EM(Expectation Maximization)期望最大化算法

转自著名的“丕子”博客

在统计计算中，最大期望（EM）算法是在概率（probabilistic）模型中寻找参数最大似然估计的算法，其中概率模型依赖于无法观测的隐藏变量（Latent Variable）。最大期望经常用在机器学习和计算机视觉的数据聚类（Data Clustering） 领域。最大期望算法经过两个步骤交替进行计算，第一步是计算期望（E），利用对隐藏变量的现有估计值，计算其最大似然估计值；第二步是最大化（M），最大 化在 E 步上求得的最大似然值来计算参数的值。M 步上找到的参数估计值被用于下一个 E 步计算中，这个过程不断交替进行。

最大期望值算法由 Arthur Dempster,Nan Laird和Donald Rubin在他们1977年发表的经典论文中提出。他们指出此方法之前其实已经被很多作者"在他们特定的研究领域中多次提出过"。

我们用  表示能够观察到的不完整的变量值，用  表示无法观察到的变量值，这样  和  一起组成了完整的数据。 可能是实际测量丢失的数据，也可能是能够简化问题的隐藏变量，如果它的值能够知道的话。例如，在混合模型（Mixture Model）中，如果“产生”样本的混合元素成分已知的话最大似然公式将变得更加便利（参见下面的例子）。

估计无法观测的数据

让  代表矢量 θ:  定义的参数的全部数据的概率分布（连续情况下）或者概率聚类函数（离散情况下），那么从这个函数就可以得到全部数据的最大似然值，另外，在给定的观察到的数据条件下未知数据的条件分布可以表示为：

EM算法有这么两个步骤E和M:

Expectation step: Choose  q to maximize  F:

Maximization step: Choose  θ to maximize  F:

举个例子吧：高斯混合

假设 x = (x₁,x₂,…,x_n) 是一个独立的观测样本，来自两个多元d维正态分布的混合, 让z=(z₁,z₂,…,z_n)是潜在变量，确定其中的组成部分，是观测的来源.

即：

 and 

where

 and 

目标呢就是估计下面这些参数了，包括混合的参数以及高斯的均值很方差:

似然函数:

where  是一个指示函数 ，f 是 一个多元正态分布的概率密度函数. 可以写成指数形式:

下面就进入两个大步骤了：

E-step

给定目前的参数估计 θ^(t),  Z_i 的条件概率分布是由贝叶斯理论得出，高斯之间用参数 τ加权:

.

因此，E步骤的结果:

M步骤

Q(θ|θ^(t))的二次型表示可以使得 最大化θ相对简单.  τ, (μ₁,Σ₁) and (μ₂,Σ₂) 可以单独的进行最大化.

首先考虑 τ, 有条件τ₁ + τ₂=1:

和MLE的形式是类似的，二项分布 , 因此:

下一步估计 (μ₁,Σ₁):

和加权的 MLE就正态分布来说类似

 and 

对称的:

 and  .

这个例子来自Answers.com的Expectation-maximization algorithm，由于还没有深入体验，心里还说不出一些更通俗易懂的东西来，等研究了并且应用了可能就有所理解和消化。另外，liuxqsmile也做了一些理解和翻译。

============

在网上的源码不多，有一个很好的EM_GM.m，是滑铁卢大学的Patrick P. C. Tsui写的，拿来分享一下：

运行的时候可以如下进行初始化：

X = zeros(600,2);
X(1:200,:) = normrnd(0,1,200,2);
X(201:400,:) = normrnd(0,2,200,2);
X(401:600,:) = normrnd(0,3,200,2);
[W,M,V,L] = EM_GM(X,3,[],[],1,[])

