\documentclass{article}
% \VignetteIndexEntry{adehabitatHR: classes and methods for home range estimation}

\usepackage{graphicx}
\usepackage[colorlinks=true,urlcolor=blue]{hyperref}

\usepackage{color}
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\usepackage{Sweave}
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\title{ Home Range Estimation in R:\\
the {\tt adehabitatHR} Package }
\author{Clement Calenge,\\
  Office national de la classe et de la faune
  sauvage\\
  Saint Benoist -- 78610 Auffargis -- France.}
\date{Mar 2019}

\begin{document}

\maketitle
\tableofcontents


<<echo=FALSE>>=
owidth <- getOption("width")
options("width"=80)
ow <- getOption("warn")
options("warn"=-1)
.PngNo <- 0
wi <- 600
pt <- 25
@

<<label=afig,echo=FALSE,eval=FALSE>>=
.PngNo <- .PngNo + 1; file <- paste("Fig-bitmap-",
          .PngNo, ".png", sep="")
png(file=file, width = wi, height = wi, pointsize = pt)
@

<<label=zfig,echo=FALSE,eval=FALSE>>=
dev.null <- dev.off()
cat("\\includegraphics[height=7cm,width=7cm]{", file, "}\n\n", sep="")
@

<<label=zfigg,echo=FALSE,eval=FALSE>>=
dev.null <- dev.off()
cat("\\includegraphics[height=14cm,width=14cm]{", file, "}\n\n", sep="")
@


\section{History of the package adehabitatHR}

The package {\tt adehabitatHR} contains functions dealing with home-range
analysis that were originally available in the package {\tt
  adehabitat} (Calenge, 2006).  The data used for such analysis are
generally relocation data collected on animals monitored using VHF or
GPS collars.\\

I developped the package {\tt adehabitat} during my PhD (Calenge, 2005)
to make easier the analysis of habitat selection by animals.  The
package {\tt adehabitat} was designed to extend the capabilities of
the package {\tt ade4} concerning studies of habitat selection by
wildlife.\\

Since its first submission to CRAN in September 2004, a lot of
work has been done on the management and analysis of spatial data in
R, and especially with the release of the package {\tt sp} (Pebesma
and Bivand, 2005).  The package {\tt sp} provides classes of data that
are really useful to deal with spatial data...\\

In addition, with the increase of both the number (more than 250
functions in Oct. 2008) and the diversity of the functions in the
package \texttt{adehabitat}, it soon became apparent that a reshaping
of the package was needed, to make its content clearer to the users.  I
decided to ``split'' the package {\tt adehabitat} into four packages:\\

\begin{itemize}
\item {\tt adehabitatHR} package provides classes and methods for dealing
  with home range analysis in R.
\item {\tt adehabitatHS} package provides classes and methods for dealing
  with habitat selection analysis in R.
\item {\tt adehabitatLT} package provides classes and methods for dealing
  with animals trajectory analysis in R.
\item {\tt adehabitatMA} package provides classes and methods for dealing
  with maps in R.\\
\end{itemize}

We consider in this document the use of the package {\tt adehabitatHR} to
deal with home-range analysis.  All the methods available in
\texttt{adehabitat} are also available in \texttt{adehabitatHR}, but the
classes of data returned by the functions of \texttt{adehabitatHR} are
completely different from the classes returned by the same functions
in \texttt{adehabitat}.  Indeed, the classes of data returned by the
functions of \texttt{adehabitatHR} have been designed to be compliant with
the classes of data provided by the package \texttt{sp}.\\


Package {\tt adehabitatHR} is loaded by
<<echo=TRUE,print=FALSE>>=
library(adehabitatHR)
@

<<echo=FALSE,print=FALSE>>=
suppressWarnings(RNGversion("3.5.0"))
set.seed(13431)
@


\section{Basic summary of the functions of the packages}

The package \texttt{adehabitatHR} implements the following home range
estimation methods:\\

\begin{enumerate}
\item The Minimum Convex Polygon (Mohr, 1947);
\item Several kernel home range methods:
  \begin{itemize}
  \item The ``classical'' kernel method (Worton, 1989)
  \item the Brownian bridge kernel method (Bullard, 1999, Horne et
    al. 2007);
  \item The Biased random bridge kernel method, also called
    ``movement-based kernel estimation'' (Benhamou and Cornelis, 2010,
    Benhamou, 2011);
  \item the product kernel algorithm (Keating and Cherry, 2009).
  \end{itemize}
\item Several home-range estimation methods relying on the calculation
  of convex hulls:
  \begin{itemize}
  \item The modification by Kenward et al. (2001) of the
    single-linkage clustering algorithm;
  \item The three LoCoH (Local Convex Hull) methods developed by Getz et
    al. (2007)\\
  \item The characteristic hull method of Downs and Horner (2009).
  \end{itemize}
\end{enumerate}

I describe these functions in this vignette.


\subsection{The classes of data required as input}

The package \texttt{adehabitatHR}, as all the brother ``adehabitat''
packages, relies on the classes of data provided by the
package \texttt{sp}.  Two types of data can be accepted as input of
the functions of the package \texttt{adehabitatHR}:\\

\begin{itemize}
\item objects of the class \texttt{SpatialPoints} (package
  \texttt{sp}).  These objects are designed to store the coordinates
  of points into space, and are especially well designed to store the
  relocations of \textbf{one} animal monitored using radio-tracking.\\

\item objects of the class \texttt{SpatialPointsDataFrame} (package
  \texttt{sp}).  These objects are also designed to store the
  coordinates of points into space, but also store additionnal
  information concerning the points (attribute data).  This class of
  data is especially well designed to store the relocations of several
  animals monitored using radio-tracking.  In this case, the attribute
  data should contain only one variable: a factor indicating the
  identity of the animals.\\
\end{itemize}

All the home-range estimation methods included in the
package can take an object belonging to one of these two classes as
the argument \texttt{xy}.  For example, generate a sample of
relocations uniformly distributed on the plane:

<<>>=
xy <- matrix(runif(60), ncol=2)
head(xy)
@

The object \texttt{xy} is a matrix containing 30 relocations of a
virtual animal.  Convert it into the class \texttt{SpatialPoints}:

<<>>=
xysp <- SpatialPoints(xy)
@

Consider the function \texttt{clusthr}, which implements the
single-linkage clustering algorithm (see next sections).  Using the
function \texttt{clusthr} on this object returns an object of class
\texttt{SpatialPolygonsDataFrame}:

<<>>=
clu <- clusthr(xysp)
class(clu)
@

Show the results:

<<label=fig1,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(clu)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<fig1>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig1>>
<<zfig>>
@
\end{center}


Now, consider the relocations of another animal:

<<>>=
xy2 <- matrix(runif(60), ncol=2)
@

We can bind the two matrices together:

<<>>=
xyt <- rbind(xy, xy2)
@

and generate a vector containing the ID of the animals for each
relocation of the object \texttt{xyt}:

<<>>=
id <- gl(2,30)
@

We can convert the object \texttt{id} into a
\texttt{SpatialPointsDataFrame}:

<<>>=
idsp <- data.frame(id)
coordinates(idsp) <- xyt
class(idsp)
@

And we can use the same function \texttt{clusthr} on this new object:

<<>>=
clu2 <- clusthr(idsp)
class(clu2)
clu2
@

An object of the class \texttt{"MCHu"} is basically a list of objects
of class \texttt{SpatialPolygonsDataFrame}, with one element per
animal.  Thus:

<<>>=
length(clu2)
class(clu2[[1]])
class(clu2[[2]])
@

Although each \texttt{SpatialPolygonsDataFrame} of the list can be
easily handled by the functions of the package \texttt{sp} and
relatives (\texttt{maptools}, etc.), the package \texttt{adehabitatHR} also
contains functions allowing to deal with such lists.  For example:

<<label=fig2,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(clu2)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<fig2>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig2>>
<<zfig>>
@
\end{center}


Therefore, the same home-range estimating function can accept an
object of class \texttt{SpatialPoints} (one animal) or
\texttt{SpatialPointsDataFrame} (several animals).  This is the case of
all home-range estimation methods in \texttt{adehabitatHR}.


