Log-Logistic Distribution

Description

The probability density function, cumulative density function, inverse cumulative density function, random generation for the log logistic distribution.

Usage

dllog(x, shape = 1, scale = 1, log = FALSE, ...)

llogSummaryStats(shape, scale)

pllog(q, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE, ...)

qllog(p, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE, ...)

rllog(n, shape = 1, scale = 1, ...)

Arguments

Details

If X is a random variable distributed according to a logistic distribution, then Y = exp(X) has a log-logistic distribution.

The log-logistic distribution with parameters shape = a and scale = s has density

f(x) = \frac{(\frac{1}{a*exp(s))})(\frac{x}{\exp{s}})^{\frac{1}{a} - 1}}{(1+(\frac{x}{\exp{s}})^{1/a})^2}

for x >= 0, a > 1, and s > 0.

The median is exp(s), mean is

\frac{a\pi*exp(s)}{sin(a*\pi)}

for 1/a > 1. The variance is

(exp(s))^2(\frac{2*\pi*a}{(sin(2*pi*a))}- \frac{(a*\pi)^2}{(sin^2(a*\pi))})

for 1/a > 2. The mode is

exp(s)(\frac{(1/a) - 1}{(1/a) + 1})^{a}

for 1/a > 1 otherwise it is zero.

Value

dllog returns vector of the densities.

pllog returns a vector of probabilities.

qllog returns a vector of quantiles.

rllog returns a vector of random log-logistic variates.

See Also

dlogis plogis qlogis rlogis

Examples

y <- rllog(5,shape=1,scale=1/3)
dllog(x=y,shape=1,scale=1/3)
dlogis(x=log(y),location=1/3,scale=1)/y

pllog(q=y,shape=1,scale=1/3)
qllog(p=seq(0,1,by=.25),shape=1,scale=1/3)

Calculate the area correction value(s) with confidence intervals

Description

Use a fitted carcass density distribution and data describing the search area to calculate area correction values and confidence intervals using a parametric bootstrap approach.

Usage

calcAC(
  distribution,
  paramVec,
  varcovVec = NULL,
  proportionSearchDF,
  distanceCol,
  proportionCol,
  additionalCol = NULL,
  nBoot = NULL,
  truncBounds = NULL,
  ciLevel = 0.9,
  randomSeed = NULL,
  ...
)

Arguments

Details

The function getDistanceProbability is used to calculate the probability (fraction of carcasses) in the intervals between distances in proportionSearchDF.

The truncBounds argument defaults to zero as a lower bound and infinity as the upper bound. If a single value is provided, it is assumed as the upper bound with zero as the lower bound. If two or more values are provided, the max(truncBounds) is the upper bound and min(truncBounds) is the lower bound.

If varcovVec is NULL, then parametric bootstrapping is impossible and a confidence interval is not estimated. The varcovVec should be in such an order that correctly fills the lower triangle including the diagonal. The first column is filled, then the second, and so on. This forms the variance-covariance matrix for the parameters.

If nBoot is greater than zero, a parametric bootstrap is done. Bootstrap parameters are generated using the rmvnorm function.

If the additionalCol argument is not NULL, separate area corrections are estimated for each unique value within the column.

Value

windAC object

See Also

weightedLikelihood weightedDistribution getDistanceProbability

Examples

## proportion of area searched data
data(proportionAreaSearched)

## no parametric bootstrap
noBootstrap <- calcAC(distribution = 'gamma',
                      paramVec = c(2.483323, 0.02495139),
                      varcovVec = NULL,
                      proportionSearchDF = proportionAreaSearched,
                      distanceCol = 'distanceFromTurbine',
                      proportionCol = 'proportionAreaSearched',
                      additionalCol = 'plotType')

## with a parametric bootstrap

withBootstrap <- calcAC(distribution = 'gamma',
                        paramVec = c(2.483323, 0.02495139),
                        varcovVec = c(0.041189428, 0.0008825275, 2.118081e-05),
                        proportionSearchDF = proportionAreaSearched,
                        distanceCol = 'distanceFromTurbine',
                        proportionCol = 'proportionAreaSearched',
                        additionalCol = 'plotType',
                        nBoot = 10)

Carcass distance example data set

Description

Example data set of carcass distance found during searches under wind turbines.

