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#' WP method background function WP_known
#'
#' This is a background/internal function called by \code{WP}. It computes the
#' maximum likelihood estimator of R0 assuming that the serial distribution is
#' known and finite discrete.
#'
#' @param NT Vector of case counts.
#' @param p Discretized version of the serial distribution.
#'
#' @return The function returns the maximum likelihood estimator of R0.
#'
#' @keywords internal
WP_known <- function(NT, p) {
k <- length(p)
TT <- length(NT) - 1
mu_t <- rep(0, TT)
for (i in 1:TT) {
Nt <- NT[i:max(1, i - k + 1)]
mu_t[i] <- sum(p[1:min(k, i)] * Nt)
}
Rhat <- sum(NT[-1]) / sum(mu_t)
return(Rhat)
}
#' WP method background function WP_unknown
#'
#' This is a background/internal function called by \code{WP}. It computes the
#' maximum likelihood estimator of R0 assuming that the serial distribution is
#' unknown but comes from a discretized gamma distribution. The function then
#' implements a simple grid search algorithm to obtain the maximum likelihood
#' estimator of R0 as well as the gamma parameters.
#'
#' @param NT Vector of case counts.
#' @param B Length of grid for shape and scale (grid search parameter).
#' @param shape.max Maximum shape value (grid \code{search} parameter).
#' @param scale.max Maximum scale value (grid \code{search} parameter).
#' @param tol cutoff value for cumulative distribution function of the serial
#' distribution (defaults to 0.999).
#'
#' @return The function returns \code{Rhat}, the maximum likelihood estimator of
#' R0, as well as the maximum likelihood estimator of the discretized
#' serial distribution given by \code{p} (the probability mass function)
#' and \code{range.max} (the distribution has support on the integers
#' one to \code{range.max}). The function also returns \code{resLL} (all
#' values of the log-likelihood) at \code{shape} (grid for shape
#' parameter) and at \code{scale} (grid for scale parameter), as well as
#' \code{resR0} (the full vector of maximum likelihood estimators),
#' \code{JJ} (the locations for the likelihood for these), and \code{J0}
#' (the location for the maximum likelihood estimator \code{Rhat}). If
#' \code{JJ} and \code{J0} are not the same, this means that the maximum
#' likelihood estimator is not unique.
#'
#' @importFrom stats pgamma qgamma
#'
#' @keywords internal
WP_unknown <- function(NT, B = 100, shape.max = 10, scale.max = 10,
tol = 0.999) {
shape <- seq(0, shape.max, length.out = B + 1)
scale <- seq(0, scale.max, length.out = B + 1)
shape <- shape[-1]
scale <- scale[-1]
resLL <- matrix(0, B, B)
resR0 <- matrix(0, B, B)
for (i in 1:B)
for (j in 1:B) {
range.max <- ceiling(qgamma(tol, shape = shape[i], scale = scale[j]))
p <- diff(pgamma(0:range.max, shape = shape[i], scale = scale[j]))
p <- p / sum(p)
mle <- WP_known(NT, p)
resLL[i, j] <- computeLL(p, NT, mle)
resR0[i, j] <- mle
}
J0 <- which.max(resLL)
R0hat <- resR0[J0]
JJ <- which(resLL == resLL[J0], arr.ind = TRUE)
range.max <- ceiling(qgamma(tol, shape = shape[JJ[1]], scale = scale[JJ[2]]))
p <- diff(pgamma(0:range.max, shape = shape[JJ[1]], scale = scale[JJ[2]]))
p <- p / sum(p)
return(list(Rhat = R0hat, J0 = J0, ll = resLL, Rs = resR0, scale = scale,
shape = shape, JJ = JJ, p = p, range.max = range.max))
}
#' WP method background function computeLL
#'
#' This is a background/internal function called by \code{WP}. It computes the
#' log-likelihood.
#'
#' @param NT Vector of case counts.
#' @param p Discretized version of the serial distribution.
#' @param R0 Basic reproductive ratio.
#'
#' @return This function returns the log-likelihood at the input variables and
#' parameters.
#'
#' @keywords internal
computeLL <- function(p, NT, R0) {
k <- length(p)
TT <- length(NT) - 1
mu_t <- rep(0, TT)
for (i in 1:TT) {
Nt <- NT[i:max(1, i - k + 1)]
mu_t[i] <- sum(p[1:min(k, i)] * Nt)
}
mu_t <- R0 * mu_t
LL <- sum(NT[-1] * log(mu_t)) - sum(mu_t)
return(LL)
}
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