The geometrical structure of PLS shrinkages is here considered. Firstly, an explicit formula for the shrinkage vector is provided. In that expression, shrinkage factors are expressed a averages of a set of basic shrinkages that depend only on the data matrix. On the other hand, the weights of that average are multilinear functions of the observed responses. That representation allows to characterise the set of possible shrinkages and identify extreme situations where the PLS estimator has an highly nonlinear behaviour. In these situations, recently proposed measures for the degrees of freedom (DoF), that directly depend on the shrinkages, fail to provide reasonable values. It is also shown that the longstanding conjecture that the DoFs of PLS always exceeds the number PLS directions does not hold.

翻译：本文考察了PLS收缩的几何结构。首先，给出了收缩向量的显式表达式。在该表达式中，收缩因子被表示为一组仅依赖于数据矩阵的基本收缩的加权平均。另一方面，该平均的权重是观测响应的多线性函数。该表示方法使得我们能够刻画可能收缩的集合，并识别出PLS估计量呈现高度非线性行为的极端情况。在这些情况下，最近提出的直接依赖于收缩的自由度度量方法无法提供合理的数值。研究还表明，关于PLS自由度始终超过PLS方向数量的长期猜想并不成立。