EM算法與混合高斯模型

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是樣本數量，j是聚類數量，w(i,j)的值表示樣本i屬於聚類j的概率，最大值表示i的聚類

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: 根據現有的（最新得到的）高斯分布和先驗概率，計算每一個樣本屬於每一個聚類的概率

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)); %%為了方便./計算，建立一個和w一樣規模的矩陣，矩陣的列是相同的，每一行是w(:,1)+w(:,2)的值

%% Maximum: 根據現有的（最新得到的）的每一個樣本屬於每一個聚類的概率，計算先驗概率和高斯參數

a = sum(w,1)./size(w,1); %%把w的每一行加起來就能得到每一個聚類的先驗概率

mu = w‘*X; %%分別得到X每一維對於每一個聚類的期望，mu(i,j),i是維數，j是聚類數

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)); %%得到一個特定聚類X每一維的方差矩陣，乘以w，（相當於選擇出屬於該聚類的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中各種高斯相關函數

matlab， 高斯函數， 高斯分布

最常見的是產生服從一維標準正態分布的隨機數

1. n=100;

   

2. x=randn(1,n)

   

實現服從任意一維高斯分布的隨機數

1. u=10;

   

2. sigma=4;

   

3. x=sigma*randn(1,n)+u

   

產生服從多元高斯分布的隨機變量函數mvnrnd,[multivarite normal random]

1. n=100; %產生隨機數的個數

   

2. mu=[1 -1];

   

3. Sigma=[.9,.4;.4,.3];

   

4. r=mvnrnd(mu,Sigma,n);

   

將產生的隨機數繪制在二維平面

1. scatter(r(:,1),r(:,2));

   

1474457688429.jpg

當然mvnrnd函數還可以產生更高維數的高斯隨機數，具體參見matlab help。

產生多元高斯分布概率密度函數
Y=mvnpdf(X,[MU,Sigma])
其中可省參數MU,Sigma默認值分別為零向量和單位陣，X是的矩陣，N是樣本個數，D是樣本維數。

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));

   

和下面代碼產生的趨勢相同

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

高斯分布函數
Y=mvncdf(X,[Mu],[Sigma]) , cumulative probability of the multivariate norm distribution with mean Mu and covariance Sigma.
具體使用看代碼

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

高斯隸屬函數
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]);

   

EM算法與混合高斯模型