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 (GP_t) 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

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 t_GP to which the Gaussian Process is being applied (rounded up to give an integer value), as is recommended^[1].

and values of λ_j given by

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

The function S_k(x) is the spectral density relating to a particular covariance function k. In the case of the Matérn 3/2 kernel (the default in EpiNow2) this is given by

and in the case of a squared exponential kernel by

The functions ϕ_j(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 ρ_min and ρ_max, μ_ρ and σ_ρ calculated from given mean m_ρ and standard deviation s_ρ, and default values (all of which can be changed by the user):