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
.
The single dimension Gaussian Processes (GPt) we use can be written as
GP(μ(t),k(t,t′))
where μ(t) and k(t,t′) are the mean and covariance functions, respectively. In our case as set out above, we have
μ(t)≡0k(t,t′)=k(|t−t′|)=k(Δt)
with the following choices available for the kernel k
k(Δt)=α2(1+√3Δtl)exp(−√3Δtl)
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.
k(Δt)=α2exp(−12(Δt2)l2)
k(Δt)=α2exp(−Δt2l2)
k(Δt)=α(1+√5Δtl+53(Δtl)2)exp(−√5Δtl)
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.
GP(0,k(Δt))≈m∑j=1(Sk(√λj))12ϕj(t)βj
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].
m=btGP
and values of λj given by
λj=(jπ2L)2
where L is a positive number termed boundary condition, and βj are regression weights with standard normal prior
βj∼Normal(0,1)
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
Sk(x)=α22√πΓ(ν+1/2)(2ν)νΓ(ν)ρ2ν(2νρ2+ω2)−(ν+12)
For ν=3/2 (the default in
EpiNow2
) this simplifies to
Sk(ω)=α24(√3/ρ)3((√3/ρ)2+ω2)2=(2α(√3/ρ)32(√3/ρ)2+ω2)2
For ν=1/2 it is
Sk(ω)=α22ρ(1/ρ2+ω2)
and for ν=5/2 it is
Sk(ω)=α23(√5/ρ)52((√5/ρ)2+ω2)3
In the case of a squared exponential the spectral kernel density is given by
Sk(ω)=α2√2πρexp(−12ρ2ω2)
The functions ϕj(x) are the eigenfunctions of the Laplace operator,
ϕj(t)=1√Lsin(√λj(t∗+L))
with time rescaled linearly to be between -1 and 1,
t∗=t−12tGP12tGP
Relevant priors are
α∼Normal(μα,σα)ρ∼LogNormal(μρ,σρ)
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):
b=0.2L=1.5mρ=21sρ=7ρmin=0ρmax=60μα=0σα=0.01