\subsection{Our example data set}

I will illustrate the use of the functions in \texttt{adehabitatHR} using
a subset of the data collected by Daniel Maillard (1996;
\textit{Office national de la chasse et de la faune sauvage},
Montpellier) on the wild boar at Puechabon (near Montpellier, South of
France).  First load the dataset \texttt{puechabonsp}:

<<>>=
data(puechabonsp)
names(puechabonsp)
@

This dataset is a list with two components: the component
\texttt{relocs} is a \texttt{SpatialPointsDataFrame} containing the
relocations of four wild boars monitored using radio-tracking.  The
following variables are available for each relocations:

<<>>=
head(as.data.frame(puechabonsp$relocs))
@

The first variable is the identity of the animals, and we will use it
in the estimation process.\\

The component \texttt{map} is a \texttt{SpatialPixelsDataFrame}
containing the maps of four environmental variables on the study area
(Elevation, slope, aspect and herbaceous cover).  Have a look at the
distribution of the relocations on an elevation map:

<<label=fig3,echo=FALSE,fig=FALSE,eval=FALSE,keep.source=TRUE>>=
## Map of the elevation
image(puechabonsp$map, col=grey(c(1:10)/10))
## map of the relocations
plot(puechabonsp$relocs, add=TRUE,
     col=as.data.frame(puechabonsp$relocs)[,1])
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE,keep.source=TRUE>>=
<<fig3>>
@

\begin{center}
<<results=tex,echo=FALSE,keep.source=TRUE>>=
<<afig>>
<<fig3>>
<<zfig>>
@
\end{center}




\section{The Minimum Convex Polygon (MCP)}

\subsection{The method}

The MCP is probably the most widely used estimation method.  This
method consists in the calculation of the smallest convex polygon
enclosing all the relocations of the animal.  This polygon is
considered to be the home range of the animal.  Because
the home range has been defined as the area traversed by the animal
during its \textbf{normal} activities of foraging, mating and caring
for young (Burt, 1943), a common operation consists to first remove a
small percentage of the relocations farthest from the centroid of the
cloud of relocations before the estimation.  Indeed, the animal may
sometimes make occasionnal large moves to unusual places outside its
home range, and this results into outliers that cannot be considered
as ``normal activities''.\\

\subsection{The function \texttt{mcp} and interaction with the package
\texttt{sp}}
\label{sec:mcpmeth}

The function \texttt{mcp} allows the MCP estimation.  For example,
using the \texttt{puechabonsp} dataset, we estimate here the MCP of
the monitored wild boars (after the removal of 5\% of extreme
points).  Remember that the first column of
\texttt{puechabonsp\$relocs} is the identity of the animals:

<<>>=
data(puechabonsp)
cp <- mcp(puechabonsp$relocs[,1], percent=95)
@

The results are stored in an object of class
\texttt{SpatialPolygonsDataFrame}:

<<>>=
class(cp)
@

Therefore the functions available in the package \texttt{sp} and
related packages (\texttt{maptools}, etc.) are available to deal with
these objects.  For example, the function \texttt{plot}:


<<label=fig4,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(cp)
plot(puechabonsp$relocs, add=TRUE)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<fig4>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig4>>
<<zfig>>
@
\end{center}

Similarly, the function \texttt{writePolyShape} from the package
\texttt{maptools} can be used to export the object \texttt{cp} to a
shapefile, so that it can be read into a GIS:


<<>>=
library(maptools)
@

<<eval=FALSE>>=
writePolyShape(cp, "homerange")
@

The resulting shapefils can be read in any GIS.

\subsection{Overlay operations}

The function \texttt{over} of the package \texttt{sp} allows to
combine points (including grid of pixels) and polygons for
points-in-polygon operations (see the help page of this function).
For example, the element \texttt{map} of the dataset
\texttt{puechabonsp} is a map of the study area according to four
variables:

<<>>=
head(as.data.frame(puechabonsp$map))
@

This object is of the class \texttt{SpatialPixelsDataFrame}.
We may identify the pixels of this map enclosed in each home range
using the function \texttt{over} from the package \texttt{sp}.
For example, to identify the pixels enclosed in the home range of the
second animal:

<<>>=
enc <- over(puechabonsp$map, cp[,2], fn=function(x) return(1))
head(enc)
@

The result is a data.frame with one variable containing the value 1 if
a pixel of the map is included in a home range, and NA if the pixel is
outside the home-range.  We may transform this vector into a
\texttt{SpatialPixelsDataFrame}, using the functions of the package
\texttt{sp}:

<<>>=
coordinates(enc) <- coordinates(puechabonsp$map)
gridded(enc) <- TRUE
@

The result is shown below, in black, on an elevation map:

<<label=fig5,echo=FALSE,fig=FALSE,eval=FALSE>>=
image(puechabonsp$map)
image(enc, add=TRUE, col="black", useRasterImage=FALSE)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<fig5>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig5>>
<<zfig>>
@
\end{center}

However, because the rasterization of home ranges is a common
operation, I included a function \texttt{hr.rast} to make this
operation more straightforward:

<<>>=
cprast <- hr.rast(cp, puechabonsp$map)
@

The resulting object is a \texttt{SpatialPixelsDataFrame} with one
column per animal:

<<label=sldkslsd,echo=FALSE,fig=FALSE,eval=FALSE>>=
par(mar=c(0,0,0,0))
par(mfrow=c(2,2))
for (i in 1:4) {
    image(cprast[,i], useRasterImage=FALSE)
    box()
}
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<sldkslsd>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<sldkslsd>>
<<zfig>>
@
\end{center}


\subsection{Computation of the home-range size}

The home range size is automatically computed by the function
\texttt{mcp}.  Thus, coercing the object returned by the function to
the class \texttt{data.frame} gives access to the home range size:

<<>>=
as.data.frame(cp)
@

The area is here expressed in hectares (the original units in the
object \texttt{puechabonsp\$relocs} were expressed in metres).  The
units of the results and of the original data are controlled by the
arguments \texttt{unin} and \texttt{unout} of the function
\texttt{mcp} (see the help page of the function \texttt{mcp}).  We did
not change the units in our example estimation, as the default
was a suitable choice.\\

In this example, we have chosen to exclude 5\% of the mst extreme
relocations, but we could have made another choice.  We may compute the
home-range size for various choices of the number of extreme
relocations to be excluded, using the function \texttt{mcp.area}:


<<label=fig6,echo=FALSE,fig=FALSE,eval=FALSE>>=
hrs <- mcp.area(puechabonsp$relocs[,1], percent=seq(50, 100, by = 5))
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<fig6>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig6>>
<<zfig>>
@
\end{center}


Here, except for Calou, the choice to exclude 5\% of the relocations
seems a sensible one.  In the case of Calou, it is disputable
to remove 20\% of the relocations to achieve an asymptote in the home
range size decrease.  20\% of the time spent in an area is no longer an
exceptionnal activity... and this area could be considered as part of
the home range.  Under these conditions, the choice to compute the MCP
with 95\% of the relocations for all animals seems a sensible one.\\

Note that the results of the function \texttt{mcp.area} are stored in
a data frame of class \texttt{"hrsize"} and can be used for further
analysis:

<<>>=
hrs
@

This class of data is common in \texttt{adehabitatHR} and we will generate
again such objects with other home range estimation methods.


\section{The kernel estimation and the utilization distribution}

\subsection{The utilization distribution}

The MCP has met a large success in the ecological literature.  However,
many authors have stressed that the definition of the home range which
is commonly used in the literature was rather imprecise:
``\textit{that area traversed by the animal during its normal
  activities of food gathering, mating and caring for
  young}'' (Burt, 1943).  Although this definition corresponds well to
the feeling of many ecologists concerning what is the home range, it
lacks formalism: what is an \textit{area traversed}? what is a
\textit{normal activity}?\\

Several authors have therefore proposed to replace this definition by
a more formal model: the \textit{utilization distribution} (UD,
van Winkle, 1975).  Under this model, we consider that the animals use
of space can be described by a bivariate probability density function,
the UD, which gives the probability density to relocate the animal at
any place according to the coordinates (x, y) of this place.  The study
of the space use by an animal could consist in the study of the
properties of the utilization distribution.  The issue is therefore to
estimate the utilization distribution from the relocation data:


<<label=fig7,echo=FALSE,fig=FALSE,eval=FALSE>>=
xy1 <- matrix(rnorm(200), ncol=2)
xy2 <- matrix(rnorm(200, mean=3), ncol=2)
xy <- rbind(xy1,xy2)
xy <- SpatialPoints(xy)
kud <- kernelUD(xy)
par(mfrow=c(1,2))
plot(xy)
title("Input: Relocations")
par(mar=c(0,0,2,0))
xxyz <- as.image.SpatialGridDataFrame(kud)
persp(xxyz, theta=30, phi=45, box=FALSE, shade=0.5, border=NA, ltheta=300)
title("Output: the UD")
@



\begin{center}
<<results=tex,echo=FALSE>>=
.PngNo <- .PngNo + 1; file <- paste("Fig-bitmap-",
          .PngNo, ".png", sep="")
png(file=file, width = 2*wi, height = wi, pointsize = pt)
<<fig7>>
dev.null <- dev.off()
cat("\\includegraphics[height=7cm,width=14cm]{", file, "}\n\n", sep="")
@
\end{center}


The seminal paper of Worton (1989) proposed the use of the kernel
method (Silverman, 1986; Wand and Jones, 1995) to estimate the UD
using the relocation data.  The kernel method was probably the most
frequently used function in the package \texttt{adehabitat}.  A lot of
useful material and information on its use can be found on the website
of the group Animove (\url{http://www.faunalia.it/animov/}).  I
give here a summary of the discussion found on this site (they include
answers to the most frequently asked questions).\\

The basic principle of the method is the following: a bivariate
\textit{kernel function} is placed over each relocation, and the
values of these functions are averaged together.\\

Many kernel functions $K$ can be used in the estimation process
provided that:

$$
\int K(\mathbf{x}) = 1
$$

and

$$
K(\mathbf{x}) > 0 \mbox{~~~~}  \forall \mathbf{x}
$$


where $\mathbf{x}$ is a vector containing the coordinates of a point
on the plane.  However, there are two common choices for the kernel
functions: the bivariate normal kernel:

$$
K(\mathbf{x}) =  \frac{1}{2 \pi} \exp(- \frac{1}{2} \mathbf{x}^t
\mathbf{x})
$$

and the Epanechnikov kernel:

$$
K(\mathbf{x}) =  \frac{3}{4} (1 - \mathbf{x}^t
\mathbf{x}) \mbox{~~~~if~} x<1 \mbox{~or~0~otherwise}
$$