Usage

data(carcassDistance)

Format

A data frame with 56 rows and 6 variables:

season

The season in which that carcass was found.

plotType

The plot type (either FULL or RP) that was being searched when the carcass was found.

proportionTurbineType

The proportion of turbines that have that plot type. For plotType=FULL the value is 0.1 and 0.9 for plotType=RP

.

distanceFromTurbine

Meters, carcass distance from the turbine.

compassBearingDegree

Degrees, the compass bearing from the turbine to the carcass location. Zero degrees is north, 90 degrees is east, etc.

probabilityDetection

The summarized probability of detection from the bias trial information.

Details

The fictitious wind farm has 100 turbines. For 90 turbines the search plot consists of the turbine pad and the road out to 100 meters (RP = road and pad). The remaining 10 turbines had full plots (FULL) that are circular with a radius of 100 meters. The wind farm was searched during two seasons: Spring and Fall.

The example carcasses are all the same size class (bats).

Bias trials are also done to account for searcher efficiency and carcass removal. These results are summarized as a probability of detection in the data set.

References

https://tethys.pnnl.gov/publications/us-fish-wildlife-service-land-based-wind-energy-guidelines

Integration of the intersection of a rectangle and a circle

Description

Calculates the area of the intersection between a rectangle and and circle.

Usage

circleBoxInt(R, S, L, ...)

Arguments

Details

The rectangle is defined with lower left corner being the origin and upper right corner at (L, S). The area returned is the intersection between the circle, centered at the origin, and the rectangle.

If R \leq S then (\pi R^2)/4 is returned.

If R \geq \sqrt{S^2 + L^2} then L*S is returned.

If R \leq L then R^2*sin^{-1}(S/R)/2 + S*\sqrt(R^2-S^2)/2 This is the area of a circle in the first quadrant between the horizontial line y=S

if R > L and R < \sqrt{S^2 + L^2} then

(R^2*sin^{-1}(S/R)/2 + S*\sqrt(R^2-S^2)/2) - (R^2*sin^{-1}(B/R)/2 + S*\sqrt(R^2-B^2)/2) + B*L

where B = \sqrt{R^2 - L^2}. In this case the there is part of the circle to the right of the rectangle. First set of parenthesis is the area of the circle below S, the second set is the area below B. Substracting the two gives the area between B and S. The rectangle defined by B and L needs to be added back in.

Value

Numeric value

Examples

radius <- 115
short <- 80
long <- 100
circleBoxInt(R=radius,S=short,L=long)

## not run
## the integral is the area inside the polygon

 x <- seq(0,max(radius,long),length=100)
outlineY <- function(x,R,S,L){
    suppressWarnings(y <- sqrt(R^2-x^2))
   y[x>R] <- 0
   y[x>L] <- 0
   y[y>=S] <- S
   return(y)
}
y <- outlineY(x=x,R=radius,S=short,L=long)
plot(x,y,type='l',ylim=c(-10,short))
text(long,0,label='L',pos=1)
text(0,short,label='S',pos=1)
text(long,sqrt(radius^2-long^2),label='B',pos=4)

Truncated Weighted Likelihood Estimation

Description

Maximum likelihood estimation of a (possibly truncated) probability density function is completed with weights on the likelihood.

Usage

estTWL(fatDist, fatW, distribution, plotBounds = NULL, ...)

weightedLikelihood(fatDist, fatW, distribution, plotBounds = NULL, ...)

Arguments

Details

The truncated likelihood for a single observation is

L^*(\theta|x_i) = \frac{f(x_i|\theta)}{\int_{a}^{b}f(y|\theta)dy}

Where x_i is fatDist, \theta is the vector of parameters to be estimated, a and b correspond to the plotBounds and f() is the distribution chosen.

The truncated weighted likelihood is

TWL(\theta|\underbar{x}) = \prod_{i=1}^{n}L^*(\theta|x_i)^{w_i}

Where n=length(fatDist) and w_i is fatW.

The truncated weighted likelihood is then estimated using standard maximum likelihood techniques.

See estTWL for examples.

