论文标题
标量输入和功能输出的神经网络
Neural Networks for Scalar Input and Functional Output
论文作者
论文摘要
功能响应对一组标量预测变量的回归可能是一项具有挑战性的任务，尤其是在有大量预测因子的情况下，或者这些预测因子与响应之间的关系是非线性的。在这项工作中，我们提出了解决此问题的解决方案：一个旨在使用标量输入来预测功能响应的进发神经网络（NN）。首先，我们将功能响应转换为有限维表示，并构建输出此表示形式的NN。然后，我们建议通过目标函数修改NN的输出，并引入网络培训的不同目标函数。所提出的模型适用于定期和不规则间隔的数据，并且可以进一步应用粗糙度的惩罚来控制预测曲线的平滑度。实现这两个功能的困难在于可以反向传播的目标函数的定义。在我们的实验中，我们证明了我们的模型在多种情况下优于传统的量表回归模型，而计算尺度则可以通过预测变量的尺寸来更好地缩放。
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.