The choice of a kernel function does not greatly affect the
estimates (Silverman, 1986; Wand and Jones, 1995).  Although the
Epanechnikov kernel is slightly more efficient, the bivariate normal
kernel is still a common choice (and is the default choice in
\texttt{adehabitatHR}).\\

Then, the kernel estimation of the UD at a given point \textbf{x} of
the plane is obtained by:

$$
\hat f(\mathbf{x}) = \frac{1}{nh^2} \sum_{i=1}^n K \left \{\frac{1}{h}
  (\mathbf{x} - \mathbf{X}_i) \right \}
$$

where $h$ is a smoothing parameter, $n$ is the number of relocations,
and $\mathbf{X}_i$ is the i$^{th}$ relocation of the sample.\\

The smoothing parameter $h$ controls the ``width'' of the kernel
functions placed over each point.  On an example dataset:

<<label=kdksks,echo=FALSE,fig=FALSE,eval=FALSE>>=
par(mfrow=c(2,2))
par(mar=c(0,0,2,0))
image(kernelUD(xy, h=0.2))
title("h = 0.2")
par(mar=c(0,0,2,0))
image(kernelUD(xy, h=0.5))
title("h = 0.5")
par(mar=c(0,0,2,0))
image(kernelUD(xy, h=1))
title("h = 1")
par(mar=c(0,0,2,0))
image(kernelUD(xy, h=2))
title("h = 2")
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<kdksks>>
<<zfig>>
@
\end{center}

There has been a considerable amount of work on the correct choice of
a smoothing parameter for home range estimation.  However, the two most
common choices are the so-called ``reference bandwidth''.  When a
bivariate normal kernel has been chosen, this ``reference bandwidth''
is equal to:

$$
h = \sigma \times n^{-1/6}
$$

\noindent where

$$
\sigma = 0.5 \times (\sigma_x + \sigma_y)
$$

\noindent and $\sigma_x$ and $\sigma_x$ are the standard deviations of
the x and y coordinates of the relocations respectively.  If an
Epanechnikov kernel is used, this value is multiplied by 1.77
(Silverman, 1986, p. 86).  The reference bandwidth supposes that the UD
is a bivariate normal distribution, which is disputable in most
ecological studies.  When the animals uses several centers of activity,
the reference smoothing parameter is often too large, which results
into a strong oversmoothing of the data (the estimated UD predicts the
frequent presence of the animal in areas which are not actually
used).\\

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.\\

Finally, a subjective visual choice for the smoothing parameter, based
on successive trials, is often a sensible choice (Silverman, 1986;
Wand and Jones, 1995).



\subsection{The function \texttt{kernelUD}: estimating the utilization
  distribution}
\label{sec:refUD}

The function \texttt{kernelUD} of the package \texttt{adehabitatHR}
implements the method described above to estimate the UD.
The UD is estimated in each pixel of a grid superposed to the
relocations.  I describe in the next sections how the grid parameters
can be controlled in the estimation.  Similarly to all the functions
allowing the estimation of the home range, the functions may take as
argument either a \texttt{SpatialPoints} object (estimation of the UD
of one animal) or a \texttt{SpatialPointsDataFrame} object with one
column corresponding to the identity of the animal (estimation of the
UD of several animals).  The argument \texttt{h} controls the value of
the smoothing parameter.  Thus, if \texttt{h = "href"}, the
``reference'' bandwidth is used in the estimation.  If \texttt{h =
  "LSCV"}, the ``LSCV'' bandwidth is used in the
estimation.  Alternatively, it is possible to pass a numeric value for
the smoothing parameter.\\

I give below a short example of the use of \texttt{kernelUD}, using
the \texttt{puechabonsp} dataset.  Remember that the first column of
the component \texttt{relocs} of this dataset contains the identity of
the animals:

<<>>=
data(puechabonsp)
kud <- kernelUD(puechabonsp$relocs[,1], h="href")
kud
@

The resulting object is a list of the class \texttt{"estUD"}.  This
class extends the class \texttt{SpatialPixelsDataFrame} (it just
contains an additional attribute storing the information about the
smoothing parameter).  Display the results:

<<label=fig8,echo=FALSE,fig=FALSE,eval=FALSE>>=
image(kud)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<fig8>>
@


\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig8>>
<<zfig>>
@
\end{center}

The values of the smoothing parameters are stored in the slot
\texttt{"h"} of each element of the list.  For example, to get the
h-value for the first animal:

<<>>=
kud[[1]]@h
@





\subsection{The Least Square Cross Validation}

Alternatively, we could have set the argument \texttt{h} equal to
\texttt{"LSCV"}.  The LSCV algorithm searches for the optimum value of
\textit{h} in the interval specified by the parameter
\texttt{hlim}.  As an example, estimate the UD with a $h$ parameter
chosen with the LSCV algorithm for the \texttt{puechabonsp} dataset:

<<label=fig9,echo=FALSE,fig=FALSE,eval=FALSE>>=
kudl <- kernelUD(puechabonsp$relocs[,1], h="LSCV")
image(kudl)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<fig9>>
@


\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig9>>
<<zfig>>
@
\end{center}

The resulting UD fits the data more closely than the use of
``href''.  However, note that the cross-validation criterion cannot be
minimized in some cases.  According to Seaman and Powell (1998)
"\textit{This is a difficult problem that has not been worked out by
  statistical theoreticians, so no definitive response is available at
  this time}".  Therefore, it is essential to look at the results of
the LSCV minimization using the function \texttt{plotLSCV}:


<<label=fig10,echo=FALSE,fig=FALSE,eval=FALSE>>=
plotLSCV(kudl)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<fig10>>
@

\begin{figure}[htbp]
\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig10>>
<<zfig>>
@
\end{center}
\end{figure}

Here, the minimization has been achieved in the specified
interval.  When the algorithm does not converge toward a solution, the
estimate should not be used in further analysis.  Solutions are
discussed on the animove website
(\url{http://www.faunalia.it/animov/}), and especially on the mailing
list.  In particular, see the following messages:

\begin{itemize}
\item \url{http://www.faunalia.com/pipermail/animov/2006-May/000137.html}
\item \url{http://www.faunalia.com/pipermail/animov/2006-May/000130.html}
\item \url{http://www.faunalia.com/pipermail/animov/2006-May/000165.html}
\end{itemize}


Finally, the user may indicate a numeric value for the smoothing
parameter.  Further discussion on the choice of h-values can be found
on the animove website, at the URL:
\url{https://www.faunalia.it/dokuwiki/doku.php?id=public:animove}.


\subsection{Controlling the grid}

The UD is estimated at the center of each pixel of a grid.  Although
the size and resolution of the grid does not have a large effect on
the estimates (see Silverman, 1986), it is sometimes useful to be
able to control the parameters defining this grid.  This is done thanks
to the parameter \texttt{grid} of the function \texttt{kernelUD}.


\subsubsection{Passing a numeric value}

When a numeric value is passed to the parameter \texttt{grid}, the
function \texttt{kernelUD} automatically computes the grid for the
estimation.  This grid is computed in the following way:\\

\begin{itemize}
\item the minimum and maximum coordinates ($X_{min}, X_{max}, Y_{min},
  Y_{max}$) of the relocations are computed.\\

\item the range of $X$ and $Y$ values is also computed (i.e., the
  difference between the maximum and minimum value).  Let $R_x$ and
  $R_Y$ be these ranges.\\

\item The coverage of the grid is defined thanks to the parameter
  \texttt{extent} of the function \texttt{kernelUD}.  Let $e$ be this
  parameter.  The minimum and maximum X coordinates of the pixels of
  the grid are equal to $X_{min} - e \times R_x$ and $X_{max} + e
  \times R_x$ respectively.  Similarly, the minimum and maximum Y
  coordinates of the pixels of the grid are equal to $Y_{min} - e
  \times R_Y$ and $Y_{max} + e \times R_Y$ respectively.\\

\item Finally, these extents are split into $g$ intervals, where $g$
  is the value of the parameter \texttt{grid}.\\
\end{itemize}

Consequently, the parameter \texttt{grid} controls the resolution of
the grid and the parameter \texttt{extent} controls its extent.  As an
illustration, consider the estimate of the first animal:

<<label=kskdkkds,echo=FALSE,fig=FALSE,eval=FALSE, keep.source=TRUE>>=
## The relocations of "Brock"
locs <- puechabonsp$relocs
firs <- locs[as.data.frame(locs)[,1]=="Brock",]

## Graphical parameters
par(mar=c(0,0,2,0))
par(mfrow=c(2,2))

## Estimation of the UD with grid=20 and extent=0.2
image(kernelUD(firs, grid=20, extent=0.2))
title(main="grid=20, extent=0.2")

## Estimation of the UD with grid=20 and extent=0.2
image(kernelUD(firs, grid=80, extent=0.2))
title(main="grid=80, extent=0.2")

## Estimation of the UD with grid=20 and extent=0.2
image(kernelUD(firs, grid=20, extent=3))
title(main="grid=20, extent=3")

## Estimation of the UD with grid=20 and extent=0.2
image(kernelUD(firs, grid=80, extent=3))
title(main="grid=80, extent=3")
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE, keep.source=TRUE>>=
<<kskdkkds>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<kskdkkds>>
<<zfig>>
@
\end{center}


Note that the parameter \texttt{same4all} is an additional parameter
allowing the control of the grid.  If it is equal to TRUE, the same
grid is used for all animals, thus for example:

<<label=kskkssqq,echo=FALSE,fig=FALSE,eval=FALSE>>=
kus <- kernelUD(puechabonsp$relocs[,1], same4all=TRUE)
image(kus)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<kskkssqq>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<kskkssqq>>
<<zfig>>
@
\end{center}

Because all the UD are estimated on the same grid, it is possible to
coerce the resulting object as a \texttt{SpatialPixelsDataFrame}:

<<>>=
ii <- estUDm2spixdf(kus)
class(ii)
@

This object can then be handed with the functions of the package
\texttt{sp}.