Value

Data frame of the parameter estimates for the distribution with fit statistics.

See Also

calcAC

Examples

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

## add proportion of area searched to each carcass
carcDist <- merge(carcassDistance,proportionAreaSearched,
by=c('plotType','distanceFromTurbine'),all.x=TRUE)

## create the weight for each carcass
carcDist$w <- with(carcDist,1/(proportionAreaSearched*probabilityDetection))

twlOutput <- with(carcDist,estTWL(fatDist=distanceFromTurbine,fatW=w,plotBounds=c(0,100),
distribution=c('norm','weibull','gamma')))

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, ...)

Arguments

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}

Value

Data frame of the parameter estimates with fit statistics.

See Also

calcAC

weightFun

Examples

## 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)


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)


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

Create proportion of area searched table for a rectangular full plot

Description

Calculate the areas of intersection of a series of nested annuli with a rectangle.

Usage

geometricRectanglePropSearchTable(
  side1,
  side2 = side1,
  mastRadius,
  annulusWidth = 1,
  ...
)

Arguments

Details

Searches are conducted around a turbine within a rectangle for bird and bat carcasses. This function creates a data frame of proportion of area searched within each annulus ring. The turbine is assumed to be centered within the rectangle.

Value

Data frame of proportion of area searched for each annulus. distanceFromTurbine column represents the outer radius of each annulus.

See Also

geometricRoadPadPropSearchTable circleBoxInt

Examples

## square 50 x 50
propSearch <- geometricRectanglePropSearchTable(side1 = 50,
                                                mastRadius = 2)


## square 50 x 70
propSearch <- geometricRectanglePropSearchTable(side1 = 50,
                                                side2 = 70,
                                                mastRadius = 2)

Calculate the areas of intersection of a series of nested annuli with an idealized access road and turbine pad.

Description

Calculate area of annulus bisected by 2 parallel lines (e.g. a road of a road/pad plot).

Usage

geometricRoadPadPropSearchTable(
  padRadius,
  roadWidth,
  maxSearchRadius,
  mastRadius,
  annulusWidth = 1,
  ...
)

Arguments

Details

Searches are conducted on the road and turbine pad around wind turbines for bird and bat fatalities. This function creates a data frame of proportion of area searched within each annulus ring on an idealized road and pad. The turbine is assumed to be centered on a perfectly circular turbine pad with radius padRadius, and a perfectly straight access road of width roadWidth is oriented from the center of the circle away from the turbine. (The resulting road and pad looks like rather like a lollipop.)

The mastRadius argument is to account for the area taken up by the turbine mast.

The arguments padRadius, roadWidth, mastRadius, and annulusWidth are all rounded to the nearest integer. The maxSearchDistance is rounded up (ceiling function) to an integer. If half units are needed, then convert to a smaller unit. See examples.

Value

Data frame of proportion of area searched for each annulus. distanceFromTurbine column represents the outer radius of each annulus.

See Also

geometricRectanglePropSearchTable circleBoxInt

Examples

pad <- 10 #meters, turbine pad radius
road <- 4 #meters, width of the road to the turbine pad
maxDistance <- 100 #meters, max distance
mast <- 2 #meters, turbine mast radius

## proportion are area searched at each annulus
propSearch <- geometricRoadPadPropSearchTable(padRadius = pad,
                                              roadWidth = road,
                                              maxSearchRadius = maxDistance,
                                              mastRadius = mast)
head(propSearch, 20)

## if half meter annulus rings are desired:
convert <- 100 # meters * 100 = centimeters

## units in centimeters
propSearchHalfMeter <- geometricRoadPadPropSearchTable(padRadius = pad * convert,
                                              roadWidth = road*convert,
                                              maxSearchRadius = maxDistance * convert,
                                              mastRadius = mast * convert,
                                              annulusWidth = 50) ##50cm = half a meter
head(propSearchHalfMeter, 30)

## convert back to meters
propSearchHalfMeter$distanceFromTurbine <- propSearchHalfMeter$distanceFromTurbine/convert
head(propSearchHalfMeter, 30)

Calculate probabilities within one unit increments.

Description

Probabilities are calculated between specified increments for a given distribution and parameter values.

