#' Convert from intrinsic to physical RSM parameters
#'
#' Convert \strong{intrinsic} RSM parameters \eqn{lambda_i} and \eqn{nu_i}
#' correspond to the \strong{physical} RSM parameters \eqn{lambda_i'}
#' and \eqn{nu_i'}. The physical parameters are more meaningful but they
#' depend on \eqn{mu}. The intrinsic parameters are independent of
#' \eqn{mu}. See book for details.
#'
#'
#' @param mu The mean of the Gaussian distribution for the ratings of latent
#' LLs, i.e. continuous ratings of lesions that were found by the search
#' mechanism ~ N(\eqn{\mu},1). The corresponding distribution for the
#' ratings of latent NLs is N(0,1).
#'
#' @param lambda_i The \emph{intrinsic} Poisson lambda_i parameter.
#'
#' @param nu_i The \emph{intrinsic} Binomial nu_i parameter.
#'
#' @return A list containing \eqn{\lambda} and \eqn{\nu}
#'
#' @details RSM is the Radiological Search Model described in the book.
#' See also \code{\link{Util2Intrinsic}}.
#'
#' @references
#' Chakraborty DP (2006) A search model and figure of merit for observer data acquired according to the free-response
#' paradigm, Phys Med Biol 51, 3449--3462.
#'
#' Chakraborty DP (2006) ROC Curves predicted by a model of visual search, Phys Med Biol 51, 3463--3482.
#'
#' Chakraborty DP (2017) \emph{Observer Performance Methods for Diagnostic Imaging - Foundations,
#' Modeling, and Applications with R-Based Examples}, CRC Press, Boca Raton, FL.
#' \url{https://www.routledge.com/Observer-Performance-Methods-for-Diagnostic-Imaging-Foundations-Modeling/Chakraborty/p/book/9781482214840}
#'
#' @examples
#' mu <- 2;lambda_i <- 20;nu_i <- 1.1512925
#' lambda <- Util2Physical(mu, lambda_i, nu_i)$lambda
#' nu <- Util2Physical(mu, lambda_i, nu_i)$nu
#' ## note that only the physical values are only constrained to be positive
#' ## the physical variable nu must obey 0 <= nu <= 1
#'
#'
#' @export
Util2Physical <- function(mu, lambda_i, nu_i) {
lambda <- lambda_i / mu
nu <- 1 - exp(-nu_i * mu)
return (list(
lambda = lambda,
nu = nu))
}
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