Model-based Smoothing with Integrated Wiener Processes and Overlapping Splines

In many applications that involve the inference of an unknown smooth function, the inference of its derivatives is also important. To make joint inferences of the function and its derivatives, a class of Gaussian processes called pth order Integrated Wiener’s Process (IWP), is considered. Methods for constructing a finite element (FEM) approximation of an IWP exist but only focus on the case p=2 and do not allow appropriate inference for derivatives. In this article, we propose an alternative FEM approximation with overlapping splines (O-spline). The O-spline approximation applies for any order p∈Z+, and provides consistent and efficient inference for all derivatives up to order p−1. It is shown both theoretically and empirically that the O-spline approximation converges to the 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, which is invariant to the order p and the knot placement. Finally, we demonstrate the practical use of the O-spline approximation through an analysis of COVID death rates where the inference of derivative has an important interpretation in terms of the course of the pandemic.

在诸多涉及未知光滑函数推断的应用场景中，对其导数的推断同样具有重要价值。为实现函数及其导数的联合推断，本文考虑一类名为p阶积分维纳过程（pth order Integrated Wiener’s Process, IWP）的高斯过程。现有针对该过程构建有限元（Finite Element Method, FEM）近似的方法，但此类方法仅针对p=2的场景，且无法对导数实现合理推断。本文提出一种基于重叠样条（overlapping splines, O-spline）的新型有限元近似方案。该重叠样条近似适用于任意正整数阶p∈Z+，可对直至p−1阶的所有导数实现一致且高效的推断。本文通过理论与实证分析证明，随着结点数量增加，重叠样条近似将收敛于积分维纳过程。此外，本文基于预测标准差的概念，提出了一种统一且可解释的平滑参数先验定义方式，该方式不受阶数p与结点布局的影响。最后，本文通过对新冠死亡率的案例分析，展示了重叠样条近似方法的实际应用价值——其中导数的推断可用于阐释疫情的发展态势，具备重要的解读意义。