ddd {dwp}

Calculate Probability Functions for Distance Distributions

Description

Calculate the standard d/p/q/r family of R probability functions for distance distributions (dd) as well as the relative carcass density (rcd). Usage broadly parallels that of the d/p/q/r probability functions like dnorm, pnorm, qnorm, and rnorm.

Usage

ddd(x, model, parms = NULL, extent = "full", zrad = 200)

pdd(q, model, parms = NULL, extent = "full", zrad = 200, silent = FALSE)

qdd(p, model, parms = NULL, extent = "full", zrad = 200, subdiv = 1000)

rdd(n, model, parms = NULL, extent = "full", zrad = 200, subdiv = 1000)

rcd(x, model, parms = NULL, extent = "full", zrad = 200)

Arguments

x, q, p, n

numeric, x \ge 0

model

either a dd object or a ddSim object

parms

model parameters; required if model is specified as a character string rather than a dd or ddSim object (otherwise optional and ignored)

extent

for a full distribution extrapolated beyond the search radius to account for all carcasses, use extent = "full"; for a distribution restricted solely to carcasses falling within the search radius, use extent = "win".

zrad

the distance at which carcass density is assumed to be zero; to be used only in simulation reps in which simulated parameters do not yield extensible distributions, essentially returning 0 rather than NA for those pathological cases.

silent

If TRUE, then console messages are suppressed.

subdiv

if the number of values to calculate with rdd or qdd is >1, the function uses breaks the PDF into subdiv subdivisions and interpolates to solve the inverse. More subdivisions gives greater accuracy but is slower.

Details

The probability density function (PDF(x) = f(x) = ddd(x, ...)) gives the probability that a carcass falls in a 1 meter ring centered at the turbine and with an outer radius of x meters. The cumulative distribution function [CDF(x) = F(x) = pdd(x, ...)] gives the probability that a carcass falls within x meters from the turbine. For a given probability, p, the inverse CDF [qdd(p,...)] gives the p quantile of carcass distances. For example, qdd(0.5,...) gives the median carcass distance, and qdd(0.9, ...) gives the radius that 90% of the carcasses are expected to fall in. Random carcass distances can be generated using rdd.

The relative carcass density function(rcd) gives relative carcass densities at a point x meters from a turbine. In general, rcd is proportional to PDF(x)/x, normalized so that the surface of rotation of rcd(x) has total volume of 1. There are more stringent contstraints on the allowable parameters in the fitted (or simulated) glm's because the integral of PDF(x)/x must converge.

Distributions may be extrapolated beyond the search radius to account for all carcasses, including those that land beyond the search radius (extent = "full"), or may be restricted to carcasses falling within the searched area (extent = "win"). Typically, in estimating dwp for a fatality estimator like eoa or GenEst, the full distributions would be used.

The probability functions have a number of purposes. A few of the more commonly used are listed below.

PDF and CDF (ddd and pdd):

• to calculate the probability that carcass lands at a distance x meters from the turbine (or, more precisely, within 0.5 meters of x) or within x meters from the turbine, use a scalar value of x and a single model (dd or ddSim) with ddd or pdd, repspectively;

• to account for uncertainty in the probabilities at x, use ddd or pdd for with scalar x and a simulated set of parameters from the fitted model (ddSim object). This would be useful for calculating confidence intervals for the probabilities;

• to calculate probabilities for a range of x values according to a single model, use a vector x with a dd object or a ddSim object with one row. This would be useful for drawing graphs of PDFs or CDFs;

• to calculate simulated probabilites for a range of x values, use a vector x and a ddSim object of simulated parameter sets. This would be useful for drawing confidence regions around a fitted PDF or CDF.

Inverse CDF (qdd):

• to calculate the distance that 100p% of the carcasses are expected to fall, use a scalar p in the interval (0, 1) and a single model (dd) or parameter set (ddSim with one row);

• to calculate account for the uncertainty in estimating the inverse CDF for a given p, use a scalar p and a ddSim object. This would be useful for calculating a confidence interval for, say, the median or the expected 90th percentile of carcass distances;

• to calculate the inverse CDF for a range of probabilities for a single model, use a vector p and a single model (dd or ddSim object with one row.

Random Carcasses Distances (rdd):

• to generate n random carcass distances for a given (fixed) model, use a dd object or a ddSim object with a single row;

• to generate n random carcass distances for a model and account for the uncertainty in estimating the model, use a ddSim object with n rows, where n is also used as the n argument in the call to rdd.

Relative Carcass Density (per m^2):

• to calculate the relative carcass density at a number of distances, use a vector x. This would be useful in generating maps of carcass density at a site.

Value

vector or matrix of values; a vector is returned unless model is a ddSim object with more than one row and is to be calculated for more than one value (x, q, p), in which case an array with dimensions length(x) by nrow(model) is returned (where "x" is x, q, or p, depending on whether ddd, pdd, or qdd is called).

Examples

data(layout_simple)
data(carcass_simple)
sitedata <- initLayout(layout_simple)
ringdata <- prepRing(sitedata)
ringsWithCarcasses <- addCarcass(carcass_simple, data_ring = ringdata)
distanceModels <- ddFit(ringsWithCarcasses)
modelEvaluations <- modelFilter(distanceModels)
bestModel <- modelEvaluations$filtered
pdd(100, model = bestModel) # estimated fraction of carcasses within 100m
ddd(1:150, model = bestModel) # estimated PDF of the carcass distances
qdd(0.9, model = bestModel) # estimated 0.9 quantile of carcass distances
rdd(1000, model = bestModel) # 1000 random draws from estimated carcass distribution