Model-Based Smoothing with Integrated Wiener Processes and Overlapping Splines
提出一种基于重叠样条的有限元近似方法，用于高阶集成维纳过程，实现对函数及其所有低阶导数的一致且高效的推断，并通过COVID死亡率分析展示其应用。
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.