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.

本文提出一种将函数型数据（functional data）整合至深度神经网络的研究方法。该模型针对带有多个函数型协变量与标量协变量的标量响应场景进行定义。该方法的附带产出为一组动态函数型权重，可在模型优化过程中实现可视化。相较于传统神经网络，该可视化方式可显著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，涵盖新数据预测以及还原函数型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模拟实验得到验证。研究团队还基于主流深度学习库Keras开发了一款R语言工具包（FuncNN），以支持该方法的通用推广应用。补充文档、数据集及R代码均可在线获取。