\subsubsection{Passing a \texttt{SpatialPixelsDataFrame}}

It is sometimes useful to estimate the UD in each pixel of a raster
map already available.  This can be useful in habitat selection
studies, when the value of the UD is to be later related to the value
of environmental variables.  In such cases, it is possible to pass a
raster map to the parameter \texttt{grid} of the function.  For
example, the dataset \texttt{puechabonsp} contains an object of class
\texttt{SpatialPixelsDataFrame} storing the environmental information
for this variable.  This map can be passed to the parameter
\texttt{grid}:

<<>>=
kudm <- kernelUD(puechabonsp$relocs[,1], grid=puechabonsp$map)
@

And as in the previous section, the object can be coerced into a
\texttt{SpatialPixelsDataFrame} using the function
\texttt{estUDm2spixdf}.  Note also that it is possible to pass a list
of \texttt{SpatialPixelsDataFrame} (with one element per animal) to
the parameter \texttt{grid} (see the help page of the function
\texttt{kernelUD}).


\subsection{Estimating the home range from the UD}

The UD gives the probability density to relocate the animal at a given
place.  The home range deduced from the UD as the minimum area
on which the probability to relocate the animal is equal to a
specified value.  For example, the 95\% home range corresponds to the
smallest area on which the probability to relocate the animal is equal
to 0.95.  The functions \texttt{getvolumeUD} and \texttt{getverticeshr}
provide utilities for home range estimation.

\subsubsection{Home ranges in vector mode}

The most useful function
is the function \texttt{getverticeshr}.  For example, to deduce the
90\% home range from the UD estimated using the LSCV algorithm:

<<>>=
homerange <- getverticeshr(kudl)
class(homerange)
@

The resulting object is of the class
\texttt{SpatialPolygonsDataFrame}.  Therefore, as for the MCP (see
previous section), the functions of the package \texttt{sp} and
\texttt{maptools} are available to deal with this object, or to export
it toward a GIS (see section 3.2 for examples).  Therefore, to display
the home range:


<<label=fig11,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(homerange, col=1:4)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<fig11>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig11>>
<<zfig>>
@
\end{center}

\subsubsection{Home ranges in raster mode}
\label{sec:hrrm}

However, the function \texttt{getverticeshr} returns an object of
class \texttt{SpatialPolygonsDataFrame}, i.e. an object in mode
vector.\\

The function \texttt{getvolumeUD} may also be useful to estimate the
home range in raster mode, from the UD.  Actually, this function
modifies the UD component of the object passed as argument, so that
the value of a pixel is equal to the percentage of the smallest home
range containing this pixel:

<<>>=
vud <- getvolumeUD(kudl)
vud
@

To make clear the differences between the output of \texttt{kernelUD}
and \texttt{getvolumeUD} look at the values on the following
contourplot:

<<label=fig12,echo=FALSE,fig=FALSE,eval=FALSE,keep.source=TRUE>>=

## Set up graphical parameters
par(mfrow=c(2,1))
par(mar=c(0,0,2,0))

## The output of kernelUD for the first animal
image(kudl[[1]])
title("Output of kernelUD")

## Convert into a suitable data structure for
## the use of contour
xyz <- as.image.SpatialGridDataFrame(kudl[[1]])
contour(xyz, add=TRUE)


## and similarly for the output of getvolumeUD
par(mar=c(0,0,2,0))
image(vud[[1]])
title("Output of getvolumeUD")
xyzv <- as.image.SpatialGridDataFrame(vud[[1]])
contour(xyzv, add=TRUE)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE,keep.source=TRUE>>=
<<fig12>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<fig12>>
<<zfigg>>
@
\end{center}

Whereas the output of \texttt{kernelUD} is the raw UD, the output of
\texttt{getvolumeUD} can be used to compute the home range (the labels
of the contour lines correspond to the home ranges computed with
various probability levels).  For example, to get the rasterized 95\%
home range of the first animal:


<<label=kdkdkdk,echo=FALSE,fig=FALSE,eval=FALSE,keep.source=TRUE>>=
## store the volume under the UD (as computed by getvolumeUD)
## of the first animal in fud
fud <- vud[[1]]

## store the value of the volume under UD in a vector hr95
hr95 <- as.data.frame(fud)[,1]

## if hr95 is <= 95 then the pixel belongs to the home range
## (takes the value 1, 0 otherwise)
hr95 <- as.numeric(hr95 <= 95)

## Converts into a data frame
hr95 <- data.frame(hr95)

## Converts to a SpatialPixelsDataFrame
coordinates(hr95) <- coordinates(vud[[1]])
gridded(hr95) <- TRUE

## display the results
image(hr95)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE,keep.source=TRUE>>=
<<kdkdkdk>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<kdkdkdk>>
<<zfig>>
@
\end{center}




\subsection{The home range size}

As for the MCP, the \texttt{SpatialPolygonsDataFrame} returned by the
function \texttt{getverticeshr} contains a column named ``area'',
which contains the area of the home ranges:

<<>>=
as.data.frame(homerange)
@

\noindent And, as for the MCP, the units of these areas are controlled
by the parameters \texttt{unin} and \texttt{unout} of the function
\texttt{getverticeshr}.\\

I also considered that it would be useful to have a function which
would compute the home-range size for several probability levels.  This
is the aim of the function \texttt{kernel.area} which takes the UD as
argument (here we use the UD estimated with the LSCV algorithm).

<<>>=
ii <- kernel.area(kudl, percent=seq(50, 95, by=5))
ii
@

The resulting object is of the class \texttt{hrsize}, and can be
plotted using the function \texttt{plot}.  Note that the home-range
sizes returned by this function are slightly different from the
home-range size stored in the \texttt{SpatialPolygonsDataFrame}
returned by the function \texttt{getverticeshr}.  Indeed, while the
former measures the area covered by the rasterized home range (area
covered by the set of pixels of the grid included in the home range),
the latter measures the area of the vector home range (with smoother
contour).  However, note that the difference between the two estimates
decrease as the resolution of the grid becomes finer.


\subsection{Taking into account the presence of physical boundary over
  the study area}

In a recent paper, Benhamou and Cornelis (2010) proposed an
interesting approach to take into account the presence of physical
boundaries on the study area. They proposed to extend to 2 dimensions
a method developed for one dimension by Silverman 1986. The principle
of this approach is described below:

<<label=figggf,echo=FALSE,fig=FALSE,eval=FALSE>>=
par(mar=c(0,0,0,0))
plot(c(0,1),asp=1, ty="n")
bound <- structure(list(x = c(1.11651014886660, 1.31884344750132, 1.49071796999748,
1.56686491034388, 1.65171435815844, 1.627782462621, 1.59079680588132,
1.63648497008916), y = c(0.819277593991856, 0.797467606139553,
0.740761637723568, 0.609901710609753, 0.417973817509493, 0.263122903758146,
0.106090991221569, -0.00732094561040263)), .Names = c("x", "y"
))
pts <- structure(list(x = c(1.14479329813812, 1.20135959668116, 1.18395458174484,
1.30143843256500, 1.29056029822980, 1.28620904449572, 1.31449219376724,
1.46896170132708, 1.39063913411364, 1.50377173119972, 1.627782462621,
1.57121616407796, 1.37758537291140, 1.21006210414932, 1.32537032810244,
1.519001119269, 1.46678607446004, 1.32754595496948, 1.17960332801076,
1.12303702946772, 1.19700834294708, 1.57121616407796, 1.57774304467908,
1.30578968629908, 1.60167494021652, 1.62995808948804, 1.5625136566098,
1.46896170132708, 1.43632729832148, 1.31449219376724), y = c(0.784381613428172,
0.732037642582647, 0.605539713039293, 0.559738738549458, 0.67315067538143,
0.749485632864488, 0.76257162557587, 0.714589652300805, 0.633892697247286,
0.330733866100284, 0.417973817509493, 0.527023756771005, 0.548833744623307,
0.378715839375349, 0.156253963281865, 0.287113890395679, 0.199873938986470,
0.354724852737816, 0.39180183208673, 0.311104877033211, 0.20423593655693,
0.106090991221569, 0.212959931697851, 0.780019615857712, 0.367810845449197,
0.424516813865184, 0.518299761630084, 0.601177715468833, 0.712408653515574,
0.703684658374653)), .Names = c("x", "y"))
lines(bound)
bound <- do.call("cbind",bound)
Slo1 <- Line(bound)
Sli1 <- Lines(list(Slo1), ID="frontier1")
barrier <- SpatialLines(list(Sli1))

points(pts, pch=16)
pts <- as.data.frame(pts)
coordinates(pts) <- c("x","y")
ii <- adehabitatHR:::.boundaryk(pts, barrier, 0.04)
points(ii, pch=3)
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<figggf>>
<<zfig>>
@
\end{center}

