EM_GM

Hi,
Thank you so much for this tool
Is it possible to save clusters membership?

Thanks
Moran

Hello,

I think the error is popping up because you are processing X with dimensions 4601*57. I would suggest try taking the transpose of X, which might solve the problem.

Let me know if it works.
Regards

Thank you!

Thx for your code. But I double checked that your likelihood calculation is wrong!

What is the time comlexity of your implementation?

I tried modifying the code a little bit, by including the following lines of code,

function [W,M,V] = Init_EM(X,k)
[n,d] = size(X);
[Ci,C] = kmeans(X,k,'Start','cluster', ...
'Maxiter',100, ...
'EmptyAction','drop', ...
'Display','off'); % Ci(nx1) - cluster indeices; C(k,d) - cluster centroid (i.e. mean)
while sum(isnan(C))>0,
[Ci,C] = kmeans(X,k,'Start','cluster', ...
'Maxiter',100, ...
'EmptyAction','drop', ...
'Display','off');
end
M = C';

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Use this routine if the cluster matrix is of variable length and contains
% Scan the cluster index matrix Ci for NaN values, and remove them
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
temp1=Ci(:,1); % Scan the entire matrix column
temp2=isnan(temp1); % Find the rows with NaN values
index_active=find(temp2~=1); % Find the rows with actual values
Ci_new=temp1(index_active); %#ok<FNDSB>
[r,c]=size(Ci_new); % Determine the size of the newly scanned matrix.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Vp = repmat(struct('count',0,'X',zeros(r,c)),1,k);
for i=1:r % Separate cluster points
Vp(Ci_new(i)).count = Vp(Ci_new(i)).count + 1;
Vp(Ci_new(i)).X(Vp(Ci_new(i)).count,:) = X(i,:);
end

V = zeros(d,d,k);
for i=1:k,
W (i) = Vp(i).count/r;
% Find the covariance ignoring the NaNs
V(:,:,i) = nancov(Vp(i).X(1:Vp(i).count,:));
end
%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Init_EM %%%%
%%%%%%%%%%%%%%%%%%%%%%%%

I worked, giving me the estimates. But still I am getting some warnings with "Empty rows of data".

I have a multidimensional file in Microsoft Excel. I imported the sheet into X, and executed [W,M,V,L] = EM_GM(X,3,[],[],1,[]);

I am getting the following errors,

[W,M,V,L] = EM_GM(X,1,[],[],1,[])
Warning: Ignoring rows of X with missing data.
> In kmeans at 127
In EM_GM>Init_EM at 261
In EM_GM at 74
??? Attempted to access C(1,2); index out of bounds because numel(C)=1.

Error in ==> kmeans>distfun at 722
D(:,i) = D(:,i) + (X(:,j) - C(i,j)).^2;

Error in ==> kmeans at 329
D = distfun(X, C, distance, 0);

Error in ==> EM_GM>Init_EM at 261
[Ci,C] = kmeans(X,k,'Start','cluster', ...

Error in ==> EM_GM at 74
[W,M,V] = Init_EM(X,k); L = 0;

Please Help....!!! thanks in advance..... :)

it might be also any other implementation of netlab. type edit kmeans to find the culprit one.

this is good.

This code computes the E-step and the M-step correctly. It is also readable and neat for the user. Nevertheless, there is a problem and an improvement to make I would like to report. The problem is that, the likelihood calculation is incorrect although the outputs of both E and M steps are correct. I have checked the reference to which the code is attributed. In fact, in that reference, the probability density function (pdf) is a single gaussian rather than a mixture. This is why there is a confusion leading to miscalculations. The actual likelihood function should be computed as [in LaTeX format]:

L = \sum^{n}_{i=1} \log(\sum^{k}_{j=1}W(j)*\mathcal{N}(X(:,i)^{T},M(:,j),V(:,:,j)))

The improvement is that there are some loops which can be replaced by matrix multiplications or some other matrix-related operations. Furthermore, the likelihood computation can easily be achieved inside the E-step.

