基于MATLAB的高斯混合模型（GMM）实现

基于MATLAB的高斯混合模型（GMM）实现，包含自动生成混合数据、EM算法实现及可视化模块：

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一、核心代码

1. 数据生成模块

function [X, trueParams] = generateGMMData(N, K, dim)
    % 生成混合高斯数据
    % 输入:
    %   N: 总样本数
    %   K: 高斯分量数
    %   dim: 数据维度
    % 输出:
    %   X: 生成的数据矩阵 (dim x N)
    %   trueParams: 真实参数结构体
    
    rng(0); % 固定随机种子
    trueParams = struct();
    
    % 随机生成参数
    trueParams.weights = rand(1,K);
    trueParams.weights = trueParams.weights / sum(trueParams.weights);
    
    trueParams.means = 10*randn(dim,K);
    trueParams.covs = repmat(eye(dim), [1,1,K]);
    for k=1:K
        trueParams.covs(:,:,k) = trueParams.covs(:,:,k) * (0.5 + 0.5*rand);
    end
    
    % 生成数据
    X = zeros(dim,N);
    for k=1:K
        Nk = round(N*trueParams.weights(k));
        X(:,(1:Nk)+(k-1)*Nk) = mvnrnd(trueParams.means(:,k), trueParams.covs(:,:,k), Nk);
    end
end

2. GMM-EM算法实现

function [mu, cov, weight, logLik] = gmm_em(X, K, maxIter, tol)
    % 输入:
    %   X: 数据矩阵 (dim x N)
    %   K: 高斯分量数
    %   maxIter: 最大迭代次数
    %   tol: 收敛阈值
    % 输出:
    %   mu: 均值矩阵 (dim x K)
    %   cov: 协方差矩阵 (dim x dim x K)
    %   weight: 权重向量 (1 x K)
    %   logLik: 对数似然序列
    
    [dim, N] = size(X);
    logLik = zeros(maxIter,1);
    
    % 初始化参数
    mu = 10*randn(dim,K);
    cov = repmat(eye(dim), [1,1,K]);
    weight = ones(1,K)/K;
    
    % 预计算对数权重
    logWeight = log(weight);
    
    for iter=1:maxIter
        % E步：计算后验概率
        logPost = zeros(N,K);
        for k=1:K
            logPost(:,k) = logWeight(k) + log_mvnpdf(X, mu(:,k), cov(:,:,k));
        end
        logSum = logsumexp(logPost, 2);
        post = exp(logPost - logSum);
        
        % M步：更新参数
        Nk = sum(post,1);
        weight = Nk/N;
        
        for k=1:K
            mu(:,k) = (X * post(:,k)) / Nk(k);
            diff = X - mu(:,k);
            cov(:,:,k) = (diff * diag(post(:,k)) * diff') / Nk(k);
            % 正则化协方差矩阵
            cov(:,:,k) = (cov(:,:,k) + cov(:,:,k)')/2 + 1e-6*eye(dim);
        end
        
        % 计算对数似然
        logLik(iter) = sum(logsumexp(logPost,2));
        
        % 检查收敛
        if iter > 1 && abs(logLik(iter)-logLik(iter-1)) < tol
            logLik = logLik(1:iter);
            break;
        end
    end
end

function y = log_mvnpdf(X, mu, Sigma)
    % 计算多元高斯分布对数概率密度
    [dim,N] = size(X);
    X_centered = X - mu;
    [R, p] = chol(Sigma);
    if p ~= 0
        y = -inf(1,N);
        return;
    end
    y = -0.5 * sum((X_centered/R).^2, 1) - 0.5*dim*log(2*pi) - sum(log(diag(R)));
end

function s = logsumexp(x, dim)
    % 稳定计算log-sum-exp
    x_max = max(x,[],dim);
    s = x_max + log(sum(exp(x - x_max), dim));
end

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二、完整示例Demo

1. 数据生成与可视化

% 生成数据
N = 1000; % 总样本数
K = 3;    % 高斯分量数
[X, trueParams] = generateGMMData(N, K, 2);

% 可视化原始数据
figure;
scatter(X(1,:), X(2,:), 10, 'filled');
hold on;
for k=1:K
    plot(trueParams.means(1,k), trueParams.means(2,k), 'kx', 'MarkerSize', 15, 'LineWidth', 2);
end
title('原始数据分布');
hold off;

2. 模型训练

% 设置参数
maxIter = 100;
tol = 1e-5;

