#' WP method
#'
#' This function implements an R0 estimation due to White and Pagano (Statistics in Medicine, 2008).
#' The method is based on maximum likelihood estimation in a Poisson transmission model.
#' See details for important implementation notes.
#'
#' This method is based on a Poisson transmission model, and hence may be most most valid at the beginning
#' of an epidemic. In their model, the serial distribution is assumed to be discrete with a finite number
#' of posible values. In this implementation, if the serial distribution is assumed known, it is taken to
#' be a discretized version of a gamma distribution with mean \code{mu}, shape parameter one, and largest
#' possible value based on parameter \code{tol}. When the serial distribution is unknown, the function
#' implements a grid search algorithm to find the maximum likelihood estimator over all possible gamma
#' distributions with unknown mean and variance, restricting these to a prespecified grid (see
#' \code{search} parameter).  
#'
#' When the serial distribution is taken to be \code{known}, sensitivity testing of the parameter \code{mu}
#' is strongly recommended. If the serial distribution is \code{unknown}, the likelihood function can be
#' flat near the maximum, resulting in numerical instability of the optimizer.  When the serial distribution
#' is \code{unkown} the implementation takes considerably longer to run.  Users should be careful about units
#' of time (e.g. are counts observed daily or weekly?) when implementing.  
#'
#' The model developed in White and Pagano (2008) is discrete, and hence the serial distribution is finite 
#' discrete. In our implementation, the input value \code{mu} is that of a continuous distribution. The
#' algorithm when \code{method="known"} disretizes this input, and hence the mean of the serial distribution
#' returned in the list \code{SD} will differ from \code{mu} somewhat. That is to say, if the user notices that
#' the input \code{mu} and out put mean of \code{SD} are different, this is to be expected, and is caused by
#' the discretization.
#'
#' @param NT Vector of case counts
#' @param mu Mean of the serial distribution (needs to match case counts in time units; for example, if case
#'           counts are weekly and the serial distribution has a mean of seven days, then \code{mu} should be
#'           set to one). The default value of \code{mu} is set to \code{NA}.
#' @param method Variable taking one of two possible values: \code{known} or \code{unknown}. If "known", the
#'               serial distribution is assumed to be gamma with  rate 1/\code{mu} and shape equal to one, if
#'               "unknown" then the serial distribution is gamma with unknown parameters. Defaults to "unknown"
#' @param search List of default values for the grid search algorithm; the list includes three elements: the
#'               first is \code{B} which is the length of the grid in one dimension, the second is
#'               \code{scale.max} which is the largest possible value of the scale parameter, and the third is
#'               \code{shape.max} which is the largest possible value of the shape parameter; defaults to
#'               \code{B=100, scale.max=10, shape.max=10}. For both shape and scale, the smallest possible
#'               value is 1/\code{B}.  
#' @param tol Cutoff value for cumulative distribution function of the pre-discretization gamma serial
#'            distribution, defaults to 0.999 (i.e. in the discretization, the maximum is chosen such that the
#'            original gamma distribution has cumulative probability of no more than 0.999 at this maximum).
#'
#' @return WP returns a list containing the following components:  \code{Rhat} is the estimate of R0, \code{SD}
#'            is either the discretized serial distribution (if \code{method="known"}) or the estimated
#'            discretized serial distribution (if \code{method="unknown"}), and \code{inputs} is a list of the
#'            original input variables \code{NT, mu, method, search, tol}. The list also returns the variable
#'            \code{check}, which is equal to the number of non-unique maximum likelihood estimators. The serial
#'            distribution \code{SD} is returned as a list made up of \code{supp} the support of the distribution
#'            and \code{pmf} the probability mass function.  
#'
#' @examples
#' ## ===================================================== ##
#' ## Illustrate on weekly data                             ##
#' ## ===================================================== ##
#'
#' NT <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)	
#' ## obtain Rhat when serial distribution has mean of five days
#' res1 <- WP(NT=NT, mu=5/7, method="known")	
#' res1$Rhat
#' ## obtain Rhat when serial distribution has mean of three days
#' res2	<- WP(NT=NT, mu=3/7, method="known")	
#' res2$Rhat
#' ## obtain Rhat when serial distribution is unknown
#' ## NOTE:  this implementation will take longer to run
#' res3	<- WP(NT=NT)	
#' res3$Rhat
#' ## find mean of estimated serial distribution
#' serial <- res3$SD
#' sum(serial$supp * serial$pmf)
#'
#' ## ========================================================= ##
#' ## Compute Rhat using only the first five weeks of data      ##
#' ## ========================================================= ##
#' 
#' res4 <- WP(NT=NT[1:5], mu=5/7, method="known") # serial distribution has mean of five days
#' res4$Rhat
#'
#' @export
WP <- function(NT, mu="NA", method="unknown", search=list(B=100, shape.max=10, scale.max=10), tol=0.999) {
    if (method == "unknown") {
        print("You have assumed that the serial distribution is unknown.")
        res <- WP_unknown(NT=NT, B=search$B, shape.max=search$shape.max, scale.max=search$scale.max, tol=tol)
        Rhat <- res$Rhat
        p <- res$p
        range.max <- res$range.max
        JJ <- res$JJ
    }

	if (method == "known") {
        if (mu=="NA") {
            res <- "NA"
            print("For method=known, the mean of the serial distribution must be specified.")
        } else {
            print("You have assumed that the serial distribution is known.")
            range.max <- ceiling(qexp(tol, rate=1/mu))
            p <- diff(pexp(0:range.max, 1/mu))
            p <- p / sum(p)
            res <- WP_known(NT=NT, p=p)
            Rhat <- res
            JJ <- NA
        }
    }

    return(list(Rhat=Rhat, check=length(JJ), SD=list(supp=1:range.max, pmf=p)))
}