Plot hotelling's confidence ellipse in PCA

Can anyone tell me how to draw an ellipse error on a pca plot. The princomp function gives us the distances from each point to the center of the group, but how can we use this information to draw the corresponding ellipse's?

[Matteo Mastropierro]

Have you found a way to do that?

Answer 1

[Aditya]

Hi Karthik,

To draw an error ellipse on a PCA plot, you can use the covariance matrix of the principal component scores to determine the shape and orientation of the ellipse.

Here is the sample code for the same:

% Sample data
data = randn(100, 3); % 100 observations with 3 variables

% Perform PCA
[coeff, score, latent] = pca(data);

% Calculate the covariance matrix of the scores
covariance = cov(score(:, 1:2)); % Use the first two principal components

% Calculate the mean of the scores
mean_score = mean(score(:, 1:2));

% Eigen decomposition of the covariance matrix
[eigenvec, eigenval] = eig(covariance);

% Calculate the ellipse parameters
chi_square_val = 2.4477; % For 95% confidence interval
theta = linspace(0, 2*pi, 100);
ellipse_x = chi_square_val * sqrt(eigenval(1,1)) * cos(theta);
ellipse_y = chi_square_val * sqrt(eigenval(2,2)) * sin(theta);

% Rotation matrix
R = [cos(atan2(eigenvec(2,1), eigenvec(1,1))), -sin(atan2(eigenvec(2,1), eigenvec(1,1)));
     sin(atan2(eigenvec(2,1), eigenvec(1,1))),  cos(atan2(eigenvec(2,1), eigenvec(1,1)))];

% Rotate the ellipse
ellipse = [ellipse_x; ellipse_y]' * R;

% Plot the PCA scores
figure;
scatter(score(:, 1), score(:, 2), 'b.');
hold on;

% Plot the error ellipse
plot(mean_score(1) + ellipse(:, 1), mean_score(2) + ellipse(:, 2), 'r-', 'LineWidth', 2);

% Add labels and title
xlabel('Principal Component 1');
ylabel('Principal Component 2');
title('PCA with Error Ellipse');
axis equal;
grid on;
hold off;