adehabitat (version 1.3)

kernelUD: Estimation of Kernel Home-Range

Description

kernelUD is used to estimate the utilization distribution (UD) of animals monitored by radio-tracking, with the classical kernel method. getvolumeUD and kernel.area provide utilities for home-range size estimation. getverticeshr stores the home range contour as objects of class area in a list of class kver, with one component per animal. plot.kver is used to display these home-range contours. kernelbb is used to fit an utilization distribution using the brownian bridge approach of the kernel method (for autocorrelated relocations; Bullard, 1991).

Usage

kernelUD(xy, id = NULL, h = "href", grid = 40, same4all = FALSE,
         hlim = c(0.1, 1.5), kern = "bivnorm")
print.khr(x, ...)
image.khr(x, axes = FALSE, mar = c(0,0,2,0),
          addcontour = TRUE, addpoints = TRUE, ...)
plotLSCV(x)
getvolumeUD(x)
kernel.area(xy, id, h = "href", grid=40,
            same4all = FALSE, hlim = c(0.1,1.5), kern = "bivnorm",
            levels = seq(20,95, by = 5),
            unin = c("m", "km"),
            unout = c("ha", "km2", "m2"))
getverticeshr(x, lev = 95)
plot.kver(x, which = names(x), colpol=rainbow(length(which)),
          colborder=rep("black", length(which)), lwd = 2,
          add=FALSE, ...)
kernelbb(tr, sig1, sig2, grid = 40, same4all=FALSE, byburst=FALSE)

Arguments

xy

a data frame with two columns (x and y coordinates of the animal relocations)

id

an optional factor giving the animals identity associated to xy

h

a character string or a number. If h is set to "href", the ad hoc method is used for the smoothing parameter (see details). If h is set to "LSCV", the least-square cross validation meth

grid

a number giving the size of the grid on which the UD should be estimated. Alternatively, this parameter may be an object of class asc (see examples)

same4all

logical. If TRUE, the same grid is used for all animals. If FALSE, one grid per animal is used

hlim

a numeric vector of length two. If h = "LSCV", the function minimizes the cross-validation criterion for values of h ranging from hlim[1]*href to hlim[2]*href, where href is the smoothing

kern

a character string. If "bivnorm", a bivariate normal kernel is used. If "epa", an Epanechnikov kernel is used.

x

an object of class khr returned by kernelUD. For plot.kver, an object of class kver returned by getverticeshr

which

a vector of character indicating the polygons to be plotted

colpol

a vector of the color for filling the polygon. The default, NA, is to leave polygons unfilled

colborder

a vector of the color to draw the border. The default. Use border = NA to omit borders

lwd

the border width, a positive number

add

logical. if TRUE, the polygons are added to a previous plot

axes

logical. Whether the axes are to be plotted

mar

the margin parameter (see help(par))

addcontour

logical. if TRUE, contours are drawn on the graphics

addpoints

logical. if TRUE, the animal relocations are drawn on the graphics

levels

a vector of percentage levels for home-range size estimation

unin

the units of the relocations coordinates. Either "m" for meters (default) or "km" for kilometers

unout

the units of the output areas. Either "m2" for square meters, "km2" for square kilometers or "ha" for hectares (default)

lev

the percentage level for home range contour estimation.

tr

an object of class traj.

sig1

first smoothing parameter for the brownian bridge method (related to the imprecision of the relocations).

sig2

second smoothing parameter for the brownian bridge method (related to the speed of the animals).

byburst

logical. Whether the brownian bridge estimation should be done by burst.

...

additionnal parameters to be passed to the generic functions print and image

Value

• The class khr is a class grouping three sub-classes, khrud, kbbhrud and khrudvol: kernelUD returns a list of the class khrud. This list has one component per animal (named as the levels of argument id). Each component is itself a list, with the following sub-components:
• UD
  an object of class asc, with the values of density probability in each cell of the grid
• h
  if LSCV is not used, the value of the smoothing parameter. if LSCV is used, a list with three components: [object Object],[object Object],[object Object]
• locs
  The relocations used in the estimation procedure.
• hmeth
  The argument h of the function kernelUD
• kernelbb returns an object of class kbbhrud, with the same components as lists of class khrud. getvolumeUD returns a list of class khrvol, with the same components as lists of class khrud. kernel.area returns a data frame of subclass "hrsize", with one column per animal and one row per level of estimation of the home range. getverticeshr returns an object of class kver.

