EM算法与混合高斯模型

EM算法与混合高斯模型

close all;

clear;

clc;

%% Sample Generate

N=5000;

a_real =[3/10,5/10,2/10];

mu_real = [7,12;12,7;14,15];

cov_real(:,:,1) = [1,0;0,1];

cov_real(:,:,2) = [3,1;1,3];

cov_real(:,:,3) = [3,1;1,3];

X_1 = mvnrnd(mu_real(1,:),cov_real(:,:,1),N*a_real(1));

X_2 = mvnrnd(mu_real(2,:),cov_real(:,:,2),N*a_real(2));

X_3 = mvnrnd(mu_real(3,:),cov_real(:,:,2),N*a_real(3));

X=[X_1;X_2;X_3];

X = X(randperm(size(X,1)),:);

%% Sample Ploting

x = 0:0.5:20;

y = 0:0.5:20;

[x y]=meshgrid(x,y);

mesh = [x(:),y(:)];

z_real = a_real(1)*mvnpdf(mesh,mu_real(1,:),cov_real(:,:,1))+...

a_real(2)* mvnpdf(mesh,mu_real(2,:),cov_real(:,:,2))+...

a_real(3)* mvnpdf(mesh,mu_real(3,:),cov_real(:,:,3));

subplot(2,3,1);

plot(X_1(:,1),X_1(:,2),'x',X_2(:,1),X_2(:,2),'o',X_3(:,1),X_3(:,2),'<')

subplot(2,3,2);

contour(x,y,reshape(z_real,size(x,2),size(y,2)));

subplot(2,3,3);

surf(x,y,reshape(z_real,size(x,2),size(y,2)));

subplot(2,3,4);

plot(X(:,1),X(:,2),'x');

%%subplot(2,2,3);

%%surf(x,y,reshape(z_real,size(x,2),size(y,2)));

%%shading interp;

%% Parameter Initialization

a = [1/3, 1/3,1/3];

cov(:,:,1) = [1,0;0,1];

cov(:,:,2) = [1,0;0,1];

cov(:,:,3) = [1,0;0,1];

mu_y_init = (max(X(:,1))+min(X(:,1)))/2;

mu_x1_init = max(X(:,2))/4+3*min(X(:,2))/4;

mu_x2_init = 2*max(X(:,2))/4+2*min(X(:,2))/4;

mu_x3_init = 3*max(X(:,2))/4+1*min(X(:,2))/4;

w = zeros(size(X,1),length(a)); %%w(i,j),i是样本数量，j是聚类数量，w(i,j)的值表示样本i属于聚类j的概率，最大值表示i的聚类

mu = [mu_x1_init,mu_y_init;mu_x2_init,mu_y_init;mu_x3_init,mu_y_init];

%% EM Implementation

iter = 40;

for i = 1:iter

%% Expectaion: 根据现有的（最新得到的）高斯分布和先验概率，计算每一个样本属于每一个聚类的概率

for j = 1 : length(a)

w(:,j)=a(j)*mvnpdf(X,mu(j,:),cov(:,:,j));

end

w=w./repmat(sum(w,2),1,size(w,2)); %%为了方便./计算，建立一个和w一样规模的矩阵，矩阵的列是相同的，每一行是w(:,1)+w(:,2)的值

%% Maximum: 根据现有的（最新得到的）的每一个样本属于每一个聚类的概率，计算先验概率和高斯参数

a = sum(w,1)./size(w,1); %%把w的每一行加起来就能得到每一个聚类的先验概率

mu = w'*X; %%分别得到X每一维对于每一个聚类的期望，mu(i,j),i是维数，j是聚类数

mu= mu./repmat((sum(w,1))',1,size(mu,2));

for j = 1 : length(a)

vari = repmat(w(:,j),1,size(X,2)).*(X- repmat(mu(j,:),size(X,1),1)); %%得到一个特定聚类X每一维的方差矩阵，乘以w，（相当于选择出属于该聚类的X）

cov(:,:,j) = (vari'*vari)/sum(w(:,j),1);

end

end

%% Estimation

[c estimate] = max(w,[],2);

