PLS2 is probably the most used algorithm to perform projection to latent structures regression in the case of multivariate response. However, several criticisms pointed to the theoretical limits of its original formulation, highlighting the need of a more robust foundation within the theory of regression analysis. The iterative deflation algorithm is here introduced as a starting point to obtain a family of regression methods, which includes PLS2, principal component regression (PCR), and elastic component regression (ECR), where different eigenvalue equations are used to calculate the weight vectors. Within this framework, an original portrait of PLS2 is drawn. The main mathematical properties useful to understand what PLS2 is and how PLS2 behaves are derived. A new regression method called iterative deflation algorithm‐based regression (IDAR) is introduced to describe the limit behaviour of PLS2, PCR, and ECR. The post‐transformation method is presented as a general property of the iterative deflation algorithm. Two data sets, one simulated and the other experimental, are investigated to illustrate the main properties of PLS2.

中文翻译：

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迭代紧缩算法、特征值方程和 PLS2

在多变量响应的情况下，PLS2 可能是最常用的算法，用于执行潜在结构回归的投影。然而，一些批评指出了其原始公式的理论局限性，突出了回归分析理论需要更强大的基础。这里引入迭代紧缩算法作为起点，得到一系列回归方法，包括PLS2、主成分回归（PCR）和弹性成分回归（ECR），其中使用不同的特征值方程来计算权重向量。在此框架内，绘制了 PLS2 的原始画像。导出了有助于理解什么是 PLS2 以及 PLS2 的行为方式的主要数学属性。引入了一种称为基于迭代通货紧缩算法的回归 (IDAR) 的新回归方法来描述 PLS2、PCR 和 ECR 的极限行为。后变换方法是作为迭代紧缩算法的一般特性提出的。研究了两个数据集，一个是模拟的，另一个是实验的，以说明 PLS2 的主要特性。