Distribution of Gaussian Process Arc Lengths

TL;DR

This paper extends the theoretical understanding of Gaussian Processes by deriving the moments of their arc length in multiple dimensions, which is crucial for trajectory modeling and path analysis.

Contribution

It introduces the first derivation of the moments of the arc length for multi-dimensional Gaussian Processes, filling a significant gap in the literature.

Findings

Derived the moments of the arc length for multi-dimensional GPs.

Proposed a new method for calculating the mean of a 1D GP over an interval.

Numerical simulations confirm the theoretical results.

Abstract

We present the first treatment of the arc length of the Gaussian Process (GP) with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical consideration of higher dimensional problems. We fill the gap in the existing literature by deriving the moments of the arc length for a stationary GP with multiple output dimensions. A new method is used to derive the mean of a one-dimensional GP over a finite interval, by considering the distribution of the arc length integrand. This technique is used to derive an approximate distribution over the arc length of a vector valued GP in $\mathbb{R}^n$ by moment matching the distribution. Numerical simulations confirm our theoretical derivations.