Neural Networks for Scalar Input and Functional Output

TL;DR

This paper introduces a neural network approach for predicting functional responses from scalar predictors, effectively handling high-dimensional data and irregular sampling, with improved performance over traditional methods.

Contribution

The authors develop a neural network model that predicts functional responses from scalar inputs, incorporating flexible objective functions and smoothness penalties, advancing functional data regression techniques.

Findings

Outperforms traditional function-on-scalar regression models.

Handles both regular and irregular data sampling.

Scales better computationally with predictor dimension.

Abstract

The regression of a functional response on a set of scalar predictors can be a challenging task, especially if there is a large number of predictors, or the relationship between those predictors and the response is nonlinear. In this work, we propose a solution to this problem: a feed-forward neural network (NN) designed to predict a functional response using scalar inputs. First, we transform the functional response to a finite-dimensional representation and construct an NN that outputs this representation. Then, we propose to modify the output of an NN via the objective function and introduce different objective functions for network training. The proposed models are suited for both regularly and irregularly spaced data, and a roughness penalty can be further applied to control the smoothness of the predicted curve. The difficulty in implementing both those features lies in the definition of objective functions that can be back-propagated. In our experiments, we demonstrate that our model outperforms the conventional function-on-scalar regression model in multiple scenarios while computationally scaling better with the dimension of the predictors.