偏最小二乘回归

PLS1是一个${\displaystyle Y}$是向量时广泛使用的算法。它估计${\displaystyle T}$是一个正交矩阵。以下是伪代码（大写字母是矩阵，带上标的小写字母是向量，带下标的小写字母和单独的小写字母都是标量）：

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

这种形式的算法不需要输入${\displaystyle X}$和${\displaystyle Y}$定中心，因为算法隐式处理了。这个算法的特点是收缩于${\displaystyle X}$ （减去${\displaystyle t_{k}t^{(k)}{p^{(k)}}^{T}}$），但向量${\displaystyle y}$不收缩，因为没有必要（可以证明收缩${\displaystyle y}$和不收缩的结果是一样的）。用户提供的变量${\displaystyle l}$是回归中隐藏因子数量的限制；如果它等于矩阵${\displaystyle X}$的秩，算法将产生${\displaystyle B}$和${\displaystyle B_{0}}$的最小二乘回归估计。