We propose a novel approach to nonlinear functional regression, called the Mapping-to-Parameter function model, which addresses complex and nonlinear functional regression problems in parameter space by employing any supervised learning technique. Central to this model is the mapping of function data from an infinite-dimensional function space to a finite-dimensional parameter space. This is accomplished by concurrently approximating multiple functions with a common set of B-spline basis functions by any chosen order, with their knot distribution determined by the Iterative Local Placement Algorithm, a newly proposed free knot placement algorithm. In contrast to the conventional equidistant knot placement strategy that uniformly distributes knot locations based on a predefined number of knots, our proposed algorithms determine knot location according to the local complexity of the input or output functions. The performance of our knot placement algorithms is shown to be robust in both single-function approximation and multiple-function approximation contexts. Furthermore, the effectiveness and advantage of the proposed prediction model in handling both function-on-scalar regression and function-on-function regression problems are demonstrated through several real data applications, in comparison with four groups of state-of-the-art methods.

翻译：本文提出了一种新的非线性函数回归方法——映射到参数函数模型，该模型通过采用任意监督学习技术，在参数空间中解决复杂非线性函数回归问题。该模型的核心在于将函数数据从无限维函数空间映射到有限维参数空间。这通过使用任意阶数的共同B样条基函数同时逼近多个函数实现，其节点分布由新提出的自由节点定位算法——迭代局部定位算法确定。与基于预设节点数量均匀分布节点位置的传统等距节点定位策略不同，我们提出的算法根据输入或输出函数的局部复杂性确定节点位置。在单函数逼近和多函数逼近两种情境中，我们的节点定位算法展现出鲁棒性。此外，通过与四组最先进方法的对比，多个实际数据应用证明了所提预测模型在处理标量函数回归和函数对函数回归问题时的有效性和优势。