Our research proposes a novel method for reducing the dimensionality of functional data, specifically for the case where the response is a scalar and the predictor is a random function. Our method utilizes distance covariance, and has several advantages over existing methods. Unlike current techniques which require restrictive assumptions such as linear conditional mean and constant covariance, our method has mild requirements on the predictor. Additionally, our method does not involve the use of the unbounded inverse of the covariance operator. The link function between the response and predictor can be arbitrary, and our proposed method maintains the advantage of being model-free, without the need to estimate the link function. Furthermore, our method is naturally suited for sparse longitudinal data. We utilize functional principal component analysis with truncation as a regularization mechanism in the development of our method. We provide justification for the validity of our proposed method, and establish statistical consistency of the estimator under certain regularization conditions. To demonstrate the effectiveness of our proposed method, we conduct simulation studies and real data analysis. The results show improved performance compared to existing methods.

翻译：本研究提出了一种针对函数型数据降维的新方法，特别适用于响应变量为标量而预测变量为随机函数的情形。该方法利用距离协方差，相较于现有技术具有多项优势。与当前需要线性条件均值与常数协方差等严格假设的技术不同，本方法对预测变量仅需较宽松的要求。此外，该方法无需涉及协方差算子的无界逆运算。响应变量与预测变量之间的连接函数可为任意形式，且所提方法保持无模型优势，无需估计该连接函数。进一步地，该方法天然适用于稀疏纵向数据。在方法开发过程中，我们采用带截断的函数型主成分分析作为正则化机制。我们论证了所提方法的有效性，并在特定正则化条件下建立了估计量的统计一致性。为验证方法的实用性，我们开展了模拟研究与真实数据分析，结果表明其性能优于现有方法。