Hi, to understand if you have successfully reduced the dimensionality of your multivariate time series using PCA, you can analyze the eigenvalues obtained from the covariance matrix. Here's how you can proceed:
- Plotting the eigenvalues:
eigenvalues = eig(covariance_matrix);
explained_variance = eigenvalues / sum(eigenvalues); % Normalize eigenvalues to calculate explained variance
cumulative_variance = cumsum(explained_variance);
% Scree plot
figure;
plot(1:length(eigenvalues), eigenvalues, 'o-');
xlabel('Principal Component');
ylabel('Eigenvalue');
title('Scree Plot');
% Cumulative variance plot
figure;
plot(1:length(cumulative_variance), cumulative_variance, 'o-');
xlabel('Number of Principal Components');
ylabel('Cumulative Explained Variance');
title('Cumulative Variance Plot');
Plotting the trajectory in 2 or 3 dimensions:
% Select the desired number of principal components for projection
num_components = 2; % Or 3 for 3D visualization
projection_matrix = eigenvectors(:, 1:num_components);
% Project the data onto the lower-dimensional space
projected_data = original_data * projection_matrix;
% Plot the trajectory in 2D or 3D
figure;
if num_components == 2
plot(projected_data(:, 1), projected_data(:, 2), 'o-');
xlabel('Principal Component 1');
ylabel('Principal Component 2');
title('Population Trajectory in 2D');
elseif num_components == 3
plot3(projected_data(:, 1), projected_data(:, 2), projected_data(:, 3), 'o-');
xlabel('Principal Component 1');
ylabel('Principal Component 2');
zlabel('Principal Component 3');
title('Population Trajectory in 3D');
end