On this figure, the relocations are located near the lower left corner
of the map. The line describes a physical boundary located on the
study area (e.g. a cliff, a pond, etc.). The animal cannot cross the
boundary. The use of the classical kernel estimation will not take
into account this boundary when estimating the UD. A classical
approach when such problem occur is to use the classical kernel
approach and then to set to 0 the value of the UD on the other side of
the boundary. However, this may lead to some bias in the estimation of
the UD near the boundary (as the kernel estimation will ``smooth'' the
use across the boundary, whereas one side of the boundary is used and
the other unused).  Benhamou and Cornelis (2010) extended an approach
developed for kernel estimation in one dimension: this approach
consists in calculating the mirror image of the points located near
the boundary, and adding these ``mirror points'' to the actual
relocations before the kernel estimation (These mirror points
correspond to the crosses on the figure). \textbf{Note that this
  approach cannot be used for all boundaries}. There are two
constraints that have to be satisfied:\\

\begin{itemize}
\item The boundary should be defined as the union of several segments,
  \textit{and each segment length should at least be equal to $3\times
    h$}, where $h$ is the smoothing parameter used for kernel
  smoothing;\\

\item The angle between two successive line segments should be greater
  that $\pi/2$ or lower than $-\pi/2$.
\end{itemize}

\noindent Therefore, the boundary shape should be fairly simple, and
not too tortuous.\\

Now, let us consider the case of the wild boar monitored in
Puechabon. Consider the map of the elevation on the study area:

<<label=mapelev,echo=FALSE,fig=FALSE,eval=FALSE>>=
image(puechabonsp$map)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<mapelev>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<mapelev>>
<<zfig>>
@
\end{center}

This area is traversed by the Herault gorges. As can be seen on this
figure, the slopes of the gorges are very steep, and it is nearly
impossible for the wild boars monitored to cross the river (although
it is possible in other places). Therefore, the gorges are a boundary
that cannot be crossed by the animal. Using the function
\texttt{locator()}, we have defined the following boundary:


<<label=figboundary,echo=FALSE,fig=FALSE,eval=FALSE>>=
bound <- structure(list(x = c(701751.385381925, 701019.24105475,
                        700739.303517889,
                        700071.760160759, 699522.651915378,
                        698887.40904327, 698510.570051342,
                        698262.932999504, 697843.026694212,
                        698058.363261028),
                        y = c(3161824.03387414,
                        3161824.03387414, 3161446.96718494,
                        3161770.16720425, 3161479.28718687,
                        3161231.50050539, 3161037.5804938,
                        3160294.22044937, 3159389.26039528,
                        3157482.3802813)), .Names = c("x", "y"))
image(puechabonsp$map)
lines(bound, lwd=3)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<figboundary>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<figboundary>>
<<zfig>>
@
\end{center}


We now convert this boundary as a SpatialLines object:

<<>>=
## We convert bound to SpatialLines:
bound <- do.call("cbind",bound)
Slo1 <- Line(bound)
Sli1 <- Lines(list(Slo1), ID="frontier1")
barrier <- SpatialLines(list(Sli1))
@

We can now set the parameter \texttt{boundary} of the function
\texttt{kernelUD}, which indicates that we want to use this approach:

<<label=imagekudbound,echo=FALSE,fig=FALSE,eval=FALSE>>=
kud <- kernelUD(puechabonsp$relocs[,1], h=100, grid=100, boundary=barrier)
image(kud)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<imagekudbound>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<imagekudbound>>
<<zfigg>>
@
\end{center}

The effect of the boundary correction is clear on the UD of Brock and
Calou.


\section{Taking into account the time dependence between relocation}

\subsection{The Brownian bridge kernel method}

\subsubsection{Description}

Bullard (1999) proposed an extension of the kernel method taking into
account the time dependence between successive relocations. Whereas
the classical kernel method places a kernel function above each
\textit{relocation}, the Brownian bridge kernel method places a kernel
function above each \textit{step} of the trajectory (a step being the
straight line connecting two successive relocations). This kernel
function is a combination of two bivariate normal pdf (two ``bumps'')
and a Brownian bridge pdf (see Bullard, 1991 and Horne et al. 2007 for
additional details):

<<label=brownbrid,echo=FALSE,fig=FALSE,eval=FALSE>>=
par(mar=c(0,0,0,0))
xx <- c(0,1)
yy <- c(0,1)
date <- c(0,1)
class(date) <- c("POSIXt", "POSIXct")
tr <- as.ltraj(data.frame(x = xx,y = yy), date, id="a")
kba <- kernelbb(tr, 0.6, 0.2, extent=0.8, grid=50)
kba <- as(kba, "SpatialPixelsDataFrame")
fullgrid(kba) <- TRUE
uu <- slot(kba, "data")[,1]
uu <- matrix(uu, nrow=50)
persp(uu, theta = 135, phi = 30, scale = FALSE,
      ltheta = -120, shade = 0.75, border = NA, box = FALSE)
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<brownbrid>>
<<zfig>>
@
\end{center}


This method takes into account not only the position of the
relocations, but also the path travelled by the animal between
successive relocations. The following figure may help to understand
the method:

<<echo=FALSE>>=
pt <- 16
@

<<label=tutorkbb,echo=FALSE,fig=FALSE,eval=FALSE>>=
suppressWarnings(RNGversion("3.5.0"))
set.seed(2098)
pts1 <- data.frame(x = rnorm(25, mean = 4.5, sd = 0.05),
                   y = rnorm(25, mean = 4.5, sd = 0.05))
pts1b <- data.frame(x = rnorm(25, mean = 4.5, sd = 0.05),
                    y = rnorm(25, mean = 4.5, sd = 0.05))
pts2 <- data.frame(x = rnorm(25, mean = 4, sd = 0.05),
                   y = rnorm(25, mean = 4, sd = 0.05))
pts3 <- data.frame(x = rnorm(25, mean = 5, sd = 0.05),
                   y = rnorm(25, mean = 4, sd = 0.05))
pts3b <- data.frame(x = rnorm(25, mean = 5, sd = 0.05),
                    y = rnorm(25, mean = 4, sd = 0.05))
pts2b <- data.frame(x = rnorm(25, mean = 4, sd = 0.05),
                    y = rnorm(25, mean = 4, sd = 0.05))
pts <- do.call("rbind", lapply(1:25, function(i) {
    rbind(pts1[i,], pts1b[i,], pts2[i,], pts3[i,],
          pts3b[i,], pts2b[i,])
}))
dat <- 1:150
class(dat) <- c("POSIXct","POSIXt")
x <- as.ltraj(pts, date=dat, id = rep("A", 150))
## Now, we suppose that there is a precision of 0.05
## on the relocations
sig2 <- 0.05
## and that sig1=0.1
sig1 <- 0.1
## Now fits the brownian bridge home range
kbb <- kernelbb(x, sig1 = sig1,
                 sig2 = sig2, grid=60)
## Now fits the classical kernel home range
coordinates(pts) <- c("x","y")
kud <- kernelUD(pts)
###### The results
opar <- par(mfrow=c(2,2), mar=c(0.1,0.1,2,0.1))
plot(pts, pch=16)
title(main="The relocation pattern")
box()
plot(x, axes=FALSE, main="The trajectory")
box()
image(kud)
title(main="Classical kernel home range")
plot(getverticeshr(kud, 95), add=TRUE)
box()
image(kbb)
title(main="Brownian bridge kernel home range")
plot(getverticeshr(kbb, 95), add=TRUE)
box()
@


\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<tutorkbb>>
<<zfigg>>
@
\end{center}

<<echo=FALSE>>=
pt <- 25
@

In this example, the relocations are located into three
patches. However, the order of use of the patches is not random. Using
the Brownian bridge method allows to identify that some areas between
the patches are actually not used by the animal.\\

The kernel Brownian bridge method is implemented in the function
\texttt{kernelbb}. Two smoothing parameters need to be set:\\

\begin{itemize}
\item \texttt{sig1}: This parameter controls the width of the
  ``bridge'' connecting successive relocations. The larger it is and
  the larger the bridge is. This parameter is therefore
  \textit{related} to the speed of the animal. But be careful: it is
  not the speed of the animal. Horne et al. (2007) call
  \texttt{sig1}$^2$ the ``Brownian motion variance parameter''. It is
  even not the variance of the position, but a parameter used to
  compute the variance (which is itself related to the speed). The
  variance of the position of the animal between two relocations at
  time $t$ is $S^2$ = \texttt{sig1}$^2 \times t \times (T-t) / T$,
  where $T$ is the duration of the trajectory between the two
  relocations and $t$ the time at the current position. Therefore,
  \texttt{sig1} is related to the speed of the animal in a very
  complex way, and its biological meaning is not that clear. The
  definition of \texttt{sig1} can be done subjectively after visual
  exploration of the result for different values, or using the
  function \texttt{liker} that implements the maximum likelihood
  approach developed by Horne et al. (2007);\\

