Gaussian Process implementation details

Overview

We make use of Gaussian Processes in several places in EpiNow2. For example, the default model for estimate_infections() uses a Gaussian Process to model the 1st order difference on the log scale of the reproduction number. This vignette describes the implementation details of the approximate Gaussian Process used in EpiNow2.

Definition

The single dimension Gaussian Processes (GPt) we use can be written as

where μ(t) and k(t, t′) are the mean and covariance functions, respectively. In our case as set out above, we have

with the following choices available for the kernel k

Matérn 3/2 covariance kernel (the default)

with l > 0 and α > 0 the length scale and magnitude, respectively, of the kernel. Note that here and later we use a slightly different definition of α compared to Riutort-Mayol et al.[1], where this is defined as our α2.

Squared exponential kernel

Ornstein-Uhlenbeck (Matérn 1/2) kernel

Matérn 5/2 covariance kernel

Hilbert space approximation

In order to make our models computationally tractable, we approximate the Gaussian Process using a Hilbert space approximation to the Gaussian Process[1], centered around mean zero.

with m the number of basis functions to use in the approximation, which we calculate from the number of time points tGP to which the Gaussian Process is being applied (rounded up to give an integer value), as is recommended[1].

and values of λj given by

where L is a positive number termed boundary condition, and βj are regression weights with standard normal prior

The function Sk(x) is the spectral density relating to a particular covariance function k. In the case of the Matérn kernel of order ν this is given by

For ν = 3/2 (the default in EpiNow2) this simplifies to

For ν = 1/2 it is

and for ν = 5/2 it is

In the case of a squared exponential the spectral kernel density is given by

The functions ϕj(x) are the eigenfunctions of the Laplace operator,

with time rescaled linearly to be between -1 and 1,

Relevant priors are

with ρ additionally constrained to be between ρmin and ρmax, μρ and σρ calculated from given mean mρ and standard deviation sρ, and default values (all of which can be changed by the user):

References

1. Riutort-Mayol, G., Bürkner, P.-C., Andersen, M. R., Solin, A., & Vehtari, A. (2020). Practical hilbert space approximate bayesian gaussian processes for probabilistic programming. https://arxiv.org/abs/2004.11408