estWD: Weighted Distribution Estimation

Description

Maximum likelihood estimatation of a weighted probability density function is completed. is done on a weighted distribution. The weighted distribution is a typical probability density distribution multiplied by a weight function. The weight function can be used to truncate the distribution by returning zero beyond some threshold value.

Usage

estWD(fatDist, weightFun, distribution, ...)
weightedDistribution(fatDist, weightFun, distribution, ...)

Value

Data frame of the parameter estimates with fit statistics.

Arguments

fatDist: Vector of fatality distanes from the turbine.
weightFun: R function that is multipled by the probability distribution, see details.
distribution: Character indicating the distribution for weightedDistribution or vector for estWD.
...: Additional arguments passed to weightFun or optim.

Details

The function estWD is a convient wrapper function to weightedDistribution, for fitting multiple distributions.

The weight function should return a (relative) probability of detection at every distance. Typically this is the proportion of area searched. The function weightFun is set up to take a table of proportion of area searched and return values in a function format.

Let $h(x)$ be the weight function (weightFun), $f(x|\theta)$ be a probability density function (specified by distribution, $x$ be the vector of carcass distances from the turbine (fatDist), and $\theta$ be the parameter vector to be estimated. The weighted distribution is $$f_{d}(x|\theta) = \frac{h(x)f(x|\theta)}{\int h(y)f(y|\theta)dy}$$ The likelihood that is maximized is $$L_{d}(\theta|\underbar{x}) = \prod_{i=1}^{n}\frac{h(x_i)f(x_i|\theta)}{\int h(y)f(y|\theta)dy}$$

Examples

Run this code

## load the data
data(carcassDistance)
data(proportionAreaSearched)

###############################################
## fit for fall carcasses found on road and pad (RP)
distanceFallRP <- subset(carcassDistance,plotType=='RP'&season=='fall',
select=distanceFromTurbine,drop=TRUE)

# \donttest{
fallRPFit <- estWD(fatDist=distanceFallRP,weightFun=weightFun,
distribution=c('norm','gamma','weibull'),propTable=proportionAreaSearched,type='RP',
typeCol='plotType',distanceCol='distanceFromTurbine',propCol='proportionAreaSearched',
maxDistance=100)
# }

###############################################
## fit for fall carcasses found on full plots
distanceFallFP <- subset(carcassDistance,plotType=='FULL'&season=='fall',
select=distanceFromTurbine,drop=TRUE)

# \donttest{
fallFPFit <- estWD(fatDist=distanceFallFP,weightFun=weightFun,
distribution=c('norm','gamma','weibull'),propTable=proportionAreaSearched,type='FULL',
typeCol='plotType',distanceCol='distanceFromTurbine',propCol='proportionAreaSearched',
maxDistance=100)
# }