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 ($\mathrm{GP}_t$) we use can be written as

$$\begin{equation} \mathcal{GP}(\mu(t), k(t, t')) \end{equation}$$

where $\mu(t)$ and $k(t,t')$ 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 $k$

Matérn 3/2 covariance kernel (the default)

$$\begin{equation} k(\Delta t) = \alpha \left( 1 + \frac{\sqrt{3} \Delta t}{l} \right) \exp \left( - \frac{\sqrt{3} \Delta t}{l}\right) \end{equation}$$

with $l>0$ and $\alpha > 0$ the length scale and magnitude, respectively, of the kernel.

Squared exponential kernel

$$\begin{equation} k(\Delta t) = \alpha \exp \left( - \frac{1}{2} \frac{(\Delta t^2)}{l^2} \right) \end{equation}$$

Ornstein-Uhlenbeck (Matérn 1/2) kernel

$$\begin{equation} k(\Delta t) = \alpha \exp{\left( - \frac{\Delta t}{2 l^2} \right)} \end{equation}$$

Matérn 5/2 covariance kernel

$$\begin{equation} k(\Delta t) = \alpha \left( 1 + \frac{\sqrt{5} \Delta t}{l} + \frac{5}{3} \left(\frac{\Delta t}{l} \right)^2 \right) \exp \left( - \frac{\sqrt{5} \Delta t}{l}\right) \end{equation}$$

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.

$$\begin{equation} \mathcal{GP}(0, k(\Delta t)) \approx \sum_{j=1}^m \left(S_k(\sqrt{\lambda_j}) \right)^\frac{1}{2} \phi_j(t) \beta_j \end{equation}$$

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

$$\begin{equation} m = b t_\mathrm{GP} \end{equation}$$

and values of $\lambda_j$ given by

$$\begin{equation} \lambda_j = \left( \frac{j \pi}{2 L} \right)^2 \end{equation}$$

where $L$ is a positive number termed boundary condition, and $\beta_{j}$ are regression weights with standard normal prior

$$\begin{equation} \beta_j \sim \mathcal{Normal}(0, 1) \end{equation}$$

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

$$\begin{equation} S_k(x) = 4 \alpha^2 \left( \frac{\sqrt{3}}{\rho}\right)^3 \left(\left( \frac{\sqrt{3}}{\rho} \right)^2 + w^2 \right)^{-2} \end{equation}$$

and in the case of a squared exponential kernel by

$$\begin{equation} S_k(x) = \alpha^2 \sqrt{2\pi} \rho \exp \left( -\frac{1}{2} \rho^2 w^2 \right) \end{equation}$$

The functions $\phi_{j}(x)$ are the eigenfunctions of the Laplace operator,

$$\begin{equation} \phi_j(t) = \frac{1}{\sqrt{L}} \sin\left(\sqrt{\lambda_j} (t^* + L)\right) \end{equation}$$

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

$$\begin{equation} t^* = \frac{t - \frac{1}{2}t_\mathrm{GP}}{\frac{1}{2}t_\mathrm{GP}} \end{equation}$$

Relevant priors are

$$\begin{align} \alpha &\sim \mathcal{Normal}(0, \sigma_{\alpha}) \\ \rho &\sim \mathcal{LogNormal} (\mu_\rho, \sigma_\rho)\\ \end{align}$$

with $\rho$ additionally constrained to be between $\rho_\mathrm{min}$ and $\rho_\mathrm{max}$, $\mu_{\rho}$ and $\sigma_\rho$ calculated from given mean $m_{\rho}$ and standard deviation $s_\rho$, and default values (all of which can be changed by the user):

$$\begin{align} b &= 0.2 \\ L &= 1.5 \\ m_\rho &= 21 \\ s_\rho &= 7 \\ \rho_\mathrm{min} &= 0\\ \rho_\mathrm{max} &= 60\\ \sigma_\alpha &= 0.05\\ \end{align}$$

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