正态分布下的最大似然估计_正态分布的最大似然估计_

% 二维正态分布的两分类问题 （ML估计） clc; clear; % 两个类别数据的均值向量 Mu = [0 0; 3 3]'; % 协方差矩阵 S1 = 0.8 * eye(2); S(:, :, 1) = S1; S(:, :, 2) = S1; % 先验概率（类别分布） P = [1/3 2/3]'; % 样本数据规模 % 收敛性：无偏或者渐进无偏，当样本数目增加时，收敛性质会更好 N = 500; % 1.生成训练和测试数据 %{ 生成训练样本 N = 500, c = 2, d = 2 μ1=[0, 0]' μ2=[3, 3]' S1=S2=[0.8, 0; 0.8, 0] p(w1)=1/3 p(w2)=2/3 %} randn('seed', 0); [X_train, Y_train] = generate_gauss_classes(Mu, S, P, N); figure(); hold on; class1_data = X_train(:, Y_train==1); class2_data = X_train(:, Y_train==2); plot(class1_data(1, :), class1_data(2, :), 'r.'); plot(class2_data(1, :), class2_data(2, :), 'g.'); grid on; title('训练样本'); xlabel('N=500'); %{ 用同样的方法生成测试样本 N = 500, c = 2, d = 2 μ1=[0, 0]' μ2=[3, 3]' S1=S2=[0.8, 0; 0.8, 0] p(w1)=1/3 p(w2)=2/3 %} randn('seed', 100); [X_test, Y_test] = generate_gauss_classes(Mu, S, P, N); figure(); hold on; test1_data = X_test(:, Y_test==1); test2_data = X_test(:, Y_test==2); plot(test1_data(1, :), test1_data(2, :), 'r.'); plot(test2_data(1, :), test2_data(2, :), 'g.'); grid on; title('测试样本'); xlabel('N=500'); % 2.用训练样本采用ML方法估计参数 % 各类样本只包含本类分布的信息，也就是说不同类别的参数在函数上是独立的 [mu1_hat, s1_hat] = gaussian_ML_estimate(class1_data); [mu2_hat, s2_hat] = gaussian_ML_estimate(class2_data); mu_hat = [mu1_hat, mu2_hat]; s_hat = (1/2) * (s1_hat + s2_hat); % 3.用测试样本和估计出的参数进行分类 % 使用欧式距离进行分类 z_euclidean = euclidean_classifier(mu_hat, X_test); % 使用贝叶斯方法进行分类 z_bayesian = bayes_classifier(Mu, S, P, X_test); % 4.计算不同方法分类的误差 err_euclidean = ( 1-length(find(Y_test == z_euclidean')) / length(Y_test) ); err_bayesian = ( 1-length(find(Y_test == z_bayesian')) / length(Y_test) ); % 输出信息 disp(['基于欧式距离分类的误分率：', num2str(err_euclidean)]); disp(['基于最小错误率贝叶斯分类的误分率：', num2str(err_bayesian)]); % 画图展示 figure(); hold on; z_euclidean = transpose(z_euclidean); o = 1; q = 1; for i = 1:size(X_test, 2) if Y_test(i) ~= z_euclidean(i) plot(X_test(1,i), X_test(2,i), 'bo'); elseif z_euclidean(i)==1 euclidean_classifier_results1(:, o) = X_test(:, i); o = o+1; elseif z_euclidean(i)==2 euclidean_classifier_results2(:, q) = X_test(:, i); q = q+1; end end plot(euclidean_classifier_results1(1, :), euclidean_classifier_results1(2, :), 'r.'); plot(euclidean_classifier_results2(1, :), euclidean_classifier_results2(2, :), 'g.'); title(['基于欧式距离分类，误分率为：', num2str(err_euclidean)]); grid on; figure(); hold on; z_bayesian = transpose(z_bayesian); o = 1; q = 1; for i = 1:size(X_test, 2) if Y_test(i) ~= z_bayesian(i) plot(X_test(1,i), X_test(2,i), 'bo'); elseif z_bayesian(i)==1 bayesian_classifier_results1(:, o) = X_test(:, i); o = o+1; elseif z_bayesian(i)==2 bayesian_classifier_results2(:, q) = X_test(:, i); q = q+1; end end plot(bayesian_classifier_results1(1, :), bayesian_classifier_results1(2, :), 'r.'); plot(bayesian_classifier_results2(1, :), bayesian_classifier_results2(2, :), 'g.'); title(['基于最小错误率的贝叶斯决策分类，误分率为：', num2str(err_bayesian)]); grid on;