\item \texttt{sig2}: This parameter controls the width of the ``bumps''
  added over the relocations (see Bullard, 1999; Horne et
  al. 2007). It is similar to the smoothing
  parameter $h$ of the classical kernel method, and is therefore
  related to the imprecision of the relocations.\\
\end{itemize}

The following figure shows the kernel function placed over the step
built by the two relocations (0,0) and (1,1), for different values of
\texttt{sig1} and \texttt{sig2}:

<<echo=FALSE>>=
wi <- 800
pt <- 18
@

<<label=ploth,echo=FALSE,fig=FALSE,eval=FALSE>>=
xx <- c(0,1)
yy <- c(0,1)
date <- c(0,1)
class(date) <- c("POSIXt", "POSIXct")
tr <- as.ltraj(data.frame(x = xx,y = yy), date, id="a")

## 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])
    }
}

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

@


\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<ploth>>
<<zfigg>>
@
\end{center}


<<echo=FALSE>>=
wi <- 600
pt <- 25
@

\subsubsection{Example}
\label{sec:ltrajj}

The Brownian bridge kernel method is implemented in the function
\texttt{kernelbb} of the package. This function takes as argument an
object of class \texttt{ltraj}. This class has been developed in the
brother package \texttt{adehabitatLT} to store animals trajectory
sampled using GPS, radio-tracking, Argos data, etc. The interested
reader should read the help page of the function \texttt{as.ltraj} as
well as the vignette of the package \texttt{adehabitatLT}.\\

First consider as an example the night monitoring (every 10 minutes)
of one wild boar:

<<>>=
data(puechcirc)
x <- puechcirc[1]
x
@

Have a look at the trajectory of the animal:

<<label=plotpuechcirc,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(x)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<plotpuechcirc>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<plotpuechcirc>>
<<zfig>>
@
\end{center}


A study has shown that the mean standard deviation of the relocations
(relocations as a sample of the actual position of the animal) is
equal to 58 meters on these data (Maillard, 1996, p. 63). Therefore,
we set \texttt{sig2 = 58} metres. Then we use the function
\texttt{liker} to find the maximum likelihood estimation of the
parameter \texttt{sig1}. We search the optimum value in the range [10,
100]:

<<label=liker1,echo=FALSE,fig=FALSE,eval=FALSE>>=
lik <- liker(x, sig2 = 58, rangesig1 = c(10, 100))
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<liker1>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<liker1>>
<<zfig>>
@
\end{center}

We expected a too large standard deviation... Let us try again, and
search in the interval [1, 10]:

<<label=liker2,echo=FALSE,fig=FALSE,eval=FALSE>>=
lik2 <- liker(x, sig2 = 58, rangesig1 = c(1, 10))
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<liker2>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<liker2>>
<<zfig>>
@
\end{center}

Actually, the optimum value is 6.23:

<<>>=
lik2
@

We can now estimate the kernel Brownian bridge home range with the
parameters \texttt{sig1=6.23} and \texttt{sig2=58}:

<<>>=
tata <- kernelbb(x, sig1 = 6.23, sig2 = 58, grid = 50)
tata
@

The result is an object of class \texttt{estUD}, and can be managed as
such (see section \ref{sec:refUD}). For example to see the utilization
distribution and the 95\% home range:

<<label=UDkbb,echo=FALSE,fig=FALSE,eval=FALSE>>=
image(tata)
plot(getverticeshr(tata, 95), add=TRUE, lwd=2)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<UDkbb>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<UDkbb>>
<<zfig>>
@
\end{center}


\subsection{The biased random bridge kernel method}

\subsubsection{Description}

Benhamou and Cornelis (2010) developed an approach similar to the
Brownian bridge method for the analysis of trajectories. They called it ``movement
based kernel estimation'' (MBKE). The principle of this method
consists in dividing each step into several sub-steps, i.e. in adding
new points at regular intervals on each step prior to a classical
kernel estimation, that is:

<<label=figexe1,echo=FALSE,fig=FALSE,eval=FALSE>>=
par(mar=c(0,0,0,0))
plot(c(0,1,2,3),c(0,1,0,1), pch=16, cex=4, xlim=c(-0.5, 3.5),
ylim=c(-0.5, 1.5))
segments(0,0,1,1)
segments(2,0,3,1)
points(seq(2,3,length=8), seq(0,1,length=8), pch=16, cex=2)
arrows(1,0.5, 2, 0.5, lwd=5)
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<figexe1>>
<<zfig>>
@
\end{center}

Then a kernel estimation is carried out with the known (large points
on the figure) and interpolated (small points) relocations, with a
variable smoothing parameter (the smoothing parameter depends on the
time spent between a known relocation and an interpolated relocation:
it is minimum when the relocation is known and larger as the
interpolated time between the known relocation and the
interpolated location increases).\\

In a later paper, Benhamou (2011) demonstrated that the movement-based
kernel approach takes place in the framework of the
biased random walk model. If we describe a trajectory as a succession
of ``steps'', each being characterized by a speed and an angle with
the east direction, the process generating the trajectory is a biased
random walk when the probability density distribution of the angles is
not an uniform distribution (i.e. there is a preferred direction of
movement). The main quality of the biased random walk is that it does
not suppose a purely diffusive movement (the brownian bridge method
supposes only a diffusion), but takes into account an advection
component in the trajectory (i.e. a ``drift'' between successive
relocations). This model is therefore more realistic when animal
movements are studied.\\

Now, consider two successive relocations $\mathbf{r}_1 = (x_1, y_1)$
and $\mathbf{r}_2 = (x_2, y_2)$, collected respectively at times $t_1$
and $t_2$, and building a step of the trajectory.  Our aim is to
estimate the function giving the probability density (pdf) that the
animal was located at a given place $\mathbf{r}=(x,y)$ at time $t_i$ (with $t_1
< t_i < t_2$), and \textit{given that the animal is moving according to a
  biased random walk}. The movement generated by a biased random walk
\textit{given} the start and end relocation is called a \textit{biased
  random   bridge} (BRB). Benhamou (2011) noted that the BRB
can be approximated by a bivariate normal distribution:

\begin{equation}
  \label{eq:brb}
  f(\mathbf{r}, t_i | \mathbf{r}_1, \mathbf{r}_2) =
  \frac{t_2-t_1}{4\pi D (t_2 - t_i)} \exp \left [ \frac{\mathbf{r}_m
      \mathbf{D} \mathbf{r}_m}{4 p_i(t_2-t_i)} \right ]
\end{equation}


\noindent where the mean location corresponds to $\mathbf{r}_m = (x_1
+ p_i(x_2-x_1), y_1 + p_i(y_2-y_1))$, and $p_i =
(t_i-t_1)/(t_2-t_1)$. The variance-covariance matrix $\mathbf{D}$ of
this distribution usually cannot be estimated in practice (Benhamou,
2011). Consequently, the biased random bridge is approximated by a
random walk with a diffusion coefficient $D$ on which a drift
$\mathbf{v}$ is appended. And, under this assumption, the required pdf
can be approximated by a circular normal distribution fully
independent of the drift, i.e. the matrix $\mathbf{D}$ in equation
\ref{eq:brb} is supposed diagonal, with both diagonal elements
corresponding to a \textit{diffusion coefficient} $D$. The figure
below illustrates how the pdf of relocating the animal changes with
the times $t_i$:

<<label=figillustrbb,echo=FALSE,fig=FALSE,eval=FALSE>>=
x1 <- c(0,0)
x2 <- c(1,1)
D <- 0.01
tt <- 0.1
foo <- function(x)
{
    m <- c(tt,tt)
    xc <- x-m
    (1/(4*pi*D*(1-tt)))*exp(-(sum(xc*xc/D))/(4*pi*tt*(1-tt)))
}
xy <- as.matrix(expand.grid(seq(-0.1, 1.1, length=100), seq(-0.1, 1.1, length=100)))
den <- sapply(1:nrow(xy), function(i) foo(xy[i,]))
df <- data.frame(x=xy[,1], y=xy[,2], den=den)
coordinates(df) <- c("x","y")
gridded(df) <- TRUE

par(mfrow=c(1,2), mar=c(0.1,0.1,0.1,0.1))
image(df)
points(c(0,1), c(0,1), pch=16, cex=4)
lines(c(0,1), c(0,1))
points(tt,tt, pch=16)
box()
tt <- 0.5
den <- sapply(1:nrow(xy), function(i) foo(xy[i,]))
df <- data.frame(x=xy[,1], y=xy[,2], den=den)
coordinates(df) <- c("x","y")
gridded(df) <- TRUE
image(df)
points(c(0,1), c(0,1), pch=16, cex=4)
lines(c(0,1), c(0,1))
points(tt,tt, pch=16)
box()
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<figillustrbb>>
<<zfig>>
@
\end{center}

We can see that the variance of this distribution is larger when the
animal is far from the known relocations.\\

Benhamou (2011) demonstrated that the MBKE can be used to estimate the
UD of the animal under the hypothesis that the animal moves according
to a biased random walk, with a drift allowed to change in strength
and direction from one step to the next. However, it is supposed that
the drift is constant during each step. For this reason, it is
required to set an upper time threshold $T_{max}$. Steps characterized
by a longer duration are not taken into account into the estimation of
the UD. This upper threshold should be based on biological grounds.\\

