标题： PLS收缩的几何学 显示英文标题

标题： The geometry of PLS shrinkages

摘要： PLS收缩的几何结构在此进行考虑。 首先，提供了一个收缩向量的显式公式。 在该表达式中，收缩因子被表示为仅依赖于数据矩阵的一组基本收缩的平均值。 另一方面，该平均值的权重是观测响应的多重线性函数。 这种表示方法允许表征可能的收缩集合，并识别出PLS估计量表现出高度非线性行为的极端情况。 在这些情况下，最近提出的度自由度（DoF）的测量方法，直接依赖于收缩，无法提供合理的值。 还表明，长期以来的猜想即PLS的自由度始终超过PLS方向的数量并不成立。 显示英文摘要

摘要： 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.