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#' 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 search parameter)
#' @param scale.max maximum scale value (grid 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.
#'
#' @export
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$R)
resR0[i,j] <- mle$R
}
# print(i)
}
J0 <- which.max(resLL)
R0hat <- resR0[J0]
JJ <- which(resLL==resLL[J0], arr.ind=TRUE)
# JJ <- which(resLL==max(resLL), 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))
}
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