Documentation of alpha

R/opts.R

@@ -454,12 +454,12 @@ backcalc_opts <- function(prior = c("reports", "none", "infections"),
 #' this is smaller.
 #'
 #' @param alpha_mean Numeric, defaults to 0. The mean of the magnitude parameter
-#' of the Gaussian process kernel. Should be approximately the expected variance
-#' of the logged Rt.
+#' of the Gaussian process kernel. Should be approximately the expected standard
+#' deviation of the logged Rt.
 #'
-#' @param alpha_sd Numeric, defaults to 0.01. The standard deviation of the
-#' magnitude parameter of the Gaussian process kernel. Should be approximately
-#' the expected standard deviation of the logged Rt.
+#' @param alpha_sd Numeric, defaults to 0.1. The standard deviation of the
+#' magnitude parameter of the Gaussian process kernel. Can be tuned to adjust
+#' the uncertainty about the expected standard deviation of the logged Rt.
 #'
 #' @param kernel Character string, the type of kernel required. Currently
 #' supporting the Matern kernel ("matern"), squared exponential kernel ("se"),
@@ -508,7 +508,7 @@ gp_opts <- function(basis_prop = 0.2,
                     ls_min = 0,
                     ls_max = 60,
                     alpha_mean = 0,
-                    alpha_sd = 0.01,
+                    alpha_sd = 0.1,
                     kernel = c("matern", "se", "ou", "periodic"),
                     matern_order = 3 / 2,
                     matern_type,

tests/testthat/test-create_gp_data.R

@@ -11,7 +11,7 @@ test_that("create_gp_data returns correct default values when GP is disabled", {
   expect_equal(gp_data$ls_sdlog, convert_to_logsd(21, 7))
   expect_equal(gp_data$ls_min, 0)
   expect_equal(gp_data$ls_max, 3.54, tolerance = 0.01)
-  expect_equal(gp_data$alpha_sd, 0.01)
+  expect_equal(gp_data$alpha_sd, 0.1)
   expect_equal(gp_data$gp_type, 2)  # Default to Matern
   expect_equal(gp_data$nu, 3 / 2)
   expect_equal(gp_data$w0, 1.0)

tests/testthat/test-gp_opts.R

@@ -6,7 +6,7 @@ test_that("gp_opts returns correct default values", {
   expect_equal(gp$ls_sd, 7)
   expect_equal(gp$ls_min, 0)
   expect_equal(gp$ls_max, 60)
-  expect_equal(gp$alpha_sd, 0.01)
+  expect_equal(gp$alpha_sd, 0.1)
   expect_equal(gp$kernel, "matern")
   expect_equal(gp$matern_order, 3 / 2)
   expect_equal(gp$w0, 1.0)

vignettes/gaussian_process_implementation_details.Rmd

@@ -44,21 +44,22 @@ 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)
+k(\Delta t) = \alpha^2 \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.
+Note that here and later we use a slightly different definition of $\alpha$ compared to Riutort-Mayol et al. [@approxGP], where this is defined as our $\alpha^2$.
 
 ## Squared exponential kernel
 
 \begin{equation}
-k(\Delta t) = \alpha \exp \left( - \frac{1}{2} \frac{(\Delta t^2)}{l^2} \right)
+k(\Delta t) = \alpha^2 \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)}
+k(\Delta t) = \alpha^2 \exp{\left( - \frac{\Delta t}{2 l^2}  \right)}
 \end{equation}
 
 ## Matérn 5/2 covariance kernel
@@ -120,7 +121,7 @@ t^* = \frac{t - \frac{1}{2}t_\mathrm{GP}}{\frac{1}{2}t_\mathrm{GP}}
 Relevant priors are
 
 \begin{align}
-\alpha &\sim \mathcal{Normal}(0, \sigma_{\alpha}) \\
+\alpha &\sim \mathcal{Normal}(\mu_\alpha, \sigma_{\alpha}) \\
 \rho   &\sim \mathcal{LogNormal} (\mu_\rho, \sigma_\rho)\\
 \end{align}
 
@@ -133,7 +134,8 @@ m_\rho &= 21 \\
 s_\rho &= 7 \\
 \rho_\mathrm{min} &= 0\\
 \rho_\mathrm{max} &= 60\\
-\sigma_\alpha &= 0.05\\
+\mu_\alpha &= 0\\
+\sigma_\alpha &= 0.1
 \end{align}
 
 # References