Usage

getDistanceProbability(
  q,
  distribution,
  param1,
  param2 = NA,
  tbound = c(-Inf, Inf),
  unitSize = 1,
  ...
)

Arguments

Details

This is a wrapper function that uses the ptrunc function. The basic calculation is ptrunc(q,...) - ptrunc(q-abs(unitSize),...)

Value

Vector of probabilities

See Also

ptrunc

Examples

## normal distribution
getDistanceProbability(q=8,distribution='norm',param1=10,param2=1)
pnorm(8,mean=10,sd=1)-pnorm(8-1,mean=10,sd=1)

## larger unitSize
getDistanceProbability(q=12,distribution='norm',param1=10,param2=1,unitSize=4)
pnorm(12,mean=10,sd=1)-pnorm(12-4,mean=10,sd=1)

Getting distribution functions

Description

Determines if the distribution functions are available. This is intended for internal use only.

Usage

getDistributionFunction(type, dist, ...)

Arguments

Details

It is determined that paste0(type, dist) is a function and returns that function. The nature of the returned function is not verified.

Value

Function, the first function in the search path that matches the name paste0(type, dist).

Examples

fun <- getDistributionFunction(type="q",dist="norm")

Summary statistics from the fitted distribution

Description

Summary statistics are calculated for the distribution with parameter estimates. Right now only the median is produced.

Usage

getDistributionSummary(distribution, paramVec, truncBounds = NULL, ...)

Arguments

Details

The truncBounds argument defaults to zero as a lower bound and infinity as the upper bound. If a single value is provided, it is assumed as the upper bound with zero as the lower bound. If two or more values are provided, the max(truncBounds) is the upper bound and min(truncBounds) is the lower bound.

Value

Data frame with the summary statistics.

Examples

getDistributionSummary('norm',c(40,25),truncBounds=c(-Inf,Inf))

getDistributionSummary('norm',c(40,25),truncBounds=NULL)

getDistributionSummary('norm',c(40,25),truncBounds=c(0,30))

Create proportion of area searched table from spatial data

Description

Calculate proportion of area searched around wind turbine based on turbine location data and polygons of search area.

Usage

getProportionAreaSearched(
  turbinePoints,
  turbineName,
  turbinePlots,
  turbineMastRadius,
  maxDistance
)

Arguments

Details

The sf package is used to calculate overlapping areas between the searched area turbinePlots and one unit annulus around the turbinePoints. The annuli increase out to a distance of maxDistance.

Caution, the function does some basic checks on the spatial objects but it is assumed that the points and polygons do not have any boundary, geometry, or other issues.

Value

Data frame of proportion of area searched for each annulus around each turbine point. distanceFromTurbine column represents outer radius of each annulus.

Examples

data(turbineSpatial)

propSearch <- getProportionAreaSearched(turbinePoints=turbineSpatial$turbinePoints,
turbineName='turbName',turbinePlots=turbineSpatial$turbinePlots,
 turbineMastRadius=2,maxDistance=10)

Calculate the start values to be passed to the optimizer.

Description

Calculate start values for weightedLikelihood or weightedDistribution.

Usage

getStartValue(x, distribution, w = rep(1, length(x)), ...)

Arguments

Details

This function is intended for internal purposes only and is called by weightedLikelihood and weightedDistribution. The function calculates the weighted mean and weighted variance and performs a method of moments approach to obtain start values for the likelihood estimation.

Value

Vector of estimated parameters for the distribution.

Examples

x <- rnorm(100,10,5)

getStartValue(x,'norm')
mean(x)
sd(x)

Data sets from Hallingstad et al. 2018

Description

List containing two data frames from Hallingstad et al. 2018

Usage

data(hallingstad)

Format

The carcass element is a data frame with 26 rows and 3 variables:

study

The name of the study for that wind farm.

species

The species of the carcass found.

distanceFromTurbine

The distance (meters) from the turbine of the location of the found carcass.