In conclusion, the following code is the one modified with respect to the aforementioned points:

function [W,M,V,L,E] = EM_GM(X,k,ltol,maxiter,pflag,Init)
% Inputs:
% X(n,d) - input data, n=number of observations, d=dimension of variable
% k - maximum number of Gaussian components allowed
% ltol - percentage of the log likelihood difference between 2 iterations ([] for none)
% maxiter - maximum number of iteration allowed ([] for none)
% pflag - 1 for plotting GM for 1D or 2D cases only, 0 otherwise ([] for none)
% Init - structure of initial W, M, V: Init.W, Init.M, Init.V ([] for none)
%
% Ouputs:
% W(1,k) - estimated weights of GM
% M(d,k) - estimated mean vectors of GM
% V(d,d,k) - estimated covariance matrices of GM
% L - log likelihood of estimates
%

%%%% Validate inputs %%%%
if nargin <= 1,
disp('EM_GM must have at least 2 inputs: X,k!/n')
return
elseif nargin == 2,
ltol = 0.1; maxiter = 1000; pflag = 0; Init = [];
err_X = Verify_X(X);
err_k = Verify_k(k);
if err_X | err_k, return; end
elseif nargin == 3,
maxiter = 1000; pflag = 0; Init = [];
err_X = Verify_X(X);
err_k = Verify_k(k);
[ltol,err_ltol] = Verify_ltol(ltol);
if err_X | err_k | err_ltol, return; end
elseif nargin == 4,
pflag = 0; Init = [];
err_X = Verify_X(X);
err_k = Verify_k(k);
[ltol,err_ltol] = Verify_ltol(ltol);
[maxiter,err_maxiter] = Verify_maxiter(maxiter);
if err_X | err_k | err_ltol | err_maxiter, return; end
elseif nargin == 5,
Init = [];
err_X = Verify_X(X);
err_k = Verify_k(k);
[ltol,err_ltol] = Verify_ltol(ltol);
[maxiter,err_maxiter] = Verify_maxiter(maxiter);
[pflag,err_pflag] = Verify_pflag(pflag);
if err_X | err_k | err_ltol | err_maxiter | err_pflag, return; end
elseif nargin == 6,
err_X = Verify_X(X);
err_k = Verify_k(k);
[ltol,err_ltol] = Verify_ltol(ltol);
[maxiter,err_maxiter] = Verify_maxiter(maxiter);
[pflag,err_pflag] = Verify_pflag(pflag);
[Init,err_Init]=Verify_Init(Init);
if err_X | err_k | err_ltol | err_maxiter | err_pflag | err_Init, return; end
else
disp('EM_GM must have 2 to 6 inputs!');
return
end

%%%% Initialize W, M, V,L %%%%
t = cputime;
if isempty(Init),
[W,M,V] = Init_EM(X,k); L = 0;
else
W = Init.W;
M = Init.M;
V = Init.V;
end
[E,Ln] = Expectation(X,k,W,M,V); % Initialize log likelihood and compute expectation
Lo = 2*Ln;

%%%% EM algorithm %%%%
niter = 0;
while (abs(100*(Ln-Lo)/Lo)>ltol) & (niter<=maxiter),
%**************************************************************************
% For the first loop, expectation is computed above (cf. Line 69). For the
% next loops, it is computed in their preceding loops. Therefore, the
% following line [Line 80] is omitted.
%**************************************************************************
% E = Expectation(X,k,W,M,V); % E-step

[W,M,V] = Maximization(X,k,E); % M-step
Lo = Ln;
[E,Ln] = Expectation(X,k,W,M,V); % Log-likelihood and expectation computation
niter = niter + 1;
LnValues(niter) = Ln;
disp(['Iteration # = ' num2str(niter) ', Ln = ' num2str(Ln) ', Stopping Criterion = ' num2str(abs(100*(Ln-Lo)/Lo)) ', Tolerance = ' num2str(ltol)])
end
L = Ln;
theTime = num2str(fix(clock));
theTime(theTime==' ') = '_';
save(['EM_CostFunction_' theTime '_MoreLoopsRemoved.mat'],'LnValues')

