Functional data such as curves and surfaces have become more and more common with modern technological advancements. The use of functional predictors remains challenging due to its inherent infinite-dimensionality. The common practice is to project functional data into a finite dimensional space. The popular partial least square (PLS) method has been well studied for the functional linear model [1]. As an alternative, quantile regression provides a robust and more comprehensive picture of the conditional distribution of a response when it is non-normal, heavy-tailed, or contaminated by outliers. While partial quantile regression (PQR) was proposed in [2], no theoretical guarantees were provided due to the iterative nature of the algorithm and the non-smoothness of quantile loss function. To address these issues, we propose an alternative PQR (APQR) formulation with guaranteed convergence. This novel formulation motivates new theories and allows us to establish asymptotic properties. Numerical studies on a benchmark dataset show the superiority of our new approach. We also apply our novel method to a functional magnetic resonance imaging (fMRI) data to predict attention deficit hyperactivity disorder (ADHD) and a diffusion tensor imaging (DTI) dataset to predict Alzheimer's disease (AD).

翻译：随着现代技术进步，曲线和曲面等函数型数据日益普遍。由于函数型预测变量固有的无限维特性，其应用仍面临挑战。常规做法是将函数型数据投影到有限维空间。偏最小二乘法已在函数型线性模型中得到深入研究[1]。作为替代方法，分位数回归能够在响应变量呈非正态、厚尾或受异常值污染时，提供更稳健且更全面的条件分布信息。尽管文献[2]提出了偏分位数回归，但由于算法的迭代特性及分位数损失函数的非光滑性，该工作未给出理论保证。针对这些问题，我们提出了一种保证收敛的替代性偏分位数回归公式。这一新公式推动了新理论发展，使我们能够建立渐近性质。基于基准数据集的数值实验证明了我们新方法的优越性。我们还将该方法应用于功能磁共振成像数据以预测注意力缺陷多动障碍，以及弥散张量成像数据集以预测阿尔茨海默病。