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基于MATLAB的支持向量数据描述算法

news 2025/9/18 6:08:48

基于MATLAB的支持向量数据描述（SVDD）算法

一、核心

function [model, decision] = mySVDD(X, kernelType, gamma, C, tol)% SVDD模型训练% 输入:% X - 训练数据 (N x D)% kernelType - 核函数类型 ('linear','poly','rbf')% gamma - RBF核参数% C - 正则化参数% tol - 支持向量容忍度[N, D] = size(X);% 计算核矩阵K = computeKernel(X, X, kernelType, gamma);% 构建二次规划参数H = (1/C) * K + eye(N);f = -ones(N, 1);Aeq = ones(1, N);beq = 1;lb = zeros(N, 1);ub = inf(N, 1);% 求解二次规划options = optimoptions('quadprog', 'Display', 'off');alpha = quadprog(H, f, [], [], Aeq, beq, lb, ub, [], options);% 提取支持向量svIdx = alpha > tol & alpha < 1 - tol;supportVectors = X(svIdx, :);alpha_sv = alpha(svIdx);% 计算偏置项bb = mean(1 - sum((alpha_sv .* (1 - alpha_sv)) .* K(svIdx, svIdx), 1));% 构建模型结构model.alpha = alpha_sv;model.supportVectors = supportVectors;model.kernelType = kernelType;model.gamma = gamma;model.b = b;% 生成决策网格[x1Grid, x2Grid] = meshgrid(linspace(min(X(:,1))-1, max(X(:,1))+1, 100), ...linspace(min(X(:,2))-1, max(X(:,2))+1, 100));decision = zeros(size(x1Grid));for i = 1:numel(x1Grid)x = [x1Grid(i), x2Grid(i)]';decision(i) = predictSVDD(x, model);end
endfunction K = computeKernel(X1, X2, type, gamma)% 核矩阵计算switch typecase 'linear'K = X1 * X2';case 'poly'K = (X1 * X2' + 1).^2;case 'rbf'dist = pdist2(X1, X2).^2;K = exp(-gamma * dist);end
endfunction y = predictSVDD(x, model)% 新样本预测y = 0;for i = 1:length(model.alpha)y = y + model.alpha(i) * kernelFunc(x, model.supportVectors(i,:), ...model.kernelType, model.gamma);endy = y + model.b;
endfunction K = kernelFunc(x1, x2, type, gamma)% 单样本核计算switch typecase 'linear'K = x1 * x2';case 'poly'K = (x1 * x2' + 1).^2;case 'rbf'K = exp(-gamma * norm(x1 - x2)^2);end
end

二、使用与可视化

%% 生成测试数据（半圆形分布）
theta = linspace(0, 2*pi, 100)';
X_normal = [cos(theta), sin(theta)];  % 正常数据
X_outliers = 0.5*[ones(50,1), ones(50,1)];  % 异常数据
X = [X_normal; X_outliers];%% 训练SVDD模型
kernelType = 'rbf';  % 核函数类型
gamma = 10;          % RBF核参数
C = 1;               % 正则化参数
tol = 1e-3;          % 支持向量容忍度[model, decision] = mySVDD(X_normal, kernelType, gamma, C, tol);%% 可视化结果
figure;
hold on;
scatter(X_normal(:,1), X_normal(:,2), 'b.', 'MarkerSize', 10);  % 正常数据
scatter(X_outliers(:,1), X_outliers(:,2), 'r.', 'MarkerSize', 10);  % 异常数据
scatter(model.supportVectors(:,1), model.supportVectors(:,2), 'ko', 'LineWidth', 2);  % 支持向量
contour(squeeze(x1Grid), squeeze(x2Grid), reshape(decision, size(x1Grid)), [0 0], 'k', 'LineWidth', 2);  % 决策边界
title('SVDD异常检测结果');
legend('正常样本', '异常样本', '支持向量', '决策边界');
axis equal;
hold off;

三、扩展

多类分类扩展
通过"one-vs-one"策略实现多类分类：

function labels = svdd_multiclass(X, Y, kernelType, gamma, C)classes = unique(Y);numClasses = length(classes);labels = zeros(size(X,1), 1);for i = 1:numClasses% 构建二分类任务idx = Y == classes(i);model = mySVDD(X(idx,:), kernelType, gamma, C);% 计算决策值[~, decision] = predictSVDD(X, model);[~, maxIdx] = max(decision);labels(idx) = classes(maxIdx);end
end

异常评分机制
输出样本到决策边界的距离作为异常分数：

function scores = svdd_anomaly_score(X, model)decision = zeros(size(X,1), 1);for i = 1:size(X,1)decision(i) = predictSVDD(X(i,:), model);endscores = 1 - decision;  % 距离越近得分越高
end

四、优化

网格搜索调参
使用交叉验证优化参数组合：

best_score = inf;
for gamma = [0.1, 1, 10]for C = [0.1, 1, 10]model = mySVDD(X_train, 'rbf', gamma, C);score = evaluate_model(model, X_val);if score < best_scorebest_gamma = gamma;best_C = C;endend
end

增量学习实现
支持在线更新模型参数：

function model = svdd_incremental(model, X_new, Y_new)% 合并新旧数据X_combined = [model.supportVectors; X_new];Y_combined = [ones(size(model.supportVectors,1),1); Y_new];% 重新训练模型model = mySVDD(X_combined, model.kernelType, model.gamma, model.C);
end

参考代码 matlab实现的svdd算法 www.youwenfan.com/contentcsh/55138.html

五、应用

工业设备故障检测

% 加载振动传感器数据
data = load('vibration_data.mat');
X_train = data.normal;  % 正常工况数据
X_test = data.faulty;   % 故障数据% 训练模型
model = mySVDD(X_train, 'rbf', 20, 0.5);% 异常检测
scores = svdd_anomaly_score(X_test, model);
anomalies = find(scores > 0.8);

网络入侵检测

% 加载网络流量数据
data = load('network_traffic.mat');% 特征标准化
scaler = fitNormalizer(data.features);
X_normalized = transform(scaler, data.features);% 训练与测试
cv = cvpartition(size(X_normalized,1),'HoldOut',0.3);
model = mySVDD(X_normalized(training(cv),:), 'rbf', 5, 1);
accuracy = sum(predictSVDD(X_normalized(test(cv),:), model) > 0)/numel(test(cv));