Partial least squares regression

PLS1 is a widely used algorithm appropriate for the vector Y case. It estimates T as an orthonormal matrix. In pseudocode it is expressed below (capital letters are matrices, lower case letters are vectors if they are superscripted and scalars if they are subscripted):

 1 function PLS1(X, y, l)
 2     ${\displaystyle X^{(0)}\gets X}$
 3     ${\displaystyle w^{(0)}\gets X^{\mathrm {T} }y/||X^{\mathrm {T} }y||}$, an initial estimate of w.
 4     for ${\displaystyle k=0}$ to ${\displaystyle l-1}$
 5         ${\displaystyle t^{(k)}\gets X^{(k)}w^{(k)}}$
 6         ${\displaystyle t_{k}\gets {t^{(k)}}^{\mathrm {T} }t^{(k)}}$ (note this is a scalar)
 7         ${\displaystyle t^{(k)}\gets t^{(k)}/t_{k}}$
 8         ${\displaystyle p^{(k)}\gets {X^{(k)}}^{\mathrm {T} }t^{(k)}}$
 9         ${\displaystyle q_{k}\gets {y}^{\mathrm {T} }t^{(k)}}$ (note this is a scalar)
10         if ${\displaystyle q_{k}=0}$
11             ${\displaystyle l\gets k}$, break the for loop
12         if ${\displaystyle k<(l-1)}$
13             ${\displaystyle X^{(k+1)}\gets X^{(k)}-t_{k}t^{(k)}{p^{(k)}}^{\mathrm {T} }}$
14             ${\displaystyle w^{(k+1)}\gets {X^{(k+1)}}^{\mathrm {T} }y}$
15     end for
16     define W to be the matrix with columns ${\displaystyle w^{(0)},w^{(1)},...,w^{(l-1)}}$.
       Do the same to form the P matrix and q vector.
17     ${\displaystyle B\gets W{(P^{\mathrm {T} }W)}^{-1}q}$
18     ${\displaystyle B_{0}\gets q_{0}-{P^{(0)}}^{\mathrm {T} }B}$
19     return ${\displaystyle B,B_{0}}$

This form of the algorithm does not require centering of the input X and Y, as this is performed implicitly by the algorithm. This algorithm features 'deflation' of the matrix X (subtraction of ${\displaystyle t_{k}t^{(k)}{p^{(k)}}^{\mathrm {T} }}$), but deflation of the vector y is not performed, as it is not necessary (it can be proved that deflating y yields the same results as not deflating[8]). The user-supplied variable l is the limit on the number of latent factors in the regression; if it equals the rank of the matrix X, the algorithm will yield the least squares regression estimates for B and ${\displaystyle B_{0}}$