Algorithm 1.

K-Factor Regularized PLS

Center the columns of X and Y. Let M̂(1) = XT Y. For k = 1 … K: Initialize uk and vk to the first left and right singular vectors of M̂(k). Repeat until convergence: Set ${\mathbf{u}}_k=\frac{{({\widehat{\mathbf{M}}}^{(k)})}^{\mathrm{T}}{\mathbf{v}}_k}{{\Vert {({\widehat{\mathbf{M}}}^{(k)})}^{\mathrm{T}}{\mathbf{v}}_k\Vert }_2}$. Set ${\widehat{\mathbf{v}}}_k={\text{argmin}}_{{\mathbf{v}}_k^{\prime }}\left\{{\Vert {\widehat{\mathbf{M}}}^{(k)}{\mathbf{u}}_k-{\mathbf{v}}_k^{\prime }\Vert }_2^2-{\lambda }_{k}P({\mathbf{v}}_k^{\prime })\right\}$. Set vk = v̂k/||v̂k||2 if ||v̂k||2 > 0, and set vk = 0 and exit the algorithm otherwise. RPLS Factor: zk = X vk. RPLS projection matrix: Set ${\mathbf{R}}^{(k)}=[\begin{array}{cc}{\mathbf{R}}^{(k-1)}& {\mathbf{X}}^{\mathrm{T}}{\mathbf{z}}_k/{\mathbf{z}}_k^{\mathrm{T}}{\mathbf{z}}_k\end{array}]$ and Pk = I − R(k) ((R(k))T R(k))−1(R(k))T. Orthogonalization Step: M̂(k+1) = Pk M̂(k) Return RPLS Factors z1 … zK and RPLS Loadings: v1 … vK.