We propose a method of sufficient dimension reduction for functional data using distance covariance. We consider the case where the response variable is a scalar but the predictor is a random function. Our method has several advantages. It requires very mild conditions on the predictor, unlike the existing methods require the restrictive linear conditional mean assumption and constant covariance assumption. It also does not involve the inverse of the covariance operator which is not bounded. The link function between the response and the predictor can be arbitrary and our method maintains the model free advantage without estimating the link function. Moreover, our method is naturally applicable to sparse longitudinal data. We use functional principal component analysis with truncation as the regularization mechanism in the development. The justification for validity of the proposed method is provided and under some regularization conditions, statistical consistency of our estimator is established. Simulation studies and real data analysis are also provided to demonstrate the performance of our method.

翻译：本文提出了一种利用距离协方差对函数型数据进行充分降维的方法。我们考虑响应变量为标量、预测变量为随机函数的情形。该方法具有多项优势：与现有方法需要严格的线性条件均值假设和常值协方差假设不同，本方法对预测变量仅需非常宽松的条件；同时无需使用有界性无法保证的协方差算子逆运算。响应变量与预测变量之间的连接函数可任意设定，本方法保持模型无先验优势且无需估计连接函数。此外，该方法天然适用于稀疏纵向数据。在技术实现中，我们采用带截断的函数型主成分分析作为正则化机制。本文证明了所提方法的有效性，并在某些正则化条件下建立了估计量的统计相合性。通过模拟研究和实际数据分析展示了本方法的性能。