In practice, data often display heteroscedasticity, making quantile regression (QR) a more appropriate methodology. Modeling the data, while maintaining a flexible nonparametric fitting, requires smoothing over a high-dimensional space which might not be feasible when the number of the predictor variables is large. This problem makes necessary the use of dimension reduction techniques for conditional quantiles, which focus on extracting linear combinations of the predictor variables without losing any information about the conditional quantile. However, nonlinear features can achieve greater dimension reduction. We, therefore, present the first nonlinear extension of the linear algorithm for estimating the central quantile subspace (CQS) using kernel data. First, we describe the feature CQS within the framework of reproducing kernel Hilbert space, and second, we illustrate its performance through simulation examples and real data applications. Specifically, we emphasize on visualizing various aspects of the data structure using the first two feature extractors, and we highlight the ability to combine the proposed algorithm with classification and regression linear algorithms. The results show that the feature CQS is an effective kernel tool for performing nonlinear dimension reduction for conditional quantiles.

在实践中，数据通常显示出异方差，使得​​分位数回归（QR）成为更合适的方法。在对数据进行建模的同时，要保持灵活的非参数拟合，需要在高维空间上进行平滑处理，而当预测变量的数量很大时，这可能是不可行的。此问题使得必须将降维技术用于条件分位数，该技术着重于提取预测变量的线性组合而不会丢失有关条件分位数的任何信息。但是，非线性特征可以实现更大的尺寸缩减。因此，我们提出了第一个非线性扩展使用内核数据估计中央分位数子空间（CQS）的线性算法的原理图。首先，我们在再现内核希尔伯特空间的框架内描述了功能CQS，其次，我们通过仿真示例和实际数据应用程序说明了其性能。具体来说，我们强调使用前两个特征提取器对数据结构的各个方面进行可视化，并且强调将拟议算法与分类和回归线性算法相结合的能力。结果表明，特征CQS是执行条件分位数非线性降维的有效内核工具。