Splinets——通过泰勒展开构造的样条、其支撑集与正交基
Splinets -- splines through the Taylor expansion, their support sets and orthogonal bases
AI总结：
本文实现了基于splinets正交基和支撑集的样条新表示方法，利用泰勒展开和二进正交化算法，为函数数据分析提供了高效且稳定的计算工具，尤其适用于稀疏数据场景。
摘要：
本文实现了一种新的样条表示方法，旨在提高函数数据分析的效率。该效率通过两个新颖特性实现：使用最近引入的正交样条基（即所谓的splinets），以及在所提出的样条对象表示中考虑样条支撑集。最近引入的正交splinets通过对$B$-splines进行二进正交化来评估。该包以Splinets对象为核心，表示一个样条集合。它将样条视为数学函数，并包含关于支撑集和节点处导数值的信息，这些信息唯一地定义了这些函数。样条的代数和微积分利用支撑集内节点处的泰勒展开。实现了多种$B$-splines正交化程序，包括推荐的二进方法（生成splinets）。该方法基于二进算法，也可视为对角化带状矩阵的高效方法。$B$-splines在支撑集方面的局部性在很大程度上被相应的splinet保留。这与利用支撑局部性的算法一起，为函数数据分析提供了有价值的计算工具。当数据稀疏性起重要作用时，其优势尤为明显。提供了多种诊断工具以维持计算的稳定性。最后，实现了到样条空间的投影操作，以促进函数数据分析。展示了使用包中工具进行简单函数数据分析的示例。该包的功能扩展到分段多项式函数，尽管样条是其重点。
英文摘要：
A new representation of splines that targets efficiency in the analysis of functional data is implemented. The efficiency is achieved through two novel features: using the recently introduced orthonormal spline bases, the so-called {\it splinets} and accounting for the spline support sets in the proposed spline object representation. The recently-introduced orthogonal splinets are evaluated by {\it dyadic orthogonalization} of the $B$-splines. The package is built around the {\it Splinets}-object that represents a collection of splines. It treats splines as mathematical functions and contains information about the support sets and the values of the derivatives at the knots that uniquely define these functions. Algebra and calculus of splines utilize the Taylor expansions at the knots within the support sets. Several orthonormalization procedures of the $B$-splines are implemented including the recommended dyadic method leading to the splinets. The method bases on a dyadic algorithm that can be also viewed as the efficient method of diagonalizing a band matrix. The locality of the $B$-splines in terms of the support sets is, to a great extend, preserved in the corresponding splinet. This together with implemented algorithms utilizing locality of the supports provides a valuable computational tool for functional data analysis. The benefits are particularly evident when the sparsity in the data plays an important role. Various diagnostic tools are provided allowing to maintain the stability of the computations. Finally, the projection operation to the space of splines is implemented that facilitates functional data analysis. An example of a simple functional analysis of the data using the tools in the package is presented. The functionality of the package extends beyond the splines to piecewise polynomial functions, although the splines are its focus.
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