GMM

什么是混合高斯模型？可以参见这篇博文，它同时给出了matlab实现代码，编者将数学表达式转化为矩阵运算的代码技巧很值得我们学习和借鉴，以下是个人对代码的剖析与理解。
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(); %初始化

Lprev = -inf; %inf表示正无究大，-inf表示为负无究大
while true
    Px = calc_prob();

    % new value for pGamma
    pGamma = Px .* repmat(pPi, N, 1);
    pGamma = pGamma ./ repmat(sum(pGamma, 2), 1, K); %求每个样本由第K个聚类，也叫“component“生成的概率

    % new value for parameters of each Component
    Nk = sum(pGamma, 1);
    pMiu = diag(1./Nk) * pGamma' * X; %重新计算每个component的均值
    pPi = Nk/N; %更新混合高斯的加权系数
    for kk = 1:K %重新计算每个component的协方差
        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')); %求混合高斯分布的似然函数
    if L-Lprev < threshold %随着迭代次数的增加，似然函数越来越大，直至不变
        break; %似然函数收敛则退出
    end
    Lprev = L;
end

if nargout == 1 %如果返回是一个参数的话，那么varargout=Px;
    varargout = {Px};
else %否则，返回[Px model],其中model是结构体
    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) + ... %distmat第j行的第i个元素表示第j个数据与第i个聚类点的距离，如果数据有4个，聚类2个，那么distmat就是4*2矩阵
        repmat(sum(pMiu.*pMiu, 2)', N, 1) - 2*X*pMiu'; %sum(A，2)结果为列向量，第i个元素是第i行的求和
    [dummy labels] = min(distmat, [], 2); %返回列向量dummy和labels，dummy向量记录distmat的每行的最小值，labels向量记录每行最小值的列号，即是第几个聚类，labels是N×1列向量，N为样本数

    for k=1:K
        Xk = X(labels == k, :); %把标志为同一个聚类的样本组合起来
        pPi(k) = size(Xk, 1)/N; %求混合高斯模型的加权系数，pPi为1*K的向量
        pSigma(:, :, k) = cov(Xk); %分别求单个高斯模型或聚类样本的协方差矩阵,pSigma为D*D*K的矩阵
    end
end

function Px = calc_prob()
    Px = zeros(N, K);
    for k = 1:K
        Xshift = X-repmat(pMiu(k, :), N, 1); %Xshift表示为样本矩阵-Uk，第i行表示xi-uk
        inv_pSigma = inv(pSigma(:, :, k)); %求协方差的逆
        tmp = sum((Xshift*inv_pSigma) .* Xshift, 2); %tmp为N*1矩阵，第i行表示（xi-uk)^T*Sigma^-1*(xi-uk)
        coef = (2*pi)^(-D/2) * sqrt(det(inv_pSigma)); %求多维正态分布中指数前面的系数
        Px(:, k) = coef * exp(-0.5*tmp); %求单独一个正态分布生成样本的概率或贡献
    end
end

end