Comprehensive Bivariate Kernel Density Estimation Assignment with R

This assignment demonstrates bivariate kernel density estimation using the R programming language. The solution includes R code for generating example data, conducting a CTR (Capture-Time-to-Recapture) analysis, and fitting different dispersal kernels (log-normal, Weibull, exponential, and normal) to the censored data. The code calculates Ordinary Least Squares (OLS) scores and uses the Akaike Information Criterion (AIC) to select the best-fitting model. The assignment also includes density plots and probability plots to compare the estimated and generating kernels. The document provides a complete and detailed solution with references to relevant literature. The R code is well-commented, making it easy to understand each step of the analysis. The assignment covers various statistical concepts and programming techniques essential for data analysis.

Running head: BIVARIATE KERNEL DENSITY ESTIMATION
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Bivariate Kernel Density Estimation Using R tool
Student Name
Institution

BIVARIATE KERNEL DENSITY ESTIMATION
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The programming tool used for this assignment is R programming tool. The R code has
detailed comment on every step taken above the code.
R code
# R code for conducting a CTR analysis using generated data
# Load dependencies
# Code requires package fdrtool & survival to be in library
# install.packages("fdrtool")
require (fdrtool);require(survival)
#create data
# Next step is to generate example data "radio tagged distance", stored as object
x
mean(x)
sd(x)
# set random seed
set.seed(2009)
# generate data from lognormal distribution, 250 tracked seeds
# meanlog=log(15), sdlog=log(1.1)
x=rlnorm(250,log(15),log(1.1))
# truncate data after 20 units to create "tracked distances"
# with 20 m search radius
xtrunc=x[x<20]
# Prepare data for CTR
CTRdata=data.frame(

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# all seeds that went beyond 20 meters are treated as censored events
# (distance > 20 meter)
d=c(xtrunc,rep(20,250-length(xtrunc))),
# classify events, found seeds = 1, censored seeds = 0
evnt=c(rep(1,length(xtrunc)),rep(0,250-length(xtrunc))))
# fitsurvival function
CTR_function=survfit(Surv(CTRdata$d, event=CTRdata$evnt) ~ 1)
# return survival probabilties (P) corresponding to distances (D)
P=summary(CTR_function)$surv;D=summary(CTR_function)$time
#Define dispersal kernels
# these kernels are then fit through OLS (Ordinary Least Squares) to objects P &D
# log normal
SSLN=function(param){
Ex=1-plnorm(D,meanlog=param[1],sdlog=param[2])
sum((Ex-P)^2)
}
# Weibull
SSW=function(param){
Ex=1-pweibull(D,shape=param[1],scale=param[2])
sum((Ex-P)^2)
}
# exponential
SSEX=function(param){

BIVARIATE KERNEL DENSITY ESTIMATION
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Ex=1-pexp(D,rate=param)
sum((Ex-P)^2)
}
# Normal
SSN=function(param){
Ex=1-phalfnorm(D,theta=param[1])
sum((Ex-P)^2)
}
#Obtain kernels with reconstructed tails
#Fit each model to the censored data and store
fitLN=optim(c(1,1),SSLN)
LNpsave=c(fitLN$par[1],fitLN$par[2])
fitW=optim(c(1.2,55),SSW)
WBpsave=c(fitW$par[1],fitW$par[2])
# Note: above the OLS function was optimized with the Nelder-Mead algorithm
# however this algorithm is optimal for optimization problems of 2 Dimensions
# or greater. The quasi-Newton method 'BFGS' is better suited for 1 D (or 1
# parameter) problems. Alternatively the function 'optimize' can be used
# however result will be the same either way.
fitN=optim(c(0.05),SSN,method="BFGS")
Npsave=c(fitN$par[1])
fitEX=optim(c(0.01),SSEX,method="BFGS")
EXpsave=c(fitEX$par[1])

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# choose best model based on AIC score
OLSscores=c(fitLN$value,fitW$value,
fitN$value,fitEX$value)
# vector with number of parameters for each model
pars=c(2,2,1,1)
# calculating AIC from sum of squares
AICscores=(1500*log(OLSscores/1500) + 2*pars)
#FINAL
#selecting bestfitting model
Bestfit=c("LN","WB","T","N","EX")[which(AICscores==min(AICscores))]
#checking difference in between estimated and generating kernel
par(cex.axis=0.9,cex.lab=1.1,las=1,mar=c(4,5,1,1),mfrow=c(2,1))
# Density plots
curve(dlnorm(x,log(25),log(3)),0,150,col='grey',xlab="distance",
ylab="probabilty density",lwd=2)
curve(dlnorm(x,fitLN$par[1],fitLN$par[2]),col='black',add=T,lty='dashed',lwd=2)
legend(100,0.010,legend=c('True', 'CTR derived'),lty=c('solid','dashed'),
col=c('grey','black'),bty='n',lwd=2)
# Probability P of dispersal beyond distance D
curve(1-plnorm(x,log(50),log(3)),0,150,col='grey',xlab="D",
ylab="P", lwd=2)
curve(1-plnorm(x,fitLN$par[1],fitLN$par[2]),col='black',add=T,lty='dashed',lwd=2)
legend (100,0.90,legend=c('True', 'CTR derived'),lty=c('solid','dashed'),

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col=c('grey','black'),bty='n',lwd=2)
R Output

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References
Gentleman, R. (2009). R programming for bioinformatics. Boca Raton: CRC Press.
Baker, K., & Trietsch, D.(2009) Principles of sequencing and scheduling.
Braun, J., & Murdoch, D.(2012). A first course in statistical programming with R.
Grolemund, G., & Wickham, H. (2006). Hands-on programming with R.