%% Estimation Plotting

z = a(1)*mvnpdf(mesh,mu(1,:),cov(:,:,1))+...

a(2)* mvnpdf(mesh,mu(2,:),cov(:,:,2))+...

a(3)* mvnpdf(mesh,mu(3,:),cov(:,:,3));

subplot(2,3,5);

contour(x,y,reshape(z,size(x,2),size(y,2)));

one = find(estimate==1);

two = find(estimate == 2);

three = find(estimate == 3);

% Plot Examples

subplot(2,3,6);

plot(X(one, 1), X(one, 2), 'x',X(two, 1), X(two, 2), 'o', X(three, 1), X(three, 2), '<');

%plot(X(neg, 1), X(neg, 2), 'o');

%% Build-in EM Implementation

%options = statset('Display','final');

%gm = gmdistribution.fit(X,2,'Options',options);

%subplot(2,2,4);

%ezcontour(@(x,y)pdf(gm,[x y]),[0 20],[0 20]);

matlab中各种高斯相关函数

matlab， 高斯函数， 高斯分布

最常见的是产生服从一维标准正态分布的随机数

1. n=100; 

   

2. x=randn(1,n) 

   

实现服从任意一维高斯分布的随机数

1. u=10; 

   

2. sigma=4; 

   

3. x=sigma*randn(1,n)+u 

   

产生服从多元高斯分布的随机变量函数mvnrnd,[multivarite normal random]

1. n=100; %产生随机数的个数 

   

2. mu=[1 -1]; 

   

3. Sigma=[.9,.4;.4,.3]; 

   

4. r=mvnrnd(mu,Sigma,n); 

   

将产生的随机数绘制在二维平面

1. scatter(r(:,1),r(:,2)); 

   

 

1474457688429.jpg

当然mvnrnd函数还可以产生更高维数的高斯随机数，具体参见matlab help。

产生多元高斯分布概率密度函数
Y=mvnpdf(X,[MU,Sigma])
其中可省参数MU,Sigma默认值分别为零向量和单位阵，X是的矩阵，N是样本个数，D是样本维数。

1. mu = [1 -1]; Sigma = [.9 .4; .4 .3]; 

   

2. [X1,X2] = meshgrid(linspace(-1,3,25)', linspace(-3,1,25)'); 

   

3. X = [X1(:) X2(:)]; 

   

4. p = mvnpdf(X, mu, Sigma); 

   

5. surf(X1,X2,reshape(p,25,25)); 

   

和下面代码产生的趋势相同

1. mu = [1 -1]; 

   

2. Sigma = [.9 .4; .4 .3]; 

   

3. [X,Y] = meshgrid(linspace(-1,3,25)', linspace(-3,1,25)'); 

   

4. for i=1:25 

   

5. for j=1:25 

   

6. XY=[X(i,j),Y(i,j)]; 

   

7. Z(i,j)=exp(-0.5*(XY-mu)/Sigma*(XY-mu)'); 

   

8. end 

   

9. end 

   

10. surf(X,Y,Z); 

    

 

1474460161935.jpg

高斯分布函数
Y=mvncdf(X,[Mu],[Sigma]) , cumulative probability of the multivariate norm distribution with mean Mu and covariance Sigma.
具体使用看代码

1. mu = [1 -1]; Sigma = [.9 .4; .4 .3]; 

   

2. [X1,X2] = meshgrid(linspace(-1,3,25)', linspace(-3,1,25)'); 

   

3. X = [X1(:) X2(:)]; 

   

4. p = mvncdf(X, mu, Sigma); 

   

5. surf(X1,X2,reshape(p,25,25)); 

   

 

1474460555428.jpg

高斯隶属函数
gaussmf(X,[Sigma,Mu])

1. x = (0:0.1:10)'; 

   

2. y1 = gaussmf(x, [0.5 5]); 

   

3. y2 = gaussmf(x, [1 5]); 

   

4. y3 = gaussmf(x, [2 5]); 

   

5. y4 = gaussmf(x, [3 5]); 

   

6. plot(x, [y1 y2 y3 y4]);