The weight element is a data frame with 2414 rows and 7 variables:

study

The name of the study for that wind farm.

distanceFromTurbine

The distance (meters) from the turbine for the outer radius of a one meter ring or annulus.

detectionProbability

The probability of detection, incorporating search efficiency and carcass persistence, for a given study and distance from turbine.

proportionAreaSearch

The proportion of area searched within the ring (annulus).

scaledNumberTurbine

The number of turbines at a study divided by the total number of turbines for all studies considered, does not change with distance from turbine.

scaledFatalityRate

The fatality rate at a study divided by the sum of the fatality rates across all studies considered, does not change with distance from turbine.

finalWeight

The product of detectionProbability*proportionAreaSearched*scaledNumberTurbine*scaledFatalityRate.

Details

A list of data frames, with two elements named carcass and weight.

References

Hallingstad EC, Rabie PA, Telander AC, Roppe JA, Nagy LR (2018) Developing an efficient protocol for monitoring eagle fatalities at wind energy facilities. PLoS ONE 13(12): e0208700. https://doi.org/10.1371/journal.pone.0208700

Calculate an area correction based on the Hull and Muir (2010) maximum distance and a triangular distribution as proposed by Huso and Dalthorp (2014).

Description

Calculate the maximum fall distance from a turbine using the regression model from Hull and Muir (2010). Calculate the carcass fall proabilities between one-unit increments of a right triangle distribution as proposed by Huso and Dalthorp (2014). Use the probabilities and proportion of area searched to calculate an area correction value.

Usage

hullMuirAreaCorrection(
  hubHeight,
  bladeRadius,
  lowerBound = 0,
  upperBound = Inf,
  proportionSearchDF,
  distanceCol,
  proportionCol,
  additionalCol = NULL,
  ...
)

Arguments

Details

The maximum Hull and Muir distances are calculated using hullMuirMaxDistance and the carcass fall probabilities are calculated using triangleProb. The probabilites are multipled by the proportion of area searched from proportionSearchDF by distance. These products are summed across distances by size class and additionalCol.

The distances in the distanceCol will be rounded to the nearest integer for matching up with the probabilities. The distances, hubHeight, and bladeRadius are assumed to be in the same units.

Value

Data frame of size class, additionalCol columns, and area correction

References

Hull, C. L., & Muir, S. (2010). Search areas for monitoring bird and bat carcasses at wind farms using a Monte-Carlo model. Australasian Journal of Environmental Management, 17(2), 77-87.

Huso, M. & Dalthorp,D (2014). Accounting for Unsearched Areas in Estimating Wind Turbine-Caused Fatality. The Journal of Wildlife Management. 78. 10.1002/jwmg.663.

See Also

hullMuirMaxDistance triangleProb

Examples

## proportion of area searched data
data(proportionAreaSearched)

hullMuirAreaCorrection(hubHeight = 87.5, bladeRadius = 62.5,
                       proportionSearchDF = proportionAreaSearched,
                       distanceCol = 'distanceFromTurbine',
                       proportionCol = 'proportionAreaSearched',
                       additionalCol = 'plotType')

## without additional columns but must separate the proportion of area searched
## data frame
hullMuirAreaCorrection(hubHeight = 87.5, bladeRadius = 62.5,
                       proportionSearchDF = subset(proportionAreaSearched, plotType == 'RP'),
                       distanceCol = 'distanceFromTurbine',
                       proportionCol = 'proportionAreaSearched')

hullMuirAreaCorrection(hubHeight = 87.5, bladeRadius = 62.5,
                       proportionSearchDF = subset(proportionAreaSearched, plotType == 'FULL'),
                       distanceCol = 'distanceFromTurbine',
                       proportionCol = 'proportionAreaSearched')

Calculate the Hull and Muir (2010) maximum distance

Description

Calculate the maximum fall distance from a turbine using the regression model from Hull and Muir (2010).

Usage

hullMuirMaxDistance(hubHeight, bladeRadius, ...)

Arguments

Details

Using the linear regression coefficients from Hull and Muir (2010), a maximum distance is calculated. This is done for three size classes (bats, small birds (SB), and large birds (LB)) separately.

It is assumed that hubHeight and bladeRadius have the same units.

Note: Hull and Muir (2010) used the range of 65 m < hubHeight < 94 m and 33 m < bladeRadius < 55 m. Anything outside of this range is extrapolation and should only be done with care.