%%%% Plot 1D or 2D %%%%
if pflag==1,
[n,d] = size(X);
if d>2,
disp('Can only plot 1 or 2 dimensional applications!/n');
else
Plot_GM(X,k,W,M,V);
end
elapsed_time = sprintf('CPU time used for EM_GM: %5.2fs',cputime-t);
disp(elapsed_time);
disp(sprintf('Number of iterations: %d',niter-1));
end
%%%%%%%%%%%%%%%%%%%%%%
%%%% End of EM_GM %%%%
%%%%%%%%%%%%%%%%%%%%%%

function [E,L] = Expectation(X,k,W,M,V)
[n,d] = size(X);
a = (2*pi)^(0.5*d);
S = zeros(1,k);
iV = zeros(d,d,k);
for j=1:k,
if V(:,:,j)==zeros(d,d), V(:,:,j)=ones(d,d)*eps; end
S(j) = sqrt(det(V(:,:,j)));
iV(:,:,j) = inv(V(:,:,j));
end
%*************************************************************************
%*************************** MODIFICATION: LOOP REMOVAL *****************
%*************************************************************************
E = zeros(n,k);
% for i=1:n,
% for j=1:k,
% dXM = X(i,:)'-M(:,j);
% pl = exp(-0.5*dXM'*iV(:,:,j)*dXM)/(a*S(j));
% E(i,j) = W(j)*pl;
% end
% E(i,:) = E(i,:)/sum(E(i,:));
% end
chunkSize = 1000;
howManyFullChunks = fix(n/chunkSize);
numberOfRemainingVectors = n - howManyFullChunks*chunkSize;
pdfValueVector = zeros(n,1);
for j = 1:k
for chunkCounter = 1:howManyFullChunks
modificationRange = (chunkCounter-1)*chunkSize+1:chunkCounter*chunkSize;
meanSubtractedChunk = X(modificationRange,:)'-repmat(M(:,j),1,chunkSize);
pdfValueVectorToExp = diag(-0.5*meanSubtractedChunk'*iV(:,:,j)*meanSubtractedChunk);
pdfValueVector(modificationRange) = exp(pdfValueVectorToExp)/(a*S(j));
end
modificationRange = howManyFullChunks*chunkSize+1:n;
meanSubtractedChunk = X(modificationRange,:)'-repmat(M(:,j),1,length(modificationRange));
pdfValueVectorToExp = diag(-0.5*meanSubtractedChunk'*iV(:,:,j)*meanSubtractedChunk);
pdfValueVector(modificationRange) = exp(pdfValueVectorToExp)/(a*S(j));
E(:,j) = W(j)*pdfValueVector';
end
sumInColumnDirection = sum(E,2);
divisor = repmat(sumInColumnDirection,1,k);
L = sum(log(sum(E,2)));
E = E./divisor;
%**************************************************************************
%*************************** MODIFICATION ENDS HERE ***********************
%**************************************************************************

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Expectation %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [W,M,V] = Maximization(X,k,E)
[n,d] = size(X);
V = zeros(d,d,k);
%*************************************************************************
%*************************** MODIFICATION: LOOP REMOVAL *****************
%*************************************************************************
% W = zeros(1,k); M = zeros(d,k);
% for i=1:k, % Compute weights
% for j=1:n,
% W(i) = W(i) + E(j,i);
% M(:,i) = M(:,i) + E(j,i)*X(j,:)';
% end
% M(:,i) = M(:,i)/W(i);
% end
W = sum(E,1);
M = X'*E;
M = M./repmat(W,d,1);
%**************************************************************************
%*************************** MODIFICATION ENDS HERE ***********************
%**************************************************************************
for i=1:k,
for j=1:n,
dXM = X(j,:)'-M(:,i);
V(:,:,i) = V(:,:,i) + E(j,i)*dXM*dXM';
end
V(:,:,i) = V(:,:,i)/W(i);
end
W = W/n;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Maximization %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%**************************************************************************
%************************ LIKELIHOOD REMOVED COMPLETELY *******************
%**************************************************************************
% function L = Likelihood(X,k,W,M,V)
% Compute L based on K. V. Mardia, "Multivariate Analysis", Academic Press, 1979, PP. 96-97
% % to enchance computational speed
% [n,d] = size(X);
% U = mean(X)';
% S = cov(X);
% L = 0;
% for i=1:k,
% iV = inv(V(:,:,i));
% L = L + W(i)*(-0.5*n*log(det(2*pi*V(:,:,i))) ...
% -0.5*(n-1)*(trace(iV*S)+(U-M(:,i))'*iV*(U-M(:,i))));
% end
%**************************************************************************
%*************************** MODIFICATION ENDS HERE ***********************
%**************************************************************************
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Likelihood %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%

