MATLAB中“fitgmdist”的用法及其GMM聚類演算法

作者：凱魯嘎吉 - 博客園 http://www.cnblogs.com/kailugaji/

高斯混合模型的基本原理：聚類——GMM，MATLAB官方文檔中有關於fitgmdist的介紹：fitgmdist。我之前寫過有關GMM聚類的演算法：GMM演算法的matlab程式。這篇文章主要應用MATLAB自帶的函數來進行聚類。

1. fitgmdist函數介紹

fitgmdist的使用形式：gmm = fitgmdist(X,k,Name,Value)

輸入

‘RegularizationValue’, 0。（取值：0， 0.1， 0.01,....，正則化繫數，防止協方差奇異）

'CovarianceType', 'full'。（取值： 'full'，協方差矩陣是非對角陣，'diagonal'，協方差矩陣為對角陣）

‘Start’, 'plus'。 （取值：‘randSample’，隨機初始化，‘plus’，k-means++初始化，‘S’，自定義初始化），其中S = struct('mu',init_Mu,'Sigma',init_Sigma,'ComponentProportion',init_Components);

‘Options’，statset('Display', 'final', 'MaxIter', MaxIter, 'TolFun', TolFun)。 （'Display'有三個取值：‘final’ 顯示最終的輸出結果、‘iter’ 顯示每次迭代的結果、‘off’ 不顯示優化參數信息；'MaxIter'：預設100，最大迭代次數；'TolFun'：預設1e-6，目標函數的終止誤差）

輸出

gmm.mu：更新完後的聚類中心（均值）

gmm.Sigma：更新完後的協方差矩陣

gmm.ComponentProportion：更新完後的混合比例

gmm.NegativeLogLikelihood：更新完後的負對數似然函數

gmm.NumIterations：實際迭代次數

gmm.BIC：貝葉斯信息準則，用於模型選擇

更多參數，請在命令行輸入properties(gmm)

2. 高斯混合模型聚類實例

generate.m

function data=generate()
%生成數據
mu1 = [1 2];
Sigma1 = [2 0; 0 0.5];
mu2 = [-1 -2];
Sigma2 = [1 0;0 1];
data = [mvnrnd(mu1,Sigma1,400), ones(400,1);mvnrnd(mu2,Sigma2,600), 2*ones(600,1)];
X=[data(:, 1), data(:, 2)];
figure(1)
plot(X(:,1), X(:,2),'bo')
title('Scatter Plot')
xlim([min(X(:)) max(X(:))]) % Make axes have the same scale
ylim([min(X(:)) max(X(:))])

具體數據

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GMM_main.m

function [accuracy,NumIterations]=GMM_main(data, K)
%主函數
[~, data_dim]=size(data);
X=data(:, 1:data_dim-1);  %數據
real_label=data(:, data_dim);
[label, ~, NumIterations]=Matlab_gmm_2(X, K);
accuracy=succeed(real_label,K,label);

Matlab_gmm.m

function [label, NegativeLogLikelihood, NumIterations]=Matlab_gmm(X, K)
%協方差矩陣為對角陣，數據獨立同分佈
[X_num,X_dim]=size(X);
para_sigma_inv=zeros(X_dim, X_dim, K);
N_pdf=zeros(X_num, K);  %單高斯分佈的概率密度函數
RegularizationValue=0.001;   %正則化繫數，協方差矩陣求逆
MaxIter=100;   %最大迭代次數
TolFun=1e-8;   %終止條件
% 自己設置初始化參數
% init_Mu = [1 1; 2 2];
% init_Sigma(:,:,1) = [1 1; 1 2];
% init_Sigma(:,:,2) = 2*[1 1; 1 2];
% init_Components = [1/2,1/2];
% S = struct('mu',init_Mu,'Sigma',init_Sigma,'ComponentProportion',init_Components);
% gmm=fitgmdist(X, K, 'RegularizationValue', RegularizationValue, 'CovarianceType', 'diagonal', 'Start', 'S', 'Options', statset('Display', 'final', 'MaxIter', MaxIter, 'TolFun', TolFun));
gmm=fitgmdist(X, K, 'RegularizationValue', RegularizationValue, 'CovarianceType', 'diagonal', 'Start', 'plus', 'Options', statset('Display', 'final', 'MaxIter', MaxIter, 'TolFun', TolFun));
NegativeLogLikelihood=gmm.NegativeLogLikelihood;
NumIterations=gmm.NumIterations;  %迭代次數
mu=gmm.mu;  %均值
Sigma=gmm.Sigma;   %協方差矩陣
ComponentProportion=gmm.ComponentProportion;  %混合比例
for k=1:K
    sigma_inv=1./Sigma(:,:,k);  %sigma的逆矩陣,(X_dim, X_dim)的矩陣
    para_sigma_inv(:, :, k)=diag(sigma_inv);  %sigma^(-1)
end
for k=1:K
    coefficient=(2*pi)^(-X_dim/2)*sqrt(det(para_sigma_inv(:, :, k)));  %高斯分佈的概率密度函數e左邊的繫數
    X_miu=X-repmat(mu(k,:), X_num, 1);  %X-miu: (X_num, X_dim)的矩陣
    exp_up=sum((X_miu*para_sigma_inv(:, :, k)).*X_miu,2);  %指數的冪，(X-miu)'*sigma^(-1)*(X-miu)
    N_pdf(:,k)=coefficient*exp(-0.5*exp_up);
end
responsivity=N_pdf.*repmat(ComponentProportion,X_num,1);  %響應度responsivity的分子，（X_num,K）的矩陣
responsivity=responsivity./repmat(sum(responsivity,2),1,K);  %responsivity:在當前模型下第n個觀測數據來自第k個分模型的概率，即分模型k對觀測數據Xn的響應度
%聚類
[~,label]=max(responsivity,[],2);
figure(2)
scatter(X(:,1),X(:,2),10,'.') % Scatter plot with points of size 10
hold on
gmPDF = @(x,y)reshape(pdf(gmm,[x(:) y(:)]),size(x));
fcontour(gmPDF,[-6 6])

