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.

我们提出了一种新的非线性函数回归方法，称为映射到参数函数模型，通过使用任何监督学习技术解决参数空间中的复杂和非线性函数回归问题。该模型的核心是将函数数据从无限维函数空间映射到有限维参数空间，通过使用任意阶的B样条基函数以及由迭代局部放置算法确定的结节分布，同时逼近多个函数。与传统的等距结节放置策略不同，我们提出的算法根据输入或输出函数的局部复杂度确定结节位置。通过多个真实数据应用与四组最先进方法进行比较，我们展示了我们的结节放置算法在单函数逼近和多函数逼近环境中的稳健性以及所提出的预测模型在处理函数对标量回归和函数对函数回归问题中的有效性和优势。

基于映射到参数的非线性函数回归与新型B样条自由结点放置算法