0 reorganize and clean the code a bit 4 Mar 2010 1.10.0.0 fix bug for 1d data 4 Mar 2010 1.8.0.0 fix bug for 1d data 27 Jan 2010 1.7.0.0 Fixed missing file 24 Jan 2010 1.3.0.0 update description 28 Dec 2009 1.2.0.0 add missing files 25 Dec 2009 1.1.0.0 fix missing files
MATLAB Release Compatibility
Created with R2009b
Compatible with any release
Platform Compatibility
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