Analytical Solution to Partial Least Squares

Partial least squares (PLS) is a widely used multivariate statistical technique, which can be used in neuroimaging, process monitoring, economics, etc.. Because the standard PLS is trained by nonlinear iterative partial least squares (NIPALS) algorithm, which can only obtain the numerical solution rather than the analytical solution. Therefore, it is hard for the subsequent theoretical analysis in PLS and the iterative optimization process is computationally intensive. This paper proposes an analytical solution to take the place of NIPALS. This determination meaningfully contributes to future theoretical analysis of PLS. In addition, the analytical solution demonstrates that the selection of initial vector in NIPALS does not affect the PLS training results. Moreover, the analytical solution avoids the multiple iterative calculations, so the divergence of the calculation process will be avoided and the computation burden will be reduced significantly.