IntegratedGPs.jl

We provide IntegratedGPs.jl, a library for performing inference with Integrated Gaussian Processes. An "integrated GP" is a Gaussian Process which is the integral of another GP. This library provides functions for evaluating the kernel of the integrated GP in terms of the base GP. In particular, we provide closed-form analytic integrated kernels for the Matérn, Squared Exponential, and Rational Quadratic kernels.

Installation

This library is currently unregistered. It can be included in a Julia project through:

(@v1.12) pkg> add https://github.com/THargreaves/IntegratedGPs.jl.git

or by cloning the repo onto your file system (see Reproducibility).

Usage

Example Julia scripts can be found in scripts/. A minimal example is shown below:

using IntegratedGPs
using LinearAlgebra
using Plots

ν = 1.5     # Matern smoothness parameter
ρ = 0.8     # Lengthscale
σk = 1.2    # Process standard deviation

gp = constructmatern(ν, ρ, σk^2)    # Construct a Matern GP with optimisations for the given ν
igp = integrate(gp)                 # Integrate the GP efficiently


N = 100
K = [kernel(igp, float(i), float(j)) for i in 1:N, j in 1:N] # Construct a covariance matrix
K = 0.5 * (K + K')

proc = cholesky(K).L * randn(N)     # Draw an iGP sample

plot(proc)

Referencing

Please cite the following paper if you use this library:

R. J. McDougall, T. Hargreaves and S. J. Godsill, "Exact Integration of Stationary Gaussian Process Kernels," in IEEE Open Journal of Signal Processing, vol. 7, pp. 257-265, 2026, doi: 10.1109/OJSP.2026.3656062

Reproducibility

The results from the paper listed above can be reproduced by running the file scripts/ojsp.jl. This script will download the multi-object tracking dataset: TeamTrack and run all analysis scripts used in the paper. The figures will be produced in scripts/figs/.

The most significant test is performed by tracking football player trajectories in the TeamTrack dataset. It is shown that the Matérn kernel is effective for highly agile trajectories:

A clean reproducibility test can be performed by running the following instructions in the terminal:

Documents> git clone https://github.com/THargreaves/IntegratedGPs.jl.git
Documents> cd IntegratedGPs.jl
Documents/IntegratedGPs.jl> julia

followed by the following instructions in the Julia REPL:

(@v1.12) pkg> activate IntegratedGPs
(IntegratedGPs) pkg> instantiate
(IntegratedGPs) pkg> activate scripts
(scripts) pkg> instantiate
julia> include("scripts/ojsp.jl")

Note that the package mode is entered by pressing the ] key, and the backspace returning to the standard REPL.