In many applications that involve the inference of an unknown smooth function, the inference of its derivatives will often be just as important as that of the function itself. To make joint inferences of the function and its derivatives, a class of Gaussian processes called $p^{\text{th}}$ order Integrated Wiener's Process (IWP), is considered. Methods for constructing a finite element (FEM) approximation of an IWP exist but have focused only on the order $p = 2$ case which does not allow appropriate inference for derivatives, and their computational feasibility relies on additional approximation to the FEM itself. In this article, we propose an alternative FEM approximation, called overlapping splines (O-spline), which pursues computational feasibility directly through the choice of test functions, and mirrors the construction of an IWP as the Ospline results from the multiple integrations of these same test functions. The O-spline approximation applies for any order $p \in \mathbb{Z}^+$, is computationally efficient and provides consistent inference for all derivatives up to order $p-1$. It is shown both theoretically, and empirically through simulation, that the O-spline approximation converges to the true IWP as the number of knots increases. We further provide a unified and interpretable way to define priors for the smoothing parameter based on the notion of predictive standard deviation (PSD), which is invariant to the order $p$ and the placement of the knot. Finally, we demonstrate the practical use of the O-spline approximation through simulation studies and an analysis of COVID death rates where the inference is carried on both the function and its derivatives where the latter has an important interpretation in terms of the course of the pandemic.

翻译：在许多涉及未知光滑函数推断的应用中，对其导数的推断往往与函数本身同样重要。为联合推断函数及其导数，本文考虑一类称为$p$阶集成维纳过程（IWP）的高斯过程。现有构建IWP有限元（FEM）近似的方法仅适用于$p=2$阶情形，这无法支持适当的导数推断，且其计算可行性依赖于对FEM本身的额外近似。本文提出一种替代性有限元近似方法——重叠样条（O-spline），该方法通过直接选择检验函数实现计算可行性，其构造方式与IWP形成镜像关系：O-spline正是这些检验函数多次积分的结果。该近似适用于任意阶数$p \in \mathbb{Z}^+$，计算效率高，且能对直至$p-1$阶的所有导数提供一致推断。理论分析与模拟实验均表明，随着节点数量增加，O-spline近似将收敛至真实IWP。我们进一步基于预测标准差（PSD）概念提出一种统一且可解释的先验设定方法，该方法对阶数$p$与节点位置均具有不变性。最后，通过模拟研究及新冠疫情死亡率分析（其中函数及其导数推断对理解疫情演变具有重要意义），我们验证了O-spline近似方法的应用价值。