Definition
The single dimension Gaussian Processes
()
we use can be written as
where
and
are the mean and covariance functions, respectively. In our case as set
out above, we have
$$\begin{equation}
\mu(t) \equiv 0 \\
k(t,t') = k(|t - t'|) = k(\Delta t)
\end{equation}$$
with the following choices available for the kernel
Matérn 3/2 covariance kernel (the default)
with
and
the length scale and magnitude, respectively, of the kernel.
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
the number of basis functions to use in the approximation, which we
calculate from the number of time points
to which the Gaussian Process is being applied (rounded up to give an
integer value), as is recommended[1].
and values of
given by
where
is a positive number termed boundary condition, and
are regression weights with standard normal prior
The function
is the spectral density relating to a particular covariance function
.
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
are the eigenfunctions of the Laplace operator,
with time rescaled linearly to be between -1 and 1,
Relevant priors are
with
additionally constrained to be between
and
,
and
calculated from given mean
and standard deviation
,
and default values (all of which can be changed by the user):