function varargout = gmm(X, K_or_centroids)
% ============================================================
% Expectation-Maximization iteration implementation of
% Gaussian Mixture Model.
%
% PX = GMM(X, K_OR_CENTROIDS)
% [PX MODEL] = GMM(X, K_OR_CENTROIDS)
%
% - X: N-by-D data matrix.
% - K_OR_CENTROIDS: either K indicating the number of
% components or a K-by-D matrix indicating the
% choosing of the initial K centroids.
%
% - PX: N-by-K matrix indicating the probability of each
% component generating each point.
% - MODEL: a structure containing the parameters for a GMM:
% MODEL.Miu: a K-by-D matrix.
% MODEL.Sigma: a D-by-D-by-K matrix.
% MODEL.Pi: a 1-by-K vector.
% ============================================================
threshold = 1e-15;
[N, D] = size(X);
if isscalar(K_or_centroids)
K = K_or_centroids;
% randomly pick centroids
rndp = randperm(N);
centroids = X(rndp(1:K), :);
else
K = size(K_or_centroids, 1);
centroids = K_or_centroids;
end
% initial values
[pMiu pPi pSigma] = init_params();
Lprev = -inf;
while true
Px = calc_prob();
% new value for pGamma
pGamma = Px .* repmat(pPi, N, 1);
pGamma = pGamma ./ repmat(sum(pGamma, 2), 1, K);
% new value for parameters of each Component
Nk = sum(pGamma, 1);
pMiu = diag(1./Nk) * pGamma' * X;
pPi = Nk/N;
for kk = 1:K
Xshift = X-repmat(pMiu(kk, :), N, 1);
pSigma(:, :, kk) = (Xshift' * ...
(diag(pGamma(:, kk)) * Xshift)) / Nk(kk);
end
% check for convergence
L = sum(log(Px*pPi'));
if L-Lprev < threshold
break;
end
Lprev = L;
end
if nargout == 1
varargout = {Px};
else
model = [];
model.Miu = pMiu;
model.Sigma = pSigma;
model.Pi = pPi;
varargout = {Px, model};
end
function [pMiu pPi pSigma] = init_params()
pMiu = centroids;
pPi = zeros(1, K);
pSigma = zeros(D, D, K);
% hard assign x to each centroids
distmat = repmat(sum(X.*X, 2), 1, K) + ...
repmat(sum(pMiu.*pMiu, 2)', N, 1) - ...
2*X*pMiu';
[dummy labels] = min(distmat, [], 2);
for k=1:K
Xk = X(labels == k, :);
pPi(k) = size(Xk, 1)/N;
pSigma(:, :, k) = cov(Xk);
end
end
function Px = calc_prob()
Px = zeros(N, K);
for k = 1:K
Xshift = X-repmat(pMiu(k, :), N, 1);
inv_pSigma = inv(pSigma(:, :, k));
tmp = sum((Xshift*inv_pSigma) .* Xshift, 2);
coef = (2*pi)^(-D/2) * sqrt(det(inv_pSigma));
Px(:, k) = coef * exp(-0.5*tmp);
end
end
end
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