As for the Brownian bridge approach, this conditional pdf based on
BRB takes an infinite value at times $t_i = t_1$ and
$t_i = t_2$ (because, at these times, the relocation of the animal is
known exactly). Benhamou (2011) proposed to circumvent this drawback
by considering that the true relocation of the animal at times $t_1$ and
$t_2$ is not known exactly. He noted: ``\textit{a GPS fix should be
  considered a punctual sample of the possible locations at which the
  animal may be observed at that time, given its current motivational
  state and history. Even if the recording noise is low, the
  relocation variance should therefore be large enough to encompass
  potential locations occurring in the same habitat patch as the
  recorded location}''. He proposed two ways to include this
``relocation uncertainty'' component in the pdf: (i) either the
relocation variance progressively merges with the movement
component, (ii) or the relocation variance has a constant weight.\\

In both cases, the minimum uncertainty over the relocation of an
animal is observed for $t_i = t_1$ or $t_2$.  This minimum standard
deviation corresponds to the parameter $h_{min}$. According to
Benhamou and Cornelis (2010), ``\textit{$h_{min}$ must be at least
  equal to the standard deviation of the localization errors and also
  must integrate uncertainty of the habitat map when UDs are computed
  for habitat preference analyses. Beyond these technical constraints,
  $h_{min}$ also should incorporate a random component inherent to
  animal behavior because any recorded location, even if accurately
  recorded and plotted on a reliable map, is just a punctual sample of
  possible locations at which the animal may be found at that time,
  given its current motivational state and history. Consequently,
  $h_{min}$ should be large enough to encompass potential locations
  occurring in the same habitat patch as the recorded location}''.\\

Therefore, the smoothing parameter used in the MKBE approach can be
calculated with:

$$
h_i^2 = h_{min}^2 + 4 p_i(1-p_i)(h_{max}^2 - h_{min}^2)T_i/T_max
$$

\noindent where $h_{max}^2 = h_{min}^2 + D \times T_{max}/2$  if we
suppose that the relocation variance has a constant weight or
$h_{max}^2 = D \times T_{max}/2$ if we suppose that it progressively
merges with the movement component.


\subsubsection{Implementation}

The BRB approach is implemented in the function \texttt{BRB} of {\tt
  adehabitatHR}. We first load the dataset \texttt{buffalo} used by
Benhamou (2011) to illustrate this approach:

<<>>=
data(buffalo)
tr <- buffalo$traj
tr
@

Note that this object has an infolocs component describing the
proportion of time between relocation $i$ and relocation $i+1$ during
which the animal was active. We show the trajectory of the animal below:

<<label=trajbuff,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(tr, spixdf=buffalo$habitat)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<trajbuff>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<trajbuff>>
<<zfig>>
@
\end{center}

The first step of the analysis consists in the estimation of the
diffusion coefficient $D$. We want to estimate one diffusion parameter
for each habitat type. As Benhamou (2011), we set $T_{max} = 180$
minutes and $L_{min} = 50$ metres (the smallest distance below which
we consider that the animal is not moving). Because it is more
sensible to consider the time of activity in the implementation of the
method, we define the argument \texttt{activity} of the function as
the name of the variable in the infolocs component storing this
information:

<<>>=
vv <- BRB.D(tr, Tmax=180*60, Lmin=50, habitat=buffalo$habitat, activity="act")
vv
@

The function \texttt{BRB.D} implements the estimation of the diffusion
coefficient described in Benhamou (2011). Note that a maximum
likelihood approach, similar to the approach used by Horne et
al. (2007) is also available in the function \texttt{BRB.likD}.
The values of the diffusion parameters are given in $m^2.s^{-1}$. Note
that the number of steps in habitat 2 was to small to allow the
calculation of a diffusion parameter for this habitat type. We can
then use the function \texttt{BRB} to estimate the UD from these
relocations. The parameter \texttt{tau} controls the time lag between
interpolated relocations. Similarly to Benhamou (2011), we set $\tau =
5$ min, and $h_{min} = 100$ metres:

<<>>=
ud <- BRB(tr, D = vv, Tmax = 180*60, tau = 300, Lmin = 50, hmin=100,
          habitat = buffalo$habitat, activity = "act", grid = 100, b=0,
          same4all=FALSE, extent=0.01)
ud
@

The missing dispersion parameter for habitat 2 was replaced
in the estimation by the global dispersion parameter.
As for all kernel methods, this function returns an object of class
\texttt{estUD} or \texttt{estUDm}, depending on the number of animals
in the object of class \texttt{ltraj} passed as argument. We show the
resulting UD:

<<label=imagebrb,echo=FALSE,fig=FALSE,eval=FALSE>>=
image(ud)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<imagebrb>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<imagebrb>>
<<zfig>>
@
\end{center}

Note that it
is also possible to specify a physical boundary that cannot be crossed
by the animals in this function.


\subsubsection{Decomposing the use of space into intensity and recursion}

Benhamou and Riotte-Lambert (2012) noted that, as a whole, the time
spent at a given location by the animal can be seen as
the product between the mean residence time per visit times the
number of visits of the location. It may be interesting, biologically,
to distinguish these two components, for example to identify the
places that the animal often uses and the places that are used only
once, but for a long time. The identification of such components can
be useful to test hypotheses concerning the habitat selection by the
animal. Thus, these authors indicate: ``Some areas, in particular
those that are more profitable, tend to be  exploited more intensively
during a given visit, involving a longer residence time. An initially
profitable area may be avoided for a  while once depleted if the
resources it encompasses renew very slowly, but should tend to be
revisited quite often otherwise, or if it is depleted only partially
at each visit''. \\

These authors therefore proposed a modification of the BRB approach to
allow the estimation of an intensity distribution (ID) reflecting this
average residence time and of a recursion distribution (RD) reflecting
the number of visits. This modification relies on the calculation of
(i) the residence time associated to each relocation interpolated
using the procedure described in the previous section (roughly, the
residence time calculated on a given relocation corresponds to the
time spent by the animal in a circle of radius $r$ centered on this
relocation, see the vignette of the package adehabitatLT), and (ii)
the number of visits associated to each relocation (i.e. the number of
times that the animal performed in a circle of radius $r$ centered on
the relocation).\\

The function \texttt{BRB} allows to calculate the ID and RD thanks to
its argument \texttt{type}. For example, consider again the dataset
\texttt{buffalo} used in the previous section. To estimate the ID or
the RD, we have to specify:\\

\begin{itemize}
\item The diffusion parameter $D$: based on the elements given in the
  previous section, we assumed a parameter $D = 7.36$. For the sake of
  conciseness, we define a unique diffusion parameter for all habitats;\\

\item As in the previous section, we set \texttt{Tmax = 180} min and
  \texttt{Lmin = 50} m, \texttt{hmin} = 100 m, and \texttt{tau =
    300} seconds;\\

\item The radius of the circle used to calculate the residence time
  and the number of visits was set to \texttt{3*hmin} = 300 m, as in
  Benhamou and Riotte-Lambert (2012);

\item The time \texttt{maxt} used to calculate the residence time (see
  the help page of this function in the package adehabitatLT, and the
  vignette of this package) and the number of visits, i.e. the maximum
  time threshold that the animal is allowed to spend outside the patch
  before that we consider that the animal actually left the patch, was
  set to 2 hours, as in Benhamou and Riotte-Lambert (2012).\\
\end{itemize}

We estimate the UD, ID and RD with these parameters below:

<<label=figidrd,echo=FALSE,fig=FALSE,eval=FALSE>>=
id <- BRB(buffalo$traj, D = 7.36, Tmax = 3*3600, Lmin = 50, type = "ID",
          hmin=100, radius = 300, maxt = 2*3600, tau = 300,
          grid = 100, extent=0.1)
rd <- BRB(buffalo$traj, D = 7.36, Tmax = 3*3600, Lmin = 50, type = "RD",
          hmin=100, radius = 300, maxt = 2*3600, tau = 300,
          grid = 100, extent=0.1)
ud <- BRB(buffalo$traj, D = 7.36, Tmax = 3*3600, Lmin = 50, type = "UD",
          hmin=100, radius = 300, maxt = 2*3600, tau = 300,
          grid = 100, extent=0.1)


par(mfrow = c(2,2), mar=c(0,0,2,0))
vid <- getvolumeUD(id)
image(vid)
contour(vid, add=TRUE, nlevels=1, levels=30,
        lwd=3, drawlabels=FALSE)
title("ID")

vrd <- getvolumeUD(rd)
image(vrd)
contour(vrd, add=TRUE, nlevels=1, levels=30,
        lwd=3, drawlabels=FALSE)
title("RD")

vud <- getvolumeUD(ud)
image(vud)
contour(vud, add=TRUE, nlevels=1, levels=95,
        lwd=3, drawlabels=FALSE)
title("UD")
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<figidrd>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<figidrd>>
<<zfig>>
@
\end{center}

Note that we used the function \texttt{getvolumeUD} for a clearer
display of the results. Remember from section \ref{sec:hrrm} that this
function modifies the UD component of the object passed as argument,
so that the value of a pixel is equal to the percentage of the
smallest home range containing this pixel. Here, we also display the
30\% isopleth calculated from the ID and RD distributions, using the
function \texttt{contour}, as well as the 95\% isopleth for the UD
(following here the recommendations of Benhamou and Riotte-Lambert
2012). In the case of the ID, the contour limits allow to identify
those patches where the animal spent a long time in average, and in
the case of the RD, those places that the animal visited often. A
comparison of the environmental composition at these places and in the
95\% home-range estimated from the UD would allow to identify the
environmental characteristics that affect both the speed of the animal
and the number of visits he performs in a patch (see the vignette of
the package adehabitatHS for additionnal details about this kind of
analysis).


