R/stage.r

In biometrics: A Package for Biometrics and Modelling

#' Function of the Stage allometric
#' model, based upon parameters (i.e., coefficients) and a variable,
#' as defined by the mathematical expression
#' \deqn{y_i= \alpha
#' \mathrm{e}^{\left(-\beta ({x_i}^{-\gamma}) \right)},
#' }
#' where: \eqn{y_i} and \eqn{x_i} are the response
#' and predictor variable, respectively, for the *i*-th observation;  
#'  and the rest are parameters (i.e., coefficients).
#' Further details on this function can be found in
#' Salas-Eljatib (2025).
#'
#' @title Function that computes the result of the Stage  allometric
#' model.
#' @param x is the predictor variable.
#' @param alpha is the coefficient-parameter  \eqn{\alpha}.
#' @param beta is the  coefficient-parameter  \eqn{\beta}.
#' @param gamma is the  coefficient-parameter  \eqn{\gamma}.
#' @param upsilon is an optional constant term that force the prediction
#' of *y* when *x=0*. Thus, the new model becomes
#' \eqn{ y_i = \Upsilon+ f(x_i,\mathbf{\theta})}, where
#' \eqn{\mathbf{\theta}} is the vector of coefficients of
#' the above described function represented by
#' \eqn{f(\cdot)}. The default
#' value for \eqn{\Upsilon} is 0.
#'
#' @return Returns the response variable based upon
#' the predictor variable and the coefficients. 
#' @author Christian Salas-Eljatib.
#' @references
#' - Stage AR. 1963. A mathematical approach to polymorphic site
#' index curves for Grand Fir. Forest Sci. 9(2):167-180.
#' \doi{10.1093/forestscience/9.2.167}
#' - Salas-Eljatib C. 2025. Funciones alométricas: reparametrizaciones
#' y características matemáticas. Documento de trabajo No. 1,
#' Serie: Cuadernos de biometría, Laboratorio de Biometría y
#' Modelación Forestal, Universidad de Chile. Santiago, Chile. 51 p.
#' \url{https://biometriaforestal.uchile.cl}
#' @examples
#' # Predictor variable values to be used
#' time<-seq(5,60,by=0.01)
#' # Using the function
#' d<-stage.fx(x=time,alpha=28,beta=56,gamma=1.19)
#' plot(time,d,type="l")
#'  
#' @rdname stage.fx
#' @export
#'
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stage.fx <- function(x,alpha, beta, gamma, upsilon=0){    
    upsilon + alpha*exp(-beta * (x^(-gamma)))
}