Details

The Utilization Distribution (UD) is the bivariate function giving the probability density that an animal is found at a point according to its geographical coordinates. Using this model, one can define the home range as the minimum area in which an animal has some specified probability of being located. The functions used here correspond to the approach described in Worton (1995). The kernel method has been recommended by many authors for the estimation of the utilization distribution (e.g. Worton, 1989, 1995). The default method for the estimation of the smoothing parameter is the ad hoc method, i.e. for a bivariate normal kernel $$h = \sigma n^{- \frac{1}{6}}$$ where $$\sigma = 0.5 (\sigma(x)+\sigma(y))$$ which supposes that the UD is bivariate normal. If an Epanechnikov kernel is used, this value is multiplied by 1.77 (Silverman, 1986, p. 86). Alternatively, the smoothing parameter h may be computed by Least Square Cross Validation (LSCV). The estimated value then minimizes the Mean Integrated Square Error (MISE), i.e. the difference in volume between the true UD and the estimated UD. Note that the cross-validation criterion cannot be minimized in some cases. According to Seaman and Powell (1998) "This is a difficult problem that has not been worked out by statistical theoreticians, so no definitive response is available at this time" (see Seaman and Powell, 1998 for further details and tricky solutions). plotLSCV allows to have a diagnostic of the success of minimization of the cross validation criterion (i.e. to know whether the minimum of the CV criterion occurs within the scanned range). Finally, the UD is then estimated over a grid. The default kernel is the bivariate normal kernel, but the Epanechnikov kernel, which requires less computer time is also available for the estimation of the UD. The function getvolumehr modifies the UD component of the object passed as argument, so that the contour of the UD displayed by the functions contour and image.khr corresponds to the different percentage levels of home-range estimation (see examples). In addition, this function is used in the function kernel.area, to compute the home-range size. Note, that the function plot.hrsize (see the help page of this function) can be used to display the home-range size estimated at various levels. The function kernelbb uses the brownian bridge approach to estimate the UD with autocorrelated relocations (Bullard, 1991). Instead of simply smoothing the relocation pattern, it takes into account the fact that between two successive relocations r1 and r2, the animal has moved through a connected path, which is not necessarily linear. A brownian bridge estimates the density of probability that this path passed through any point of the study area, given that the animal was located at the point r1 at time t1 and at the point r2 at time t2, with a certain amount of inaccuracy (see Examples). Brownian bridges are placed over the different sections of the traject, and these functions are then summed over the area. The brownian bridge approach therefore smoothes a trajectory. The brownian bridge estimation relies on two smoothing parameters, sig1 and sig2. The parameter sig1 is related to the speed of the animal, and describes how far from the line joining two successive relocations the animal can go during one time unit (here the time is measured in second). The larger this parameter is, and the more wiggly the trajectory is likely to be. The parameter sig2 is equivalent to the parameter h of the classical kernel method: it is related to the inaccuracy of the relocations (See examples for an illustration of the smoothing parameters).

References

Bullard, F. (1991) Estimating the home range of an animal: a Brownian bridge approach. Master of Science, University of North Carolina, Chapel Hill. Silverman, B.W. (1986) Density estimation for statistics and data analysis. London: Chapman & Hall. Worton, B.J. (1989) Kernel methods for estimating the utilization dirstibution in home-range studies. Ecology, 70, 164--168. Worton, B.J. (1995) Using Monte Carlo simulation to evaluate kernel-based home range estimators. Journal of Wildlife Management, 59,794--800. Seaman, D.E. and Powell, R.A. (1998) Kernel home range estimation program (kernelhr). Documentation of the program. ftp://ftp.im.nbs.gov/pub/software/CSE/wsb2695/KERNELHR.ZIP.

