#' seqB method
#'
#' This function implements a sequential Bayesian estimation method of R0 due to
#' Bettencourt and Riberio (PloS One, 2008). See details for important
#' implementation notes.
#'
#' The method sets a uniform prior distribution on R0 with possible values
#' between zero and \code{kappa}, discretized to a fine grid. The distribution
#' of R0 is then updated sequentially, with one update for each new case count
#' observation. The final estimate of R0 is \code{Rhat}, the mean of the (last)
#' posterior distribution. The prior distribution is the initial belief of the
#' distribution of R0, which is the uninformative uniform distribution with
#' values between zero and \code{kappa}. Users can change the value of
#' /code{kappa} only (i.e., the prior distribution cannot be changed from the
#' uniform). As more case counts are observed, the influence of the prior
#' distribution should lessen on the final estimate \code{Rhat}.
#'
#' This method is based on an approximation of the SIR model, which is most
#' valid at the beginning of an epidemic. The method assumes that the mean of
#' the serial distribution (sometimes called the serial interval) is known. The
#' final estimate can be quite sensitive to this value, so sensitivity testing
#' is strongly recommended. Users should be careful about units of time (e.g.,
#' are counts observed daily or weekly?) when implementing.
#'
#' Our code has been modified to provide an estimate even if case counts equal
#' to zero are present in some time intervals. This is done by grouping the
#' counts over such periods of time. Without grouping, and in the presence of
#' zero counts, no estimate can be provided.
#'
#' @param NT Vector of case counts.
#' @param mu Mean of the serial distribution. This 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. If case counts are daily and the serial distribution has a
#'           mean of seven days, then \code{mu} should be set to seven.
#' @param kappa Largest possible value of uniform prior (defaults to 20). This
#'              describes the prior belief on ranges of R0, and should be set to
#'              a higher value if R0 is believed to be larger.
#'
#' @return \code{seqB} returns a list containing the following components:
#'         \code{Rhat} is the estimate of R0 (the posterior mean),
#'         \code{posterior} is the posterior distribution of R0 from which
#'         alternate estimates can be obtained (see examples), and \code{group}
#'         is an indicator variable (if \code{group=TRUE}, zero values of NT
#'         were input and grouping was done to obtain \code{Rhat}). The variable
#'         \code{posterior} is returned as a list made up of \code{supp} (the
#'         support of the distribution) and \code{pmf} (the probability mass
#'         function).
#'
#' @examples
#' # Weekly data.
#' NT <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)
#'
#' ## Obtain R0 when the serial distribution has a mean of five days.
#' res1 <- seqB(NT, mu = 5 / 7)
#' res1$Rhat
#'
#' ## Obtain R0 when the serial distribution has a mean of three days.
#' res2 <- seqB(NT, mu = 3 / 7)
#' res2$Rhat
#'
#' # Compute posterior mode instead of posterior mean and plot.
#'
#' Rpost <- res1$posterior
#' loc <- which(Rpost$pmf == max(Rpost$pmf))
#' Rpost$supp[loc] # Posterior mode.
#' res1$Rhat # Compare with the posterior mean.
#'
#' par(mfrow = c(2, 1), mar = c(2, 2, 1, 1))
#'
#' plot(Rpost$supp, Rpost$pmf, col = "black", type = "l", xlab = "", ylab = "")
#' abline(h = 1 / (20 / 0.01 + 1), col = "red")
#' abline(v = res1$Rhat, col = "blue")
#' abline(v = Rpost$supp[loc], col = "purple")
#' legend("topright",
#'   legend = c("Prior", "Posterior", "Posterior mean", "Posterior mode"),
#'   col = c("red", "black", "blue", "purple"), lty = 1)
#'
#' @export
seqB <- function(NT, mu, kappa = 20) {
  if (length(NT) < 2) {
    print("Warning: length of NT should be at least two.")
  } else {
    if (min(NT) > 0) {
      times <- 1:length(NT)
      tau <- diff(times)
    }
    group <- FALSE
    if (min(NT) == 0) {
      times <- which(NT > 0)
      NT <- NT[times]
      tau <- diff(times)
      group <- TRUE
    }

    R <- seq(0, kappa, 0.01)
    prior0 <- rep(1, kappa / 0.01 + 1)
    prior0 <- prior0 / sum(prior0)
    k <- length(NT) - 1
    R0.post <- matrix(0, nrow = k, ncol = length(R))
    prior <- prior0
    posterior <- seq(0, length(prior0))
    gamma <- 1 / mu

    for (i in 1:k) {
      mm1 <- NT[i]
      mm2 <- NT[i + 1]
      lambda <- tau[i] * gamma * (R - 1)
      lambda <- log(mm1) + lambda
      loglik <- mm2 * lambda - exp(lambda)
      maxll <- max(loglik)
      const <- 0

      if (maxll > 700)
        const <- maxll - 700

      loglik <- loglik - const
      posterior <- exp(loglik) * prior
      posterior <- posterior / sum(posterior)
      prior <- posterior
    }

    Rhat <- sum(R * posterior)

    return(list(Rhat = Rhat,
                posterior = list(supp = R, pmf = posterior),
                group = group))
  }
}