function err_X = Verify_X(X)
err_X = 1;
[n,d] = size(X);
if n<d,
disp('Input data must be n x d!/n');
return
end
err_X = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_X %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%

function err_k = Verify_k(k)
err_k = 1;
if ~isnumeric(k) | ~isreal(k) | k<1,
disp('k must be a real integer >= 1!/n');
return
end
err_k = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_k %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%

function [ltol,err_ltol] = Verify_ltol(ltol)
err_ltol = 1;
if isempty(ltol),
ltol = 1e-2;
elseif ~isreal(ltol) | ltol<=0,
disp('ltol must be a positive real number!');
return
end
err_ltol = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_ltol %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [maxiter,err_maxiter] = Verify_maxiter(maxiter)
err_maxiter = 1;
if isempty(maxiter),
maxiter = 1000;
elseif ~isreal(maxiter) | maxiter<=0,
disp('ltol must be a positive real number!');
return
end
err_maxiter = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_maxiter %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [pflag,err_pflag] = Verify_pflag(pflag)
err_pflag = 1;
if isempty(pflag),
pflag = 0;
elseif pflag~=0 & pflag~=1,
disp('Plot flag must be either 0 or 1!/n');
return
end
err_pflag = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_pflag %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [Init,err_Init] = Verify_Init(Init)
err_Init = 1;
if isempty(Init),
% Do nothing;
elseif isstruct(Init),
[Wd,Wk] = size(Init.W);
[Md,Mk] = size(Init.M);
[Vd1,Vd2,Vk] = size(Init.V);
if Wk~=Mk | Wk~=Vk | Mk~=Vk,
disp('k in Init.W(1,k), Init.M(d,k) and Init.V(d,d,k) must equal!/n')
return
end
if Md~=Vd1 | Md~=Vd2 | Vd1~=Vd2,
disp('d in Init.W(1,k), Init.M(d,k) and Init.V(d,d,k) must equal!/n')
return
end
else
disp('Init must be a structure: W(1,k), M(d,k), V(d,d,k) or []!');
return
end
err_Init = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_Init %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [W,M,V] = Init_EM(X,k)
[n,d] = size(X);
[Ci,C] = kmeans(X,k,'Start','cluster', ...
'Maxiter',100, ...
'EmptyAction','drop', ...
'Display','off'); % Ci(nx1) - cluster indeices; C(k,d) - cluster centroid (i.e. mean)
while sum(isnan(C))>0,
[Ci,C] = kmeans(X,k,'Start','cluster', ...
'Maxiter',100, ...
'EmptyAction','drop', ...
'Display','off');
end
M = C';
Vp = repmat(struct('count',0,'X',zeros(n,d)),1,k);
for i=1:n, % Separate cluster points
Vp(Ci(i)).count = Vp(Ci(i)).count + 1;
Vp(Ci(i)).X(Vp(Ci(i)).count,:) = X(i,:);
end
V = zeros(d,d,k);
for i=1:k,
W(i) = Vp(i).count/n;
V(:,:,i) = cov(Vp(i).X(1:Vp(i).count,:));
end
%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Init_EM %%%%
%%%%%%%%%%%%%%%%%%%%%%%%

