高斯混合模型（matlab代码+注释）

这里我学习的是Statistical Patte7rn Recognition Toolbox中的emgmm代码，代码中的主要知识点在之前的GMM文档中基本解释清楚，包括EM算法中的两个步骤。我自己先看原理，再去看代码，在给代码注释的过程中我又重新把整个理论体系梳理了一遍，还是很感谢这种方式，踏踏实实地做一件事情。

主函数 emgmm

子函数

pdfgauss 多元高斯分布概率密度估计

knnrule knnclass 模型参数初始化过程中使用的数据分类方法

mlcgmm 对于数据类别已知的GMM参数估计

melgmm EM算法中M步参数估计

function model=emgmm(X,options,init_model)
% EMGMM Expectation-Maximization Algorithm for Gaussian mixture model.
%
% Synopsis:
%  model = emgmm(X)
%  model = emgmm(X,options)
%  model = emgmm(X,options,init_model)
%
% Description:
%  This function implements the Expectation-Maximization algorithm
%  (EM) [Schles68][DLR77] which computes the maximum-likelihood
%  estimate of the paramaters of the Gaussian mixture model (GMM).
%  The EM algorithm is an iterative procedure which monotonically
%  increases log-likelihood of the current estimate until it reaches
%  a local optimum.
%
%  The number of components of the GMM is given in options.ncomp
%  (default 2).
%
%%%%EM算法迭代停止条件
%  The following three stopping are condition used:
%   1. Improvement of the log-likelihood is less than given
%      threshold
%                logL(t+1)  - logL(t) < options.eps_logL
%%%logL(t)单调递增，随着下界的提升，逐渐逼近其最大值。当它不再变化或者变化幅度很小时，停止迭代.

%   2. Change of the squared differences of a estimated posteriory
%      probabilities is less than given threshold
%               ||alpha(t+1) - alpha(t)||^2 < options.eps_alpha
%%%%E步给定参数，求出隐含变量的期望；M步根据期望重新估计参数，返回到E步，得出新的隐含变量的期望。通过求取二者向量的L2范数，当其
%%%%不变或者变化很小时，停止迭代。

%   3. Number of iterations exceeds given threshold.
%               t >= options.tmax
%%%%设置迭代次数上限

%  The type of estimated covariance matrices is optional:
%    options.cov_type = 'full'      full covariance matrix (default)
%    options.cov_type = 'diag'      diagonal covarinace matrix
%    cov_options.type = 'spherical' spherical covariance matrix

%%%%在模型参数初始化中，采用knn给样本数据分类并采用极大似然估计法进行参数估计，下面三种就是knn中心点初始化方法

%  The initial model (estimate) is selected:
%    1. randomly (options.init = 'random')
%    2. using C-means (options.init = 'cmeans')
%    3. using the user specified init_model.
%
% Input:
%  X [dim x num_data] Data sample.
%
%  options [struct] Control paramaters:
%   .ncomp [1x1] Number of components of GMM (default 2).
%   .tmax [1x1] Maximal number of iterations (default inf).
%   .eps_logL [1x1] Minimal improvement in log-likelihood (default 0).
%   .eps_alpha [1x1] Minimal change of Alphas (default 0).
%   .cov_type [1x1] Type of estimated covarince matrices (see above).
%   .init [string] 'random' use random initial model (default);
%                  'cmeans' use K-means to find initial model.
%   .verb [1x1] If 1 then info is displayed (default 0).
%
%  init_model [struct] Initial model:
%   .Mean [dim x ncomp] Mean vectors.
%   .Cov [dim x dim x ncomp] Covariance matrices.
%   .Priors [1 x ncomp] Weights of mixture components.
%   .Alpha [ncomp x num_data] (optional) Distribution of hidden state.
%   .t [1x1] (optional) Counter of iterations.
%
% Output:
%  model [struct] Estimated Gaussian mixture model:
%   .Mean [dim x ncomp] Mean vectors.
%   .Cov [dim x dim x ncomp] Covariance matrices.
%   .Prior [1 x ncomp] Weights of mixture components.
%   .t [1x1] Number iterations.
%   .options [struct] Copy of used options.
%   .exitflag [int] 0      ... maximal number of iterations was exceeded.
%                   1 or 2 ... EM has converged; indicates which stopping
%                              was used (see above).
%
% Example:
% Note: if EM algorithm does not converge run it again from different
% initial model.
%
% EM is used to estimate parameters of mixture of 2 Guassians:
%  true_model = struct('Mean',[-2 2],'Cov',[1 0.5],'Prior',[0.4 0.6]);
%  sample = gmmsamp(true_model, 100);
%  estimated_model = emgmm(sample.X,struct('ncomp',2,'verb',1));
%
%  figure; ppatterns(sample.X);
%  h1=pgmm(true_model,struct('color','r'));
%  h2=pgmm(estimated_model,struct('color','b'));
%  legend([h1(1) h2(1)],'Ground truth', 'ML estimation');
%  figure; hold on; xlabel('iterations'); ylabel('log-likelihood');
%  plot( estimated_model.logL );
%
% See also
%  MLCGMM, MMGAUSS, PDFGMM, GMMSAMP.
%

