论文标题
通过深神经网络估计功能数据的平均功能
Estimation of the Mean Function of Functional Data via Deep Neural Networks
论文作者
论文摘要
在这项工作中，我们提出了一种深层神经网络方法，以对功能数据进行非参数回归。所提出的估计量基于具有Relu激活函数的稀疏连接的深神经网络。通过正确选择网络体系结构，我们的估计器实现了经验规范中最佳的非参数收敛率。在某些情况下，例如三角多项式内核和足够大的采样频率，收敛速率甚至比根本$ n $ rate的速度更快。通过蒙特卡洛模拟研究，我们检查了所提出方法的有限样本性能。最后，提出的方法用于分析从阿尔茨海默氏病神经影像学计划数据库获得的阿尔茨海默氏病患者的正电子发射断层扫描图像。
In this work, we propose a deep neural network method to perform nonparametric regression for functional data. The proposed estimators are based on sparsely connected deep neural networks with ReLU activation function. By properly choosing network architecture, our estimator achieves the optimal nonparametric convergence rate in empirical norm. Under certain circumstances such as trigonometric polynomial kernel and a sufficiently large sampling frequency, the convergence rate is even faster than root-$n$ rate. Through Monte Carlo simulation studies we examine the finite-sample performance of the proposed method. Finally, the proposed method is applied to analyze positron emission tomography images of patients with Alzheimer disease obtained from the Alzheimer Disease Neuroimaging Initiative database.