function Plot_GM(X,k,W,M,V)
[n,d] = size(X);
if d>2,
disp('Can only plot 1 or 2 dimensional applications!/n');
return
end
S = zeros(d,k);
R1 = zeros(d,k);
R2 = zeros(d,k);
for i=1:k, % Determine plot range as 4 x standard deviations
S(:,i) = sqrt(diag(V(:,:,i)));
R1(:,i) = M(:,i)-4*S(:,i);
R2(:,i) = M(:,i)+4*S(:,i);
end
Rmin = min(min(R1));
Rmax = max(max(R2));
R = [Rmin:0.001*(Rmax-Rmin):Rmax];
% clf, hold on
figure; hold on
if d==1,
Q = zeros(size(R));
for i=1:k,
P = W(i)*normpdf(R,M(:,i),sqrt(V(:,:,i)));
Q = Q + P;
plot(R,P,'r-'); grid on,
end
plot(R,Q,'k-');
xlabel('X');
ylabel('Probability density');
else % d==2
% plot(X(:,1),X(:,2),'r.');
plot(X(1:200,1),X(1:200,2),'r.');
plot(X(201:400,1),X(201:400,2),'g.');
plot(X(401:600,1),X(401:600,2),'b.');
for i=1:k,
Plot_Std_Ellipse(M(:,i),V(:,:,i));
end
xlabel('1^{st} dimension');
ylabel('2^{nd} dimension');
axis([Rmin Rmax Rmin Rmax])
end
title('Gaussian Mixture estimated by EM');
%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Plot_GM %%%%
%%%%%%%%%%%%%%%%%%%%%%%%

function Plot_Std_Ellipse(M,V)
[Ev,D] = eig(V);
d = length(M);
if V(:,:)==zeros(d,d),
V(:,:) = ones(d,d)*eps;
end
iV = inv(V);
% Find the larger projection
P = [1,0;0,0]; % X-axis projection operator
P1 = P * 2*sqrt(D(1,1)) * Ev(:,1);
P2 = P * 2*sqrt(D(2,2)) * Ev(:,2);
if abs(P1(1)) >= abs(P2(1)),
Plen = P1(1);
else
Plen = P2(1);
end
count = 1;
step = 0.001*Plen;
Contour1 = zeros(2001,2);
Contour2 = zeros(2001,2);
for x = -Plen:step:Plen,
a = iV(2,2);
b = x * (iV(1,2)+iV(2,1));
c = (x^2) * iV(1,1) - 1;
Root1 = (-b + sqrt(b^2 - 4*a*c))/(2*a);
Root2 = (-b - sqrt(b^2 - 4*a*c))/(2*a);
if isreal(Root1),
Contour1(count,:) = [x,Root1] + M';
Contour2(count,:) = [x,Root2] + M';
count = count + 1;
end
end
Contour1 = Contour1(1:count-1,:);
Contour2 = [Contour1(1,:);Contour2(1:count-1,:);Contour1(count-1,:)];
plot(M(1),M(2),'k+');
plot(Contour1(:,1),Contour1(:,2),'k-');
plot(Contour2(:,1),Contour2(:,2),'k-');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Plot_Std_Ellipse %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Thanks for this well written piece of code

Nice code. Its very easy to use - no variations required and the usuage of the function given makes the whole process of integration of the function simpler. Thank you.

here's an approximation:
http://nlp.cs.byu.edu/mediawiki/index.php/Log_Domain_Computations

I found some error prones : the code can't deal when covariance matrix becomes singular.

Thank you... well documented, easy to use and flexible interface

The algorithm converts very well to a good solution and is reasonable stable if applied multiple times to the same data. It is even faster than the greedy EM algorithm. My compliments

I tried the one from the classification toolbox mentioned in the Duda, Hart, and Stork text. It was impossible to figure out.

This one worked right off. Thanks.

Mr. Tsui's implementation does indeed resolve several errors found in several implementations that I have reviewed.

I thougth I'd had to writte something like this my-self. I would then have spend a lot of time, and it would never have been as perfect.

Thank you so much !!!

Really Compact and professionally written, extremely useful . Thanks for sharing !

This is the first time that I could find a stand alone function for Gaussian mixture estimation.Compact and professionally written, extremely useful. It saved my day at a critical time in my research. Thank you Patrick!

As the author suggests, this routine estimates the parameters for a mixture of multivariate normal distributions (ala Aaron D'Souza's article from USC, "Using EM To Estimate A Probablity[sic] Density With A Mixture of Gaussians"). Runs quickly and produces reasonable answers.
Wish that the functions were factored into separate files so I could use the nice ellipse plotting and other functions as stand-alones. But that's nitpicking.