下面是程序源码：

function[W,M,V,L] = EM_GM(X,k,ltol,maxiter,pflag,Init)
% [W,M,V,L] = EM_GM(X,k,ltol,maxiter,pflag,Init)
%
% EM algorithm for k multidimensional Gaussian mixture estimation
%
% Inputs:
%   X(n,d) - input data, n=number of observations, d=dimension of variable
%   k - maximum number of Gaussian components allowed
%   ltol - percentage of the log likelihood difference between 2 iterations ([] for none)
%   maxiter - maximum number of iteration allowed ([] for none)
%   pflag - 1 for plotting GM for 1D or 2D cases only, 0 otherwise ([] for none)
%   Init - structure of initial W, M, V: Init.W, Init.M, Init.V ([] for none)
%
% Ouputs:
%   W(1,k) - estimated weights of GM
%   M(d,k) - estimated mean vectors of GM
%   V(d,d,k) - estimated covariance matrices of GM
%   L - log likelihood of estimates
%
% Written by
%   Patrick P. C. Tsui,
%   PAMI research group
%   Department of Electrical and Computer Engineering
%   University of Waterloo,
%   March, 2006
%

%%%% Validate inputs %%%%
ifnargin <= 1,
 disp('EM_GM must have at least 2 inputs: X,k!/n')
 return
elseifnargin == 2,
 ltol = 0.1; maxiter = 1000; pflag = 0; Init = [];
 err_X = Verify_X(X);
 err_k = Verify_k(k);
 iferr_X | err_k, return;end
elseifnargin == 3,
 maxiter = 1000; pflag = 0; Init = [];
 err_X = Verify_X(X);
 err_k = Verify_k(k);
 [ltol,err_ltol] = Verify_ltol(ltol);
 iferr_X | err_k | err_ltol, return;end
elseifnargin == 4,
 pflag = 0;  Init = [];
 err_X = Verify_X(X);
 err_k = Verify_k(k);
 [ltol,err_ltol] = Verify_ltol(ltol);
 [maxiter,err_maxiter] = Verify_maxiter(maxiter);
 iferr_X | err_k | err_ltol | err_maxiter, return;end
elseifnargin == 5,
 Init = [];
 err_X = Verify_X(X);
 err_k = Verify_k(k);
 [ltol,err_ltol] = Verify_ltol(ltol);
 [maxiter,err_maxiter] = Verify_maxiter(maxiter);
 [pflag,err_pflag] = Verify_pflag(pflag);
 iferr_X | err_k | err_ltol | err_maxiter | err_pflag, return;end
elseifnargin == 6,
 err_X = Verify_X(X);
 err_k = Verify_k(k);
 [ltol,err_ltol] = Verify_ltol(ltol);
 [maxiter,err_maxiter] = Verify_maxiter(maxiter);
 [pflag,err_pflag] = Verify_pflag(pflag);
 [Init,err_Init]=Verify_Init(Init);
 iferr_X | err_k | err_ltol | err_maxiter | err_pflag | err_Init, return;end
else
 disp('EM_GM must have 2 to 6 inputs!');
 return
end

%%%% Initialize W, M, V,L %%%%
t = cputime;
ifisempty(Init),
 [W,M,V] = Init_EM(X,k); L = 0;
else
 W = Init.W;
 M = Init.M;
 V = Init.V;
end
Ln = Likelihood(X,k,W,M,V); % Initialize log likelihood
Lo = 2*Ln;

%%%% EM algorithm %%%%
niter = 0;
while(abs(100*(Ln-Lo)/Lo)>ltol) & (niter<=maxiter),
 E = Expectation(X,k,W,M,V); % E-step
 [W,M,V] = Maximization(X,k,E);  % M-step
 Lo = Ln;
 Ln = Likelihood(X,k,W,M,V);
 niter = niter + 1;
end
L = Ln;

%%%% Plot 1D or 2D %%%%
ifpflag==1,
 [n,d] = size(X);
 ifd>2,
 disp('Can only plot 1 or 2 dimensional applications!/n');
 else
 Plot_GM(X,k,W,M,V);
 end
 elapsed_time = sprintf('CPU time used for EM_GM: %5.2fs',cputime-t);
 disp(elapsed_time);
 disp(sprintf('Number of iterations: %d',niter-1));
end
%%%%%%%%%%%%%%%%%%%%%%
%%%% End of EM_GM %%%%
%%%%%%%%%%%%%%%%%%%%%%