Value

data frame of maximum distance by size class, hubHeight, and bladeRadius. Distance will be in the same units as were provided for hubHeight and bladeRadius

References

Hull, C. L., & Muir, S. (2010). Search areas for monitoring bird and bat carcasses at wind farms using a Monte-Carlo model. Australasian Journal of Environmental Management, 17(2), 77-87.

Examples

hubHeights <- rnorm(10, mean = 87.5, sd = 10)
bladeRadii <- rnorm(10, mean = 62.5, sd = 10)

hullMuirMaxDistance(hubHeight = hubHeights, bladeRadius = bladeRadii)

Print windAC object

Description

Print windAC object.

Usage

## S3 method for class 'windAC'
print(x, ...)

Arguments

Details

see calcAC

Value

Print for windAC object

Proportion of area searched example data set

Description

An example data set of the proportion of area searched within 1 m annuli around turbines at a wind farm.

Usage

data(proportionAreaSearched)

Format

A data frame with 400 rows and 3 variables:

plotType

Either FULL or RP, the type of plot searched at the turbine

distanceFromTurbine

The outer radius distance from the turbine, for the one unit annulus.

proportionAreaSearched

The proportion of area searched within the annulus at the turbine.

Details

Plots around turbines are searched for bird and bat carcasses. The area searched is summarized into the proportion of area searched with one unit annuli (or rings). The distance here corresponds to the outer radius of the annuli.

For example: plotType distanceFromTurbine portionAreaSearched RP 11 0.12441226

This row is corresponds to the annulus with outer radius of 11 and inner radius of 10. The proportion of area search on RP (road and pad) plot types is 0.12441226, or approximately 12 percent.

secondDerivative

Description

Computes numeric second derivatives (hessian) of an arbitrary multidimensional function at a particular location.

Usage

secondDerivative(loc, FUN, ..., eps = 1e-07)

Arguments

Details

This function uses the "5-point" numeric second derivative method advocated in numerous numerical recipe texts. During computation of the second derivative, FUN will be evaluated at locations within a hyper-elipsoid with cardinal radius 2*loc*(eps)^0.25.

A handy way to use this function is to call an optimization routine like nlminb with FUN, then call this function with the optimized values (solution) and FUN. This will yeild the hessian at the solution rather than the hessian at the previous step of the optimization.

Value

Matrix

Examples

func <- function(x){-x*x} # second derivative should be -2
secondDerivative(0,func)
secondDerivative(3,func)

func <- function(x){3 + 5*x^2 + 2*x^3} # second derivative should be 10+12x
secondDerivative(0,func)
secondDerivative(2,func)

func <- function(x){x[1]^2 + 5*x[2]^2} # should be rbind(c(2,0),c(0,10))
secondDerivative(c(1,1),func)
secondDerivative(c(4,9),func)

Calculate probabilities from a triangle distribution based on Hull and Muir (2010) maximum distance as proposed by Huso and Dalthorp (2014).

Description

Calculate the probabilities between one-unit increments of a right triangle distribution.

Usage

triangleProb(hubHeight, bladeRadius, lowerBound = 0, upperBound = Inf, ...)

Arguments

Details

A right triangle is constructed with the 90 degree corner at the origin in the first quadrant of the cartesian plane. The lowerBound will move the left edge of the triangle to the right. The upperBound will truncate the triangle distribution at that value.

The maximum horizontal distance is calculated using hullMuirMaxDistance. This is typically not a whole number and the ceiling is used. The maximum vertical distance is such that the area under the hypotenuse edge of triangle integrates to one. This is done using the equation for the area of a triangle.

The two points that make up the hypotenuse are used to calculate the slope and intercept of the line. The area under the line in one-unit increments is calculated using

\int_{x-1}^{x}mZ+b dZ = m(x-.5)+b

where m is the slope, b is the intercept, and x is a distance. Integrating between x-1 and x gives the probability between the one-unit increments.

All of this is done for three size classes (bats, small birds (SB), and large birds (LB)) separately. An additional size class (RAPTOR) is included and identical to the large bird result.

The floor function is applied to lowerBound.

It is assumed that hubHeight and bladeRadius have the same units.

