#' IDEA method
#'
#' This function implements a least squares estimation method of R0 due to
#' Fisman et al. (PloS One, 2013). See details for implementation notes.
#'
#' This method is closely related to that implemented in \code{ID}. The method
#' is based on an incidence decay model. The estimate of R0 is the value which
#' minimizes the sum of squares between observed case counts and cases counts
#' expected under the model.
#'
#' 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.
#'
#' @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.
#'
#' @return \code{IDEA} returns a single value, the estimate of R0.
#'
#' @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.
#' IDEA(NT, mu = 5 / 7)
#'
#' # Obtain R0 when the serial distribution has a mean of three days.
#' IDEA(NT, mu = 3 / 7)
#'
#' @export
IDEA <- function(NT, mu) {
  if (length(NT) < 2)
    print("Warning: length of NT should be at least two.")
  else {
    NT <- as.numeric(NT)
    TT <- length(NT)
    s <- (1:TT) / mu

    y1 <- log(NT) / s
    y2 <- s^2
    y3 <- log(NT)

    IDEA1 <- sum(y2) * sum(y1) - sum(s) * sum(y3)
    IDEA2 <- TT * sum(y2) - sum(s)^2
    IDEA <- exp(IDEA1 / IDEA2)

    return(IDEA)
  }
}