% About: Statistical Patte7rn Recognition Toolbox
% (C) 1999-2003, Written by Vojtech Franc and Vaclav Hlavac
% <a href="http://www.cvut.cz">Czech Technical University Prague</a>
% <a href="http://www.feld.cvut.cz">Faculty of Electrical Engineering</a>
% <a href="http://cmp.felk.cvut.cz">Center for Machine Perception</a>

% Modifications:
% 26-may-2004, VF, initialization by K-means added
% 1-may-2004, VF
% 19-sep-2003, VF
% 16-mar-2003, VF

% processing input arguments
% -----------------------------------------
if nargin < 2, options=[]; else options=c2s(options); end

if ~isfield( options, 'ncomp'), options.ncomp = 2; end
if ~isfield( options, 'tmax'), options.tmax =1000; end
if ~isfield( options, 'eps_alpha'), options.eps_alpha = 0; end
if ~isfield( options, 'eps_logL'), options.eps_logL = 0; end
if ~isfield( options, 'cov_type'), options.cov_type = 'full'; end
if ~isfield( options, 'init'), options.init = 'cmeans'; end
if ~isfield( options, 'verb'), options.verb = 0; end

[dim,num_data] = size(X);      %%%%%数据输入格式要注意

% setup initial model
% ---------------------------------

if nargin == 3,

% take model from input
%-----------------------------

model = init_model;
if ~isfield(model,'t'), model.t = 0; end
if ~isfield(model,'Alpha'),
model.Alpha=-inf*ones(options.num_gauss,num_data);
end
if ~isfield(model,'logL'), model.logL=-inf; end
else

% compute initial model
%------------------------------------
switch options.init,
% random model
case 'random'
% takes randomly first num_gauss trn. vectors as mean vectors
inx = randperm(num_data);
inx=inx(1:options.ncomp);
centers_X = X(:,inx);

% K-means clustering
case 'cmeans'
tmp = cmeans( X, options.ncomp );
centers_X = tmp.X;

otherwise
error('Unknown initialization method.');
end

knn = knnrule({'X',centers_X,'y',[1:options.ncomp]},1);
y = knnclass(X,knn);                                         %%%%%%作者默认1最近邻将数据分类成n个类别（n个component）

% uses ML estimation of complete data
model = mlcgmm( {'X',X,'y',y}, options.cov_type );           %%%%%%数据类别已知采用极大似然估计

model.Alpha = zeros(options.ncomp,num_data);
for i = 1:options.ncomp,
model.Alpha(i,find(y==i)) = 1;                               %%%%在每一列中将对应类别k的第k行标记为1
end
model.logL= -inf;
model.t = 1;
model.options = options;
model.fun = 'pdfgmm';

end

% Main loop of EM algorithm
% -------------------------------------
model.exitflag = 0;
while model.exitflag == 0 & model.t < options.tmax,

% counter of iterations
model.t = model.t + 1;

%----------------------------------------------------
% E-Step
% The distribution of hidden states is computed based
% on the current estimate.
%----------------------------------------------------
a=pdfgauss(X, model);
newAlpha = (model.Prior(:)*ones(1,num_data)).*pdfgauss(X, model);          %%%%根据E步期望公式求出估计值
newLogL = sum(log(sum(newAlpha,1)));                                       %%%%%求取新的似然概率值
newAlpha = newAlpha./(ones(options.ncomp,1)*sum(newAlpha,1));              %%%%%满足数据点在每个conponent中的概率之和为1
b=a.*newAlpha;
%------------------------------------------------------
% Stopping conditions.
%------------------------------------------------------

% 1) change in distribution of hidden state Alpha
model.delta_alpha = sum(sum((model.Alpha - newAlpha).^2));                 %%%%更新隐状态分布

% 2) change in log-Likelihood
model.delta_logL = newLogL - model.logL(end);                              %%%%新计算出的似然概率值减去上一次的似然概率估计值，保证在迭代终止条件内
model.logL = [model.logL newLogL];

if options.verb,
fprintf('%d: logL=%f, delta_logL=%f, delta_alpha=%f\n',...
model.t, model.logL(end), model.delta_logL, model.delta_alpha );
end

if options.eps_logL >= model.delta_logL,
model.exitflag = 1;
elseif options.eps_alpha >= model.delta_alpha,
model.exitflag = 2;
else

model.Alpha = newAlpha;

%----------------------------------------------------
% M-Step
% The new parameters maximizing expectation of
% log-likelihood are computed.
%----------------------------------------------------

tmp_model = melgmm(X,model.Alpha,options.cov_type);                       %%%%%根据新的隐状态分布估计参数，同样采用极大似然估计法 ，根据M步中的公式计算估计值

model.Mean = tmp_model.Mean;
model.Cov = tmp_model.Cov;
model.Prior = tmp_model.Prior;

end
end % while main loop

return;

备注：子函数不再贴出，如果需要整套代码，可去下载工具箱http://cmp.felk.cvut.cz/cmp/software/stprtool/