如何在 Matlab 中绘制 PCA 后线性 SVM 的决策边界

Question

我已经在一个大型数据集上进行了线性SVM，但是为了减少维度的数量，我执行了PCA，而不是对组件分数的子集(前650个组件解释了99.5%的方差(进行了SVM。现在，我想使用在 PCA 空间中创建的 SVM 的 beta 权重和偏差绘制原始变量空间中的决策边界。但是我不知道如何将 SVM 中的偏置项投影到原始变量空间中。我用费舍尔虹膜数据写了一个演示来说明：

clear; clc; close all
% load data
load fisheriris
inds = ~strcmp(species,'setosa');
X = meas(inds,3:4);
Y = species(inds);
mu = mean(X)
% perform the PCA
[eigenvectors, scores] = pca(X);
% train the svm
SVMModel = fitcsvm(scores,Y);
% plot the result
figure(1)
gscatter(scores(:,1),scores(:,2),Y,'rgb','osd')
title('PCA space')
% now plot the decision boundary
betas = SVMModel.Beta; 
m = -betas(1)/betas(2); % my gradient
b = -SVMModel.Bias;     % my y-intercept
f = @(x) m.*x + b;      % my linear equation
hold on
fplot(f,'k')
hold off
axis equal
xlim([-1.5 2.5])
ylim([-2 2])
% inverse transform the PCA
Xhat = scores * eigenvectors';
Xhat = bsxfun(@plus, Xhat, mu);
% plot the result
figure(2)
hold on
gscatter(Xhat(:,1),Xhat(:,2),Y,'rgb','osd')
% and the decision boundary
betaHat = betas' * eigenvectors';
mHat = -betaHat(1)/betaHat(2);
bHat = b * eigenvectors';
bHat = bHat + mu;    % I know I have to add mu somewhere...
bHat = bHat/betaHat(2);
bHat = sum(sum(bHat)); % sum to reduce the matrix to a single value
% the correct value of bHat should be 6.3962
f = @(x) mHat.*x + bHat;
fplot(f,'k')
hold off
axis equal
title('Recovered feature space')
xlim([3 7])
ylim([0 4])

关于我如何错误地计算 bHat 的任何指导将不胜感激。

Answer 1

以防万一其他人遇到这个问题，解决方案是偏差项可用于查找 y 截距，b = -SVMModel.Bias/betas(2) 。而y截距只是空间[0 b]中的另一个点，可以通过PCA逆变换来恢复/取消旋转。然后，这个新点可用于求解线性方程 y = mx + b(即 b = y - mx(。所以代码应该是：

% and the decision boundary 
betaHat = betas' * eigenvectors'; 
mHat = -betaHat(1)/betaHat(2);
yint = b/betas(2);                   % y-intercept in PCA space
yintHat = [0 b] * eigenvectors';     % recover in original space
yintHat = yintHat + mu;    
bHat = yintHat(2) - mHat*yintHat(1); % solve the linear equation
% the correct value of bHat is now 6.3962