Model-based Smoothing with Integrated Wiener Processes and Overlapping Splines

Ziang Zhang, Alex Stringer, Patrick Brown, Jamie Stafford

Abstract:

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 $p^{\text{th}}$ 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 $p \in \mathbb{Z}^+$, 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.