Every point in this cloud lives in a 3D feature space, and its three raw axes (x, y, z) are correlated with each other — knowing one tells you something about the others. Principal Component Analysis finds a new, rotated set of axes — the principal components — ranked so that PC1 points along the direction of maximum spread in the data, PC2 the next-most-spread direction orthogonal to PC1, and so on.
Points shown in brighter rose have the most "information" being thrown away by the current reduction — their true position sits far from the kept subspace. That per-point residual is exactly what PCA minimizes in aggregate when it chooses the top components: no other choice of that many orthogonal directions captures more total variance.
A correlated 3D data cloud with its principal axes computed live in the browser via eigendecomposition of the covariance matrix, letting you drag a slider and watch every point collapse onto a lower-dimensional summary.
PCA rotates the data onto new, uncorrelated axes ranked by how much variance each one explains. Keeping only the top axes gives the best possible lower-dimensional approximation of the data in a least-squares sense.
Choose how many components to keep, then drag the reduction blend slider to morph points from their true position onto that subspace. Adjust axis correlation and variance flatness to see how the principal axes and the variance they capture change.
Points that turn brighter rose as you increase the blend are the ones losing the most information — their true position sits farthest from the kept subspace, which is exactly the quantity PCA minimizes in aggregate.