functionE = Expectation(X,k,W,M,V)
[n,d] = size(X);
a = (2*pi)^(0.5*d);
S = zeros(1,k);
iV = zeros(d,d,k);
forj=1:k,
 ifV(:,:,j)==zeros(d,d), V(:,:,j)=ones(d,d)*eps; end
 S(j) = sqrt(det(V(:,:,j)));
 iV(:,:,j) = inv(V(:,:,j));
end
E = zeros(n,k);
fori=1:n,
 forj=1:k,
 dXM = X(i,:)'-M(:,j);
 pl = exp(-0.5*dXM'*iV(:,:,j)*dXM)/(a*S(j));
 E(i,j) = W(j)*pl;
 end
 E(i,:) = E(i,:)/sum(E(i,:));
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Expectation %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[W,M,V] = Maximization(X,k,E)
[n,d] = size(X);
W = zeros(1,k); M = zeros(d,k);
V = zeros(d,d,k);
fori=1:k,  % Compute weights
 forj=1:n,
 W(i) = W(i) + E(j,i);
 M(:,i) = M(:,i) + E(j,i)*X(j,:)';
 end
 M(:,i) = M(:,i)/W(i);
end
fori=1:k,
 forj=1:n,
 dXM = X(j,:)'-M(:,i);
 V(:,:,i) = V(:,:,i) + E(j,i)*dXM*dXM';
 end
 V(:,:,i) = V(:,:,i)/W(i);
end
W = W/n;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Maximization %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

functionL = Likelihood(X,k,W,M,V)
% Compute L based on K. V. Mardia, "Multivariate Analysis", Academic Press, 1979, PP. 96-97
% to enchance computational speed
[n,d] = size(X);
U = mean(X)';
S = cov(X);
L = 0;
fori=1:k,
 iV = inv(V(:,:,i));
 L = L + W(i)*(-0.5*n*log(det(2*pi*V(:,:,i))) ...
 -0.5*(n-1)*(trace(iV*S)+(U-M(:,i))'*iV*(U-M(:,i))));
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Likelihood %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%

functionerr_X = Verify_X(X)
err_X = 1;
[n,d] = size(X);
ifn<d,
 disp('Input data must be n x d!/n');
 return
end
err_X = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_X %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%

functionerr_k = Verify_k(k)
err_k = 1;
if~isnumeric(k) | ~isreal(k) | k<1,
 disp('k must be a real integer >= 1!/n');
 return
end
err_k = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_k %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%

function[ltol,err_ltol] = Verify_ltol(ltol)
err_ltol = 1;
ifisempty(ltol),
 ltol = 0.1;
elseif~isreal(ltol) | ltol<=0,
 disp('ltol must be a positive real number!');
 return
end
err_ltol = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_ltol %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[maxiter,err_maxiter] = Verify_maxiter(maxiter)
err_maxiter = 1;
ifisempty(maxiter),
 maxiter = 1000;
elseif~isreal(maxiter) | maxiter<=0,
 disp('ltol must be a positive real number!');
 return
end
err_maxiter = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_maxiter %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[pflag,err_pflag] = Verify_pflag(pflag)
err_pflag = 1;
ifisempty(pflag),
 pflag = 0;
elseifpflag~=0 & pflag~=1,
 disp('Plot flag must be either 0 or 1!/n');
 return
end
err_pflag = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_pflag %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[Init,err_Init] = Verify_Init(Init)
err_Init = 1;
ifisempty(Init),
 % Do nothing;
elseifisstruct(Init),
 [Wd,Wk] = size(Init.W);
 [Md,Mk] = size(Init.M);
 [Vd1,Vd2,Vk] = size(Init.V);
 ifWk~=Mk | Wk~=Vk | Mk~=Vk,
 disp('k in Init.W(1,k), Init.M(d,k) and Init.V(d,d,k) must equal!/n')
 return
 end
 ifMd~=Vd1 | Md~=Vd2 | Vd1~=Vd2,
 disp('d in Init.W(1,k), Init.M(d,k) and Init.V(d,d,k) must equal!/n')
 return
 end
else
 disp('Init must be a structure: W(1,k), M(d,k), V(d,d,k) or []!');
 return
end
err_Init = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Verify_Init %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[W,M,V] = Init_EM(X,k)
[n,d] = size(X);
[Ci,C] = kmeans(X,k,'Start','cluster', ...
 'Maxiter',100, ...
 'EmptyAction','drop', ...
 'Display','off');% Ci(nx1) - cluster indeices; C(k,d) - cluster centroid (i.e. mean)
whilesum(isnan(C))>0,
 [Ci,C] = kmeans(X,k,'Start','cluster', ...
 'Maxiter',100, ...
 'EmptyAction','drop', ...
 'Display','off');
end
M = C';
Vp = repmat(struct('count',0,'X',zeros(n,d)),1,k);
fori=1:n, % Separate cluster points
 Vp(Ci(i)).count = Vp(Ci(i)).count + 1;
 Vp(Ci(i)).X(Vp(Ci(i)).count,:) = X(i,:);
end
V = zeros(d,d,k);
fori=1:k,
 W(i) = Vp(i).count/n;
 V(:,:,i) = cov(Vp(i).X(1:Vp(i).count,:));
end
%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Init_EM %%%%
%%%%%%%%%%%%%%%%%%%%%%%%