Value

List of two data frames: the first has distances in one-unit increments (the outer distance), the probabilities between the distances, and a column indicating size class; the second gives the maximum distance of each size class.

References

Hull, C. L., & Muir, S. (2010). Search areas for monitoring bird and bat carcasses at wind farms using a Monte-Carlo model. Australasian Journal of Environmental Management, 17(2), 77-87.

Huso, M. & Dalthorp,D (2014). Accounting for Unsearched Areas in Estimating Wind Turbine-Caused Fatality. The Journal of Wildlife Management. 78. 10.1002/jwmg.663.

See Also

hullMuirMaxDistance

Examples

triResult <- triangleProb(hubHeight = 100, bladeRadius = 50, lowerBound = 0)
names(triResult) ## list names
triResult$maxDist ## max distance for each size class
head(triResult$triDistProb)

Truncated Distributions

Description

Truncated probability density function, truncated cumulative density function, inverse truncated cumulative density function, and random variates from a truncated distribution.

Usage

dtrunc(x, distribution, tbound = c(-Inf, Inf), ..., log = FALSE)

ptrunc(
  q,
  distribution,
  tbound = c(-Inf, Inf),
  ...,
  lower.tail = TRUE,
  log.p = NULL
)

qtrunc(
  p,
  distribution,
  tbound = c(-Inf, Inf),
  ...,
  lower.tail = TRUE,
  log.p = NULL
)

rtrunc(n, distribution, tbound = c(-Inf, Inf), ...)

Arguments

Details

The non truncated distribution functions are assumed to be available. For example if the normal distribution is desired then used distribution='norm', the functions then look for 'qnorm', 'pnorm', etc.

The max(tbound) and min(tbound) are considered the upper and lower truncation bounds, respectively.

The random variates are produced using the direct method (see Casella and Berger 2002).

Value

dtrunc returns a vector of densities.

ptrunc returns a vector of probabilities.

qtrunc returns a vector of quantiles.

rtrunc returns a vector of random variates.

References

G. Casella and R. L. Berger. Statistical inference. Vol. 2. Duxbury Pacific Grove, CA, 2002.

Examples

## dtrunc
# not truncted
dnorm(5,mean=5)
# truncated
dtrunc(x=5,distribution='norm',tbound=c(4,5.5),mean=5)



## ptrunc
#not truncated
pgamma(2,shape=3,rate=2)
# truncated
ptrunc(2, distribution = 'gamma', tbound=c(1,5),shape=3,rate=2)

## upper tail
# not truncated
pgamma(2,shape=3,rate=2,lower.tail=FALSE)
# truncated
ptrunc(2,distribution='gamma',tbound=c(1,5),shape=3,rate=2,lower.tail=FALSE)

## qtrunc
#not truncated
qnorm(p=.975)
# truncted
qtrunc(p=.975,distribution='norm',tbound=c(0,1))

## upper tail
# not truncted
qnorm(p=.975,lower.tail=FALSE)
# truncated
qtrunc(p=.975,distribution='norm',tbound=c(0,1),lower.tail=FALSE)

## rtrunc
rtrunc(n=5, distribution = 'gamma', tbound=c(2,5),shape=3,rate=2)

Data sets of spatial area searched at a wind farm

Description

List containing two spatial data frames

Usage

data(turbineSpatial)

Format

An object of class list of length 2.

Details

A list of data frames, with two elements named turbinePlots and turbinePoints. These are fictious data for example purposes only. The turbinePlots element is a spatial polygon data frame with the polygons being the area searched round the turbines. The turbinePoints element is a spatial points data frame with the turbine locations.

weight function

Description

Generic weight function for use with estWD.

Usage

weightFun(
  x,
  propTable,
  type,
  typeCol,
  distanceCol,
  propCol,
  xFun = ceiling,
  maxDistance = NULL,
  ...
)

Arguments

Details

The weightedDistribution function requires the weights be described using a function. This allows integration to happen.

Typically propTable has integer values for the distances, but the function needs to take in any numeric values, the xFun function is how any numeric value can be matched up to the values in propTable. If the distances in propTable correspond to the outer radius of the annuli, for calculating proportion of area searched, then the ceiling is appropriate. If the distances in propTable correspond to the inner radius of the annuli then the floor might be more appropriate.