See Also

import.asc for additionnal informations on objects of class asc, mcp for estimation of home ranges using the minimum convex polygon, and for help on the function plot.hrsize.

Examples

data(puechabon)
loc <- puechabon$locs[, c("X", "Y")]
id <- puechabon$locs[, "Name"]

## Estimation of UD for the four animals
ud <- kernelUD(loc, id)
ud

image(ud) ## Note that the contours
          ## corresponds to values of probability density
udvol <- getvolumeUD(ud)
image(udvol)
## Here, the contour corresponds to the
## home ranges estimated at different probability
## levels (i.e. the contour 90 corresponds to the 90 percent
## kernel home-range)
## udvol describes, for each cell of the grid,
## the smaller home-range to which it belongs 

## Calculation of the 95 percent home range
ver <- getverticeshr(ud, 95)
elev <- getkasc(puechabon$kasc, "Elevation") # Map of the area
image(elev)
plot(ver, add=TRUE)
legend(696500, 3166000, legend = names(ver), fill = rainbow(4))


## Example of estimation using LSCV
udbis <- kernelUD(loc, id, h = "LSCV")
image(udbis)

## Compare the estimation with ad hoc and LSCV method
## for the smoothing parameter
(cuicui1 <- kernel.area(loc, id)) ## ad hoc
plot(cuicui1)
(cuicui2 <- kernel.area(loc, id, h = "LSCV")) ## LSCV
plot(cuicui2)

## Diagnostic of the cross-validation
plotLSCV(udbis)


## Use of the same4all argument: the same grid
## is used for all animals
udbis <- kernelUD(loc, id, same4all = TRUE)
image(udbis)

## And finally, estimation of the UD on a map
## (e.g. for subsequent analyses on habitat selection)
elev <- getkasc(puechabon$kasc, "Elevation")
opar <- par(mfrow = c(2, 2), mar = c(0, 0, 2, 0))
cont <- getcontour(elev)

for (i in 1:length(udbis)) {
   image(elev, main = names(udbis)[i], axes = FALSE)
   points(udbis[[i]]$locs, pch = 21, bg = "white", col = "black")
}


## Measures the UD in each pixel of the map
udbis <- kernelUD(loc, id, grid = elev)
opar <- par(mfrow = c(2, 2), mar = c(0, 0, 2, 0))
for (i in 1:length(udbis)) {
  image(udbis[[i]]$UD, main = names(udbis)[i], axes = FALSE)
  box()
  polygon(cont[, 2:3])
}
par(opar)




###############################################
###############################################
###############################################
###
###         Kernel estimation: a brownian
###               bridge approach

## loads the data
data(puechcirc)

## gets one circuit
x <- getburst(puechcirc, burst = "CH930824")

## fits the home range
(tata <- kernelbb(x, 10, 10, same4all = TRUE))
image(tata)
lines(x$x, x$y, lwd = 2, col = "red")

## Image of a brownian bridge. Fit with two relocations:
fac <- factor(c("a", "a"))
xx <- c(0,1)
yy <- c(0,1)
date <- c(0,1)
class(date) <- c("POSIXt", "POSIXct")
tr <- as.traj(fac, data.frame(x = xx,y = yy), date)

## Use of different smoothing parameters
sig1 <- c(0.05, 0.1, 0.2, 0.4, 0.6)
sig2 <- c(0.05, 0.1, 0.2, 0.5, 0.7)

y <- list()
for (i in 1:5) {
  for (j in 1:5) {
     k <- paste("s1=", sig1[i], ", s2=", sig2[j], sep = "")
     y[[k]]<-kernelbb(tr, sig1[i], sig2[j])[[1]]$UD
   }
 }

## Displays the results
opar <- par(mar = c(0,0,2,0), mfrow = c(5,5))
foo <- function(x)
   {
     image(y[[x]], main = names(y)[x], axes = F)
     points(tr[,c("x","y")], pch = 16, col = "red")
   }
lapply(1:length(y), foo)

par(opar)