Modern Non-Linear Function-on-Function Regression
This work addresses regression problems in functional data analysis, offering a novel method for capturing non-linear relationships, but it appears incremental as it builds on existing neural network techniques.
The authors tackled non-linear function-on-function regression for functional data by introducing neural network models with a continuous hidden layer, achieving effective modeling of complex relations between functional predictors and responses through simulation and real data examples.
We introduce a new class of non-linear function-on-function regression models for functional data using neural networks. We propose a framework using a hidden layer consisting of continuous neurons, called a continuous hidden layer, for functional response modeling and give two model fitting strategies, Functional Direct Neural Network (FDNN) and Functional Basis Neural Network (FBNN). Both are designed explicitly to exploit the structure inherent in functional data and capture the complex relations existing between the functional predictors and the functional response. We fit these models by deriving functional gradients and implement regularization techniques for more parsimonious results. We demonstrate the power and flexibility of our proposed method in handling complex functional models through extensive simulation studies as well as real data examples.