Calculating Principal component analysis (PCA), step by step using a simple dataset.

In this video, I explain what a PCA is and how to calculate it step by step, both manually and using Matlab (no prior knowledge of the language is necessary). I will use a simple and intuitive 3D dataset, which I will use PCA to transform into a 2D dataset. This example is suitable for biologists and non-biologists alike. --------------------------- The Matlab code that I used: % Dataset A = double([1 0 0; 0 1 0; 0 0 1; 0 0 0]); % Calculating PCA manually, step-by-step [n,m] = size(A) AMean = mean(A) B = (A-repmat(AMean, [n 1])) C = (B'*B)./(size(B,1)-1) [V_eigenvectors D_eigenvalues] = eig(C) round(V_eigenvectors,2) round(D_eigenvalues,2) % Using matlab's function to calculate PCA [coeff1, SCORE1, V1] = pca(A, 'Algorithm','eig') % Two vecotrs are siad to be "mutually orthogonal" if the dot product of % any pair of distinct vectors in the set is 0 V_eigenvectors(:,1)'*V_eigenvectors(:2) V_eigenvectors(:,1)'*V_eigenvectors(:3) V_eigenvectors(:,2)'*V_eigenvectors(:3) % plotting the eigenvectors e1 = [zeros(size(V_eigenvectors(:,1))), V_eigenvectors(:,1)]' e1 = e1/norm(e1) e2 = [zeros(size(V_eigenvectors(:,2))), V_eigenvectors(:,2)]' e2 = e2/norm(e2) e3 = [zeros(size(V_eigenvectors(:,3))), V_eigenvectors(:,3)]' e3 = e3/norm(e3) plot3(e1(:,1), e1(:,2), e1(:,3), 'k', 'LineWidth',2) hold on; plot3(e2(:,1), e2(:,2), e2(:,3), 'b', 'LineWidth',2) plot3(e3(:,1), e3(:,2), e3(:,3), 'g', 'LineWidth',2) --------------------------- This video supports the paper: https://www.nature.com/articles/s41598-022-14395-4