R/negexp.like.R

In Rdistance: Distance-Sampling Analyses for Density and Abundance Estimation

#' @title Negative exponential distance function for distance analyses
#' 
#' @description Computes likelihood contributions for off-transect sighting distances, 
#' scaled appropriately, for use as a distance likelihood.
#' 
#' @param a A vector of likelihood parameter values. Length and meaning depend on \code{series} and \code{expansions}. If no expansion terms were called for
#'   (i.e., \code{expansions = 0}), the distance likelihoods contains only one canonical parameter, which 
#'   is the first element of \code{a} (see Details). If one or more expansions are called for,
#'   coefficients for the expansion terms follow coefficients for the canonical parameter.  
#'   Coefficients
#'   for the expansion terms, if present, are \code{a[2:length(a)]}.
#' @param dist A numeric vector containing the observed distances.
#' @param covars Data frame containing values of covariates at each observation in \code{dist}.
#' @param w.lo Scalar value of the lowest observable distance.  This is the \emph{left truncation} of sighting distances in \code{dist}. Same units as \code{dist}.
#'   Values less than \code{w.lo} are allowed in \code{dist}, but are ignored and their contribution to the likelihood is set to \code{NA} in the output.
#' @param w.hi Scalar value of the largest observable distance.  This is the \emph{right truncation} of sighting distances in \code{dist}.  Same units as \code{dist}.
#'   Values greater than \code{w.hi} are allowed in \code{dist}, but are ignored and their contribution to the likelihood is set to \code{NA} in the output.
#' @param series A string specifying the type of expansion to use.  Currently, valid values are 'simple', 'hermite', and 'cosine'; but, see 
#'   \code{\link{dfuncEstim}} about defining other series.
#' @param expansions A scalar specifying the number of terms in \code{series}. Depending on the series, this could be 0 through 5.
#'   The default of 0 equates to no expansion terms of any type.
#' @param scale Logical scalar indicating whether or not to scale the likelihood so it integrates to 1. This parameter is used to stop recursion in other functions.
#'   If \code{scale} equals TRUE, a numerical integration routine (\code{\link{integration.constant}}) is called, which in turn calls this likelihood function again
#'   with \code{scale} = FALSE. Thus, this routine knows when its values are being used to compute the likelihood and when its value is being used to compute the 
#'   constant of integration.  All user defined likelihoods must have and use this parameter.
#' @param pointSurvey Boolean. TRUE if \code{dist} is point transect data, FALSE if line transect data.
#' 
#' @details The negative exponential likelihood is \deqn{f(x|a) = \exp(-ax)}{f(x|a) = exp( -a*x )} where \eqn{a} is a slope parameter to be estimated. 
#'   \bold{Expansion Terms}: If \code{expansions} = k (k > 0), the expansion function specified by \code{series} is called (see for example
#'   \code{\link{cosine.expansion}}). Assuming \eqn{h_{ij}(x)}{h_ij(x)} is the \eqn{j^{th}}{j-th} expansion term for the \eqn{i^{th}}{i-th} distance and that 
#'   \eqn{c_1, c_2, \dots, c_k}{c(1), c(2), ..., c(k)}are (estimated) coefficients for the expansion terms, the likelihood contribution for the \eqn{i^{th}}{i-th} 
#'   distance is, \deqn{f(x|a,b,c_1,c_2,\dots,c_k) = f(x|a,b)(1 + \sum_{j=1}^{k} c_j h_{ij}(x)).}
#'   {f(x|a,b,c_1,c_2,...,c_k) = f(x|a,b)(1 + c(1) h_i1(x) + c(2) h_i2(x) + ... + c(k) h_ik(x)). }
#'   
#' @return A numeric vector the same length and order as \code{dist} containing the likelihood contribution for corresponding distances in \code{dist}. 
#'   Assuming \code{L} is the returned vector from one of these functions, the full log likelihood of all the data is \code{-sum(log(L), na.rm=T)}. Note that the
#'   returned likelihood value for distances less than \code{w.lo} or greater than \code{w.hi} is \code{NA}, and thus it is prudent to use \code{na.rm=TRUE} in the
#'   sum. If \code{scale} = TRUE, the integral of the likelihood from \code{w.lo} to \code{w.hi} is 1.0. If \code{scale} = FALSE, the integral of the likelihood is
#'   arbitrary.
#'   
#' @author Trent McDonald, WEST Inc. \email{tmcdonald@west-inc.com}
#'         Aidan McDonald, WEST Inc. \email{aidan@mcdcentral.org}
#'         
#' @seealso \code{\link{dfuncEstim}},
#'          \code{\link{halfnorm.like}},
#'          \code{\link{uniform.like}},
#'          \code{\link{hazrate.like}},
#'          \code{\link{Gamma.like}}
#'          
#' @examples \dontrun{
#' set.seed(238642)
#' x <- seq(0, 100, length=100)
#' 
#' # Plots showing effects of changes in parameter Beta
#' plot(x, negexp.like(0.01, x), type="l", col="red")
#' plot(x, negexp.like(0.05, x), type="l", col="blue")
#' 
#' # Estimate 'negexp' distance function
#' Beta <- 0.01
#' x <- rexp(1000, rate=Beta)
#' dfunc <- dfuncEstim(x~1, likelihood="negexp")
#' plot(dfunc)
#' }
#'          
#' @keywords models
#' @export

negexp.like <- function (a, 
                         dist, 
                         covars = NULL, 
                         w.lo = 0, 
                         w.hi = max(dist),
                         series = "cosine", 
                         expansions = 0, 
                         scale = TRUE,
                         pointSurvey = FALSE){

    dist[ (dist < w.lo) | (dist > w.hi) ] <- NA

    if(!is.null(covars)){
      
      q <- ncol(covars)
      # not necessary, in all negexp norm cases, no extra params hanging off the end 
      # but, I'll leave it here so it's compatible with other likelihoods and 
      # just in case we want to allow expansions with covariates later.
      beta <- a[1:q] 
      s <- drop( covars %*% matrix(beta,ncol=1) )      
      beta <- exp(s)
    } else {
      beta <- a[1]
    }
    
	  key = exp(-beta*dist)
    dfunc <- key
    w <- w.hi - w.lo


    if(expansions > 0){

        nexp <- min(expansions,length(a)-1)  # should be equal. If not, fire warning next
        
        if( length(a) != (expansions+1) ) {
            #warning("Wrong number of parameters in expansion. Should be (expansions+1). High terms ignored.")
        }

		if (series=="cosine"){
            dscl = dist/w
            exp.term <- cosine.expansion( dscl, nexp )
		} else if (series=="hermite"){
            dscl = dist/w
            exp.term <- hermite.expansion( dscl, nexp )
		} else if (series == "simple") {
            dscl = dist/w
            exp.term <- simple.expansion( dscl, nexp )
        } else {
            stop( paste( "Unknown expansion series", series ))
        }

        dfunc <- key * (1 + c(exp.term %*% a[(length(a)-(nexp-1)):(length(a))]))


    } else if(length(a) > 1){
        #warning("Wrong number of parameters in halfnorm. Only 1 needed if no expansions. High terms ignored.")
    }

    if( scale ){
        dfunc = dfunc / integration.constant(dist=dist, 
                                             density=negexp.like, 
                                             a=a,
                                             covars = covars, 
                                             w.lo=w.lo,
                                             w.hi=w.hi,
                                             series=series,
                                             expansions=expansions, 
                                             pointSurvey = pointSurvey)  # makes integral from w.lo to w.hi = 1.0
    }
    
    c(dfunc)
}