23-12-2025, 01:43 AM
(23-12-2025, 12:55 AM)ReneZ wrote: You are not allowed to view links. Register or Login to view.PCA is only used to find the largest dimensions in a multi-dimensional cloud of points. The result can be used for visualisation.
Consider for example two populations of points A and B. Point set A has a distribution in 3-space that looks like an ellipsoid aligned with the X, Y, and Z axis, with diameters proportional to 4, 2, and 1 along these axes, respectively. Point set B has a similar distribution, but displaced in Z and Y by +2 units each.
PCA should give three axes which are roughly u = X = (+1,0,0) , v = (0,+1,-1), and w = (0,+1,+1), with u having the largest eigenvalue. Projecting the points on the uv (or uw) plane will show the two sets as overlapping. But in fact they are well-separated by a horizontal plane. To see this fact, the best axes are Y and any axis orthogonal to it. Or (in this case) the v and w axes, skipping u.
The point is that the first 2 or 3 PCA axes are the directions along which the whole set has maximum extent. They are not necessarily the directions along which the centroids of the clusters of interest are maximally far apart, and do not necessarily include the direction(s) along which the clusters can be linearly separated.
All the best, stolfi.