The geometry of partial least squares
Aloke Phatak, Sijmen de Jong · Journal of Chemometrics · 1997
Our objective in this article is to clarify partial least squares (PLS) regression by illustrating the geometry of NIPALS and SIMPLS, two algorithms for carrying out PLS, in both object and variable space. We introduce the notion of the tangent rotation of a vector on an ellipsoid and show how it is intimately related to the power method of finding the eigenvalues and eigenvectors of a symmetric matrix. We also show that the PLS estimate of the vector of coefficients in the linear model turns out to be an oblique projection of the ordinary least squares estimate. With two simple building blocks—tangent rotations and orthogonal and oblique projections—it becomes possible to visualize precisely how PLS functions. © 1997 John Wiley & Sons, Ltd.