Neural Networks for Scalar Input and Functional Output
This work addresses a challenging regression task for researchers and practitioners dealing with functional data, but it is incremental as it adapts existing neural network methods to a specific domain.
The authors tackled the problem of predicting functional responses from scalar predictors using a feed-forward neural network, demonstrating that their model outperforms conventional function-on-scalar regression in multiple scenarios and scales better computationally with predictor dimensions.
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.