% 运行EM算法
[mu, cov, weight, logLik] = gmm_em(X, K, maxIter, tol);

% 显示训练结果
disp('估计参数:');
disp('均值:');
disp(mu');
disp('协方差:');
disp(cov(:,:,1));
disp(['对数似然: ', num2str(logLik(end))]);

3. 结果可视化

% 绘制等高线
figure;
hold on;
scatter(X(1,:), X(2,:), 10, 'filled');
[x1, x2] = meshgrid(linspace(min(X(1,:)), max(X(1,:)),50), ...
                     linspace(min(X(2,:)), max(X(2,:)),50));
X_grid = [x1(:)'; x2(:)'];
post = zeros(size(X_grid,2), K);

% 计算后验概率
for k=1:K
    post(:,k) = weight(k) * mvnpdf(X_grid', mu(:,k), cov(:,:,k));
end
post = post ./ sum(post,2);

% 绘制概率密度等高线
contour(x1, x2, reshape(post(:,1), size(x1)), [0.01,0.1,0.5,1], 'LineColor', 'r');
contour(x1, x2, reshape(post(:,2), size(x1)), [0.01,0.1,0.5,1], 'LineColor', 'g');
contour(x1, x2, reshape(post(:,3), size(x1)), [0.01,0.1,0.5,1], 'LineColor', 'b');
title('GMM概率密度估计');
hold off;

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三、关键优化策略

1. 协方差矩阵正则化

% 在M步更新协方差时添加正则化
cov(:,:,k) = (diff * diag(post(:,k)) * diff') / Nk(k) + 1e-6*eye(dim);

2. 并行计算加速

% 使用parfor加速E步计算
parfor k=1:K
    logPost(:,k) = logWeight(k) + log_mvnpdf(X, mu(:,k), cov(:,:,k));
end

3. 收敛性增强

% 添加参数变化监控
mu_prev = mu;
for iter=1:maxIter
    % ... M步更新参数 ...
    if norm(mu - mu_prev, 'fro') < tol
        break;
    end
    mu_prev = mu;
end

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四、性能评估

% 计算估计误差
trueMu = trueParams.means;
estMu = mu;

trueCov = trueParams.covs;
estCov = cov;

% 均值误差
mu_err = mean(sqrt(sum((trueMu - estMu).^2, 1)));

% 协方差误差
cov_err = mean(vecnorm(trueCov(:) - estCov(:)));

disp(['均值平均误差: ', num2str(mu_err)]);
disp(['协方差平均误差: ', num2str(cov_err)]);

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五、扩展应用场景

1. 图像分割

% 加载图像
img = imread('peppers.png');
img_gray = rgb2gray(img);
img_double = im2double(img_gray);

% 将图像转换为数据矩阵
[X_img, Y_img] = meshgrid(1:size(img_double,2), 1:size(img_double,1));
X_img = X_img(:);
Y_img = Y_img(:);
img_data = [X_img, Y_img]';

% 运行GMM聚类
K = 5;
[mu_img, cov_img, weight_img, ~] = gmm_em(img_data, K, 100, 1e-5);

% 生成分割结果
segmented = reshape(mode(bsxfun(@plus, mu_img', cov_img(:,:,1)), 2), size(img));
imshow(label2rgb(segmented));

2. 语音特征聚类

% 加载语音MFCC特征
load('mfcc_data.mat'); % 假设包含1000个20维MFCC特征向量

% 运行GMM聚类
K = 10;
[mu, cov, weight, ~] = gmm_em(mfcc_data, K, 200, 1e-5);

% 计算软分类
post = zeros(size(mfcc_data,2), K);
for k=1:K
    post(:,k) = weight(k) * mvnpdf(mfcc_data', mu(:,k), cov(:,:,k));
end
post = post ./ sum(post,2);

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六、注意事项

1. 初始化策略：建议先用K-means初始化均值

   [idx, kmeans_mu] = kmeans(X', K);
   mu = kmeans_mu';

2. 维度灾难：高维数据建议使用对角协方差矩阵

   cov(:,:,k) = diag(diag(cov(:,:,k))); % 强制对角

3. 数值稳定性：计算概率密度时使用对数域

   log_prob = log_mvnpdf(X, mu, cov);

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七、参考

1. Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer.
2. Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective. MIT Press.
3. 代码 matlab 实现GMM------EM算法 www.youwenfan.com/contentcsi/63293.html
4. MathWorks官方网页：fitgmdist函数说明