Deep Learning With Functional Inputs

We present a methodology for integrating functional data into deep neural networks. The model is defined for scalar responses with multiple functional and scalar covariates. A by-product of the method is a set of dynamic functional weights that can be visualized during the optimization process. This visualization leads to a greater interpretability of the relationship between the covariates and the response relative to conventional neural networks. The model is shown to perform well in a number of contexts including prediction of new data and recovery of the true underlying relationship between the functional covariate and scalar response; these results were confirmed through real data applications and simulation studies. An R package (FuncNN) has also been developed on top of Keras, a popular deep learning library—this allows for general use of the approach. A supplemental document, the data and R codes are available online.

本研究提出一种将功能数据（functional data）集成至深度神经网络的方法论。该模型针对带有多个功能协变量与标量协变量的标量响应场景进行定义。本方法的一项副产物为一组动态功能权重，可在优化过程中进行可视化展示。相较于传统神经网络，该可视化方式可提升协变量与响应之间关联关系的可解释性。实验表明，该模型在多种场景下均表现优异，包括新数据预测以及还原功能协变量与标量响应间的真实内在关联；上述结论已通过真实数据集应用与仿真实验得到验证。研究团队还基于热门深度学习库Keras开发了一款R语言工具包（FuncNN），以此实现该方法的通用化应用。补充文档、数据集及R语言代码均可在线获取。