Matlab_gmm_2.m

function [label, NegativeLogLikelihood, NumIterations]=Matlab_gmm_2(X, K)
%協方差矩陣為非對角陣，數據不獨立
[X_num,X_dim]=size(X);
N_pdf=zeros(X_num, K);  %單高斯分佈的概率密度函數
RegularizationValue=0.001;   %正則化繫數，協方差矩陣求逆
MaxIter=100;   %最大迭代次數
TolFun=1e-8;   %終止條件
% 自己設置初始化參數
% init_Mu = [1 1; 2 2];
% init_Sigma(:,:,1) = [1 1; 1 2];
% init_Sigma(:,:,2) = 2*[1 1; 1 2];
% init_Components = [1/2,1/2];
% S = struct('mu',init_Mu,'Sigma',init_Sigma,'ComponentProportion',init_Components);
% gmm=fitgmdist(X, K, 'RegularizationValue', RegularizationValue, 'CovarianceType', 'diagonal', 'Start', 'S', 'Options', statset('Display', 'final', 'MaxIter', MaxIter, 'TolFun', TolFun));
gmm=fitgmdist(X, K, 'RegularizationValue', RegularizationValue, 'CovarianceType', 'full', 'Start', 'plus', 'Options', statset('Display', 'final', 'MaxIter', MaxIter, 'TolFun', TolFun));
NegativeLogLikelihood=gmm.NegativeLogLikelihood;
NumIterations=gmm.NumIterations;  %迭代次數
mu=gmm.mu;  %均值
Sigma=gmm.Sigma;   %協方差矩陣
ComponentProportion=gmm.ComponentProportion;  %混合比例
for k=1:K
    X_miu=X-repmat(mu(k,:), X_num, 1);  %X-miu: (X_num, X_dim)的矩陣
    sigma_inv=inv(Sigma(:,:,k));  %sigma的逆矩陣,(X_dim, X_dim)的矩陣
    exp_up=sum((X_miu*sigma_inv).*X_miu,2);  %指數的冪，(X-miu)'*sigma^(-1)*(X-miu)
    coefficient=(2*pi)^(-X_dim/2)*sqrt(det(sigma_inv));  %高斯分佈的概率密度函數e左邊的繫數
    N_pdf(:,k)=coefficient*exp(-0.5*exp_up);
end
responsivity=N_pdf.*repmat(ComponentProportion,X_num,1);  %響應度responsivity的分子，（X_num,K）的矩陣
responsivity=responsivity./repmat(sum(responsivity,2),1,K);  %responsivity:在當前模型下第n個觀測數據來自第k個分模型的概率，即分模型k對觀測數據Xn的響應度
%聚類
[~,label]=max(responsivity,[],2);
figure(2)
scatter(X(:,1),X(:,2),10,'.') % Scatter plot with points of size 10
hold on
gmPDF = @(x,y)reshape(pdf(gmm,[x(:) y(:)]),size(x));
fcontour(gmPDF,[-6 6])

succeed.m

function accuracy=succeed(real_label,K,id)
%輸入K：聚的類，id：訓練後的聚類結果，N*1的矩陣
N=size(id,1);   %樣本個數
p=perms(1:K);   %全排列矩陣
p_col=size(p,1);   %全排列的行數
new_label=zeros(N,p_col);   %聚類結果的所有可能取值，N*p_col
num=zeros(1,p_col);  %與真實聚類結果一樣的個數
%將訓練結果全排列為N*p_col的矩陣，每一列為一種可能性
for i=1:N
    for j=1:p_col
        for k=1:K
            if id(i)==k
                new_label(i,j)=p(j,k);  %iris資料庫，1 2 3
            end
        end
    end
end
%與真實結果比對，計算精確度
for j=1:p_col
    for i=1:N
        if new_label(i,j)==real_label(i)
                num(j)=num(j)+1;
        end
    end
end
accuracy=max(num)/N;

結果

以第二種情況為例，數據不獨立，協方差矩陣不是只在對角線上有元素。

>> [accuracy,NumIterations]=GMM_main(data, 2)
32 iterations, log-likelihood = -3449.42

accuracy =

   0.995000000000000


NumIterations =

    32