Value

Numeric vector of weights with length equal to length(x), and with a 1:1 relationship to the values in x.

See Also

estWD

Examples

data(proportionAreaSearched)

d <- c(-300.23,14.3,16,75)

## RP proportion of area searched
weightFun(x=d,propTable=proportionAreaSearched,type='RP',typeCol='plotType',
distanceCol='distanceFromTurbine',propCol='proportionAreaSearched')
#[1] 0.00000000 0.08896480 0.08308577 0.01709869

## FULL plot proportion of area searched
weightFun(x=d,propTable=proportionAreaSearched,type='FULL',typeCol='plotType',
distanceCol='distanceFromTurbine',propCol='proportionAreaSearched')
# [1] 0 1 1 1

### with a max distance restriction
## RP proportion of area searched
weightFun(x=d,propTable=proportionAreaSearched,type='RP',typeCol='plotType',
distanceCol='distanceFromTurbine',propCol='proportionAreaSearched',maxDistance=40)
# [1] 0.00000000 0.08896480 0.08308577 0.00000000

## FULL plot proportion of area searched
weightFun(x=d,propTable=proportionAreaSearched,type='FULL',typeCol='plotType',
distanceCol='distanceFromTurbine',propCol='proportionAreaSearched',maxDistance=40)
# [1] 0 1 1 0

A package for calculating area correction values for fatality estimation at wind farms.

Description

Post-construction fatality monitoring studies at wind facilities are based on data from searches for bird and bat carcasses in plots beneath turbines. Bird and bat carcasses can fall outside of the search plot. Bird and bat carcasses from wind turbines often fall outside of the searched area. To compensate, area correction (AC) estimations are calculated to estimate the percentage of fatalities that fall within the searched area versus those that fall outside of it. This package provides two likelihood based methods and one physics based method (Hull and Muir (2010), Huso and Dalthorp (2014)) to estimate the carcass fall distribution. There are also functions for calculating the proportion of area searched within one unit annuli, log logistic distribution functions, and truncated distribution functions.

The two likelihood methods are the truncated weighted likelihood (estTWL) and the weighted distribution (estWD). Both use carcass distances from the turbine, accounting for unequal detection by distance, to estimate the distance distribution. Alternatively, a right triangle distribution can be used for the carcass density distribution with the max distance estimated (hullMuirMaxDistance from the regression from Hull and Muir (2010) as proposed by Huso and Dalthorp (2014).

The area correction value is calculated from the combination of the carcass distance density and the proportion of area searched at each distance. The function getProportionAreaSearched uses the sf package to do this from turbine points spatial data and search area polygons. The functions geometricRectanglePropSearchTable and geometricRoadPadPropSearchTable also calculate proportion of area searched but assuming perfect geometric shapes, meaning no spatial data is required.

Search areas are often irregular. proportionAreaSearched summarizes the area searched into the proportion of area searched with one unit annuli (or ring).

Two sets of distribution functions are available. Log logistic distribution functions (see dllog). These are a transformation of the logistic distribution and use the base R functions (see dlogis). The second is truncation functions (see dtrunc), that provide truncation for R function distributions.

Example data sets:

• carcassDistance: Example data set of carcass distances found during searches under wind turbines.

• hallingstad: List containing two data frames from Hallingstad et al. 2018. Data frames list carcass and detection probability data for a few wind farm studies.

• proportionAreaSearched: An example data set of the proportion of area searched around turbines at a wind farm.

References

Hallingstad EC, Rabie PA, Telander AC, Roppe JA, Nagy LR (2018) Developing an efficient protocol for monitoring eagle fatalities at wind energy facilities. PLoS ONE 13(12): e0208700. https://doi.org/10.1371/journal.pone.0208700

Huso, M. & Dalthorp,D (2014). Accounting for Unsearched Areas in Estimating Wind Turbine-Caused Fatality. The Journal of Wildlife Management. 78. 10.1002/jwmg.663.

Hull, C. L., & Muir, S. (2010). Search areas for monitoring bird and bat carcasses at wind farms using a Monte-Carlo model.Australasian Journal of Environmental Management, 17(2), 77-87.