Abstract The paper contains a discussion about the null spaces associated to linear prediction models for the particular case of Partial Least Squares regression models. The discussion separately considers the two existing null spaces: the one in the input space related to the projection onto the latent space and the null space, coming from the projection space, corresponding to the mapping of the scores onto the predicted responses. The paper also explores the impact of such null spaces in the definition of the design space around some feasible solutions obtained by inverting the prediction model, via several cases with simulated and real data from the literature. The case-studies serve to illustrate the discussion and the need of considering points in the two null spaces, rather than just take into account the null space within the latent space. They also serve to show how to address the use of the resulting vectors in the design space to maintain the desired quality by modifying the tunable (maneuverable) process variables to compensate for variations due to some other feature variables not so easy to control.

中文翻译：

---

潜变量模型反演中的残差空间及其对给定质量特性的设计空间的影响

摘要 本文讨论了与偏最小二乘回归模型特定情况下的线性预测模型相关的零空间。讨论分别考虑了两个现有的零空间：输入空间中的一个与潜在空间的投影有关，而零空间来自投影空间，对应于分数到预测响应的映射。本文还通过几个案例，利用文献中的模拟和真实数据，探讨了这种零空间在通过反转预测模型获得的一些可行解的设计空间定义中的影响。案例研究用于说明讨论和考虑两个零空间中的点的必要性，而不仅仅是考虑潜在空间内的零空间。