functionPlot_GM(X,k,W,M,V)
[n,d] = size(X);
ifd>2,
 disp('Can only plot 1 or 2 dimensional applications!/n');
 return
end
S = zeros(d,k);
R1 = zeros(d,k);
R2 = zeros(d,k);
fori=1:k,  % Determine plot range as 4 x standard deviations
 S(:,i) = sqrt(diag(V(:,:,i)));
 R1(:,i) = M(:,i)-4*S(:,i);
 R2(:,i) = M(:,i)+4*S(:,i);
end
Rmin = min(min(R1));
Rmax = max(max(R2));
R = [Rmin:0.001*(Rmax-Rmin):Rmax];
clf, hold on
ifd==1,
 Q = zeros(size(R));
 fori=1:k,
 P = W(i)*normpdf(R,M(:,i),sqrt(V(:,:,i)));
 Q = Q + P;
 plot(R,P,'r-'); grid on,
 end
 plot(R,Q,'k-');
 xlabel('X');
 ylabel('Probability density');
else% d==2
 plot(X(:,1),X(:,2),'r.');
 fori=1:k,
 Plot_Std_Ellipse(M(:,i),V(:,:,i));
 end
 xlabel('1^{st} dimension');
 ylabel('2^{nd} dimension');
 axis([Rmin Rmax Rmin Rmax])
end
title('Gaussian Mixture estimated by EM');
%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Plot_GM %%%%
%%%%%%%%%%%%%%%%%%%%%%%%

functionPlot_Std_Ellipse(M,V)
[Ev,D] = eig(V);
d = length(M);
ifV(:,:)==zeros(d,d),
 V(:,:) = ones(d,d)*eps;
end
iV = inv(V);
% Find the larger projection
P = [1,0;0,0];  % X-axis projection operator
P1 = P * 2*sqrt(D(1,1)) * Ev(:,1);
P2 = P * 2*sqrt(D(2,2)) * Ev(:,2);
ifabs(P1(1)) >= abs(P2(1)),
 Plen = P1(1);
else
 Plen = P2(1);
end
count = 1;
step = 0.001*Plen;
Contour1 = zeros(2001,2);
Contour2 = zeros(2001,2);
forx = -Plen:step:Plen,
 a = iV(2,2);
 b = x * (iV(1,2)+iV(2,1));
 c = (x^2) * iV(1,1) - 1;
 Root1 = (-b + sqrt(b^2 - 4*a*c))/(2*a);
 Root2 = (-b - sqrt(b^2 - 4*a*c))/(2*a);
 ifisreal(Root1),
 Contour1(count,:) = [x,Root1] + M';
 Contour2(count,:) = [x,Root2] + M';
 count = count + 1;
 end
end
Contour1 = Contour1(1:count-1,:);
Contour2 = [Contour1(1,:);Contour2(1:count-1,:);Contour1(count-1,:)];
plot(M(1),M(2),'k+');
plot(Contour1(:,1),Contour1(:,2),'k-');
plot(Contour2(:,1),Contour2(:,2),'k-');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% End of Plot_Std_Ellipse %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%