[Speech recognition] speaker recognition based on matlab Gaussian Mixture Model (GMM) [including Matlab source code 574]

1. Introduction

1 Overview of Gaussian Mixture Model

Gaussian density function estimation is a parametric model. The Gaussian Mixture Model (GMM) is an extension of a single Gaussian probability density function. GMM can smoothly approximate the density distribution of any shape. There are two types of Gaussian Mixture Models: Single Gaussian Model (SGM) and Gaussian Mixture Model (GMM). Similar to clustering, according to the different parameters of the Gaussian probability density function (PDF), each Gaussian model can be regarded as a category, input a sample x, the value can be calculated by PDF, and then judged by a threshold Whether the sample belongs to the Gaussian model. Obviously, SGM is suitable for the division of problems with only two categories, while GMM is more refined because it has multiple models, is suitable for multi-category division, and can be applied to complex object modeling.
1.1 Single Gaussian model
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1.2 Gaussian mixture model

2 Gaussian mixture model parameter estimation

2.1 GMM with known sample classification
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Second, the source code

function mix=gmm_init(ncentres,data,kiter,covar_type)
%% 输入：
% ncentres:混合模型数目
% train_data:训练数据
% kiter:kmeans的迭代次数
%% 输出：
% mix:gmm的初始参数集合


[dim,data_sz]=size(data');

mix.priors=ones(1,ncentres)./ncentres;
mix.centres=randn(ncentres,dim);
switch covar_type
case 'diag'
  % Store diagonals of covariance matrices as rows in a matrix
  mix.covars=ones(ncentres,dim);
case 'full'
  % Store covariance matrices in a row vector of matrices
  mix.covars=repmat(eye(dim),[1 1 ncentres]);
otherwise
  error(['Unknown covariance type ', mix.covar_type]);  
end


% Arbitrary width used if variance collapses to zero: make it 'large' so
% that centre is responsible for a reasonable number of points.
GMM_WIDTH=1.0;

%kmeans算法
% [mix.centres,options,post]=k_means(mix.centres,data);
[mix.centres,post]=k_means(mix.centres,data,kiter);

% Set priors depending on number of points in each cluster
cluster_sizes = max(sum(post,1),1);  % Make sure that no prior is zero
mix.priors = cluster_sizes/sum(cluster_sizes); % Normalise priors

switch covar_type
case 'diag'
  for j=1:ncentres
   % Pick out data points belonging to this centre
   c=data(find(post(:,j)),:);
   diffs=c-(ones(size(c,1),1)*mix.centres(j,:));
   mix.covars(j,:)=sum((diffs.*diffs),1)/size(c,1);
   % Replace small entries by GMM_WIDTH value
   mix.covars(j,:)=mix.covars(j,:)+GMM_WIDTH.*(mix.covars(j,:)<eps);
  end 
case 'full'
  for j=1:ncentres
   % Pick out data points belonging to this centre
   c=data(find(post(:,j)),:);
   diffs=c-(ones(size(c,1),1)*mix.centres(j,:));
   mix.covars(:,:,j)=(diffs'*diffs)/(size(c,1)+eps);
   % Add GMM_WIDTH*Identity to rank-deficient covariance matrices
   if rank(mix.covars(:,:,j))<dim
	mix.covars(:,:,j)=mix.covars(:,:,j)+GMM_WIDTH.*eye(dim);
   end
  end
otherwise
  error(['Unknown covariance type ', mix.covar_type]);
end

mix.ncentres=ncentres;
mix.covar_type=covar_type;

%=============================================================
function [centres,post]=k_means(centres,data,kiter)

[dim,data_sz]=size(data');
ncentres=size(centres,1); %簇的数目
[ignore,perm]=sort(rand(1,data_sz)); %产生任意顺序的随机数
perm = perm(1:ncentres); %取前ncentres个作为初始簇中心的序号
centres=data(perm,:); %指定初始中心点
id=eye(ncentres); %Matrix to make unit vectors easy to construct
for n=1:kiter
  % Save old centres to check for termination
  old_centres=centres; %存储旧的中心,便于计算终止条件
  
  % Calculate posteriors based on existing centres
  d2=(ones(ncentres,1)*sum((data.^2)',1))'+...
     ones(data_sz,1)* sum((centres.^2)',1)-2.*(data*(centres')); %计算距离
 
  % Assign each point to nearest centre
  [minvals, index]=min(d2', [], 1);
  post=id(index,:);

  num_points = sum(post, 1);
  % Adjust the centres based on new posteriors
  for j = 1:ncentres
    if (num_points(j) > 0)
      centres(j,:) = sum(data(find(post(:,j)),:), 1)/num_points(j);
    end
  end

Three, running results

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Four, remarks