\subsection{The product kernel algorithm}

\subsubsection{Description}

Until now, we have only considered utilization distribution in
space. However, Horne et al. (2007) noted that it could also be useful
to explore utilization distributions in space and time. That is, to
smooth the animals distribution not only in space, but also in
time. This approach is done by placing a three dimensional kernel
function (X and Y coordinates, and date) over each relocation, and
then, by summing these kernel functions. The three-dimensional kernel
function $K$ correspond to the combination of three one-dimensional
kernel functions $K_j$:

$$
K(\mathbf{x}) = \frac{1}{h_xh_yh_t} \sum_{i=1}^n \left ( \prod_{j=1}^3
  K_j \left ( \frac{x_j - X_{ij}}{h_j} \right ) \right )
$$

\noindent where $h_x, h_y, h_t$ are the smoothing parameters for the
$X$, $Y$ and time dimension, $n$ is the number of relocations, and
$X_{ij}$ is the coordinate of the $i^{th}$ relocation on the $j^{th}$
dimension ($X, Y$ or time).\\

This method is implemented in the function \texttt{kernelkc} of the
package \texttt{adehabitatHR}. The univariate kernel functions
associated to the spatial coordinates are the biweight kernel, that
is:

$$
K(v_i) = \frac{15}{16} (1-v_i^2)^2
$$

when $v_i < 1$ and 0 otherwise. For the time, the user has the choice
between two kernel functions. Either the time is considered to be a
linear variable, and the biweight kernel is used, or the time is
considered to be a circular variable (e.g. January 3$^{th}$ 2001 is
considered to be the same date as January 3$^{th}$ 2002 and January
3$^{th}$ 2003, etc.) and the wrapped Cauchy kernel is used. The
wrapped Cauchy kernel is:

$$
K(v_i) = \frac{h \left (1 - (1-h)^2 \right )}{2 \pi \left (1+ (1-h)^2
    - 2 (1-h) \cos (v_ih) \right )}
$$

\noindent where $0 < h < 1$.

\subsubsection{Application}

We now illustrate the product kernel algorithm on an example. We load
the \texttt{bear} dataset. This dataset describes the trajectory of a
brown bear monitored using GPS (from the package
\texttt{adehabitatLT}):

<<>>=
data(bear)
bear
@

The data are stored as an object of class \texttt{ltraj} (see section
\ref{sec:ltrajj}). Let us compute the UD of this bear for the date
2004-05-01. After a visual exploration, we noted that a smoothing
parameter of 1000 meters for X and Y coordinates, and of 72 hours
for the time returned interesting results. We can compute the UD with:

<<>>=
uu <- kernelkc(bear, h = c(1000,1000,72*3600),
               tcalc= as.POSIXct("2004-05-01"), grid=50)
uu
@

The result is an object of class \texttt{estUD}, and can be managed as
such (see section \ref{sec:refUD}). For example to see both the trajectory and
the UD, we type:

<<label=plotuu,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(bear, spixdf=uu)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<plotuu>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<plotuu>>
<<zfig>>
@
\end{center}


To compute the 95\% home-range, there is a subtlety. Note that the
UD integral over the whole plane is not equal to 1:

<<>>=
sum(slot(uu,"data")[,1])*(gridparameters(uu)[1,2]^2)
@

This is because the UD is defined in space \textit{and time}. That is,
the integral over the $X,Y$ and time direction is equal to 1, and not
the integral over the $X,Y$ direction at a given time $t$. Therefore,
the 95\% home range at a given time $t$ relies on a
``restandardization'':

$$
UD'(x,y |t_i) = \frac{UD(x,y, t_i)}{\int_{x',y'} UD(x', y', t_i) }
$$

This is done by setting to \texttt{TRUE} the argument
\texttt{standardize} of the functions \texttt{getvolumeUD} or
\texttt{getverticeshr}. Thus, to calculate the home range of the bear
at this date, type:

<<label=plotver,echo=FALSE,fig=FALSE,eval=FALSE>>=
ver <- getverticeshr(uu, percent=95, standardize=TRUE)
image(uu)
plot(ver, add=TRUE)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<plotver>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<plotver>>
<<zfig>>
@
\end{center}


Note that it is possible to estimate the UD at a succession of $t_i$
close in time, then to draw the images corresponding and save them in
a file, and finally to combine these images into a movie. The movie
can then be explored to identify patterns. For example, the following
code can be used to generate image files (because it is very long, we
do not execute it in this vignette:

<<eval=FALSE>>=
## compute the sequence of dates at which the UD is to be
## estimated
vv <- seq(min(bear[[1]]$date), max(bear[[1]]$date), length=50)
head(vv)

## estimates the UD at each time point
re <- lapply(1:length(vv), function(i) {

    ## estimate the UD. We choose a smoothing parameter of
    ## 1000 meters for X and Y coordinates, and of 72 hours
    ## for the time (after a visual exploration)
    uu <- kernelkc(bear, h = c(1000,1000,72*3600),
                   tcalc= vv[i], grid=50)

    ## To save the result in a file, type
    jpeg(paste("UD", i, ".jpg", sep=""))


    ## draw the image
    image(uu, col=grey(seq(1,0,length=10)))
    title(main=vv[i])

    ## highlight the 95 percent home range
    ## we set standardize = TRUE because we want to estimate
    ## the home range in space from a UD estimated in space and
    ## time
    plot(getverticeshr(uu, 95, standardize=TRUE), lwd=2,
         border="red", add=TRUE)

    ## close the device
    dev.off()
})
@

The images can then be combined using the program \texttt{mencoder}
(Linux) or \texttt{ImageMagick} (Windows). For Linux users, the
following command line can be used:\\

\begin{scriptsize}
\texttt{mencoder mf://*.jpg -mf w=320:h=240:fps=15:type=jpeg -ovc lavc -o UD.avi}
\end{scriptsize}



\subsubsection{Application with time considered as a circular variable}

Now, consider the time as a circular variable. We consider a time
cycle of 24 hour. We will estimate the UD at 03h00. We have to define
the beginning of the cycle (argument \texttt{t0} of the function
\texttt{kernelkc}). We consider that the time cycle begins at midnight
(there is no particular reason, it is just to illustrate the
function). We have to set, as the argument \texttt{t0}, any
object of class POSIXct corresponding to a date at this hour, for
example the 12/25/2012 at 00H00:

<<>>=
t0 <- as.POSIXct("2012-12-25 00:00")
@

We define $h=0.2$ for the time dimension (remember that $h$ should be
comprised between 0 and 1). We can have a look at the amount of weight
given to the neighbouring time points of the point 3h00:

<<label=exwc,echo=FALSE,fig=FALSE,eval=FALSE>>=
exwc(0.2)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<exwc>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<exwc>>
<<zfig>>
@
\end{center}

The function \texttt{exwc} draws a graph of the wrapped Cauchy
distribution for the chosen h parameter (for circular time), so
that it is possible to make one's mind concerning the weight that
can be given to the neighbouring points of a given time point. Now use
the function \texttt{kernelkc} to estimate the UD:

<<label=imagekernelkc1,echo=FALSE,fig=FALSE,eval=FALSE>>=
uu <- kernelkc(bear, h=c(1000,1000,0.2), cycle=24*3600,
               tcalc=3*3600, t0=t0, circular=TRUE)
image(uu)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<imagekernelkc1>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<imagekernelkc1>>
<<zfig>>
@
\end{center}

\noindent \textit{Remark}: In these examples, we have used the function
\texttt{kernelkc} to estimate the UD at a given date, with an object
of class \texttt{ltraj} as argument. However, note that the product
kernel algorithm can be used on other kind of data. Thus,
the function \texttt{kernelkcbase} accepts as argument a data frame
with three columns (x, y and time). The example section on the help
page of this function provides examples of use.


\section{Convex hulls methods}

Several methods relying on the calculation of multiple convex hulls
have been developed to estimate the home range from relocation data.
The package \texttt{adehabitatHR} implements three such methods.  I
describe these three methods in this section.\\

\noindent \textit{Remark:} note that the function
\texttt{getverticeshr} is generic, and can be used for all
home-range estimation methods of the package \texttt{adehabitatHR}
(except for the minimum convex polygon).\\


\subsection{The single linkage algorithm}

\subsubsection{The method}

Kenward et al. (2001) proposed a modification of the single-linkage
cluster analysis for the estimation of the home range.   Whereas this
method does not rely on a probabilistic definition of the home range
(contrary to the kernel method), it can still be useful to identify
the structure of the home range.\\

The clustering process is the following (also described on the help
page of the function \texttt{clusthr}): the three locations with the
minimum mean of nearest-neighbour joining distances (NNJD) form the
first cluster.  At each step of the clustering process, two distances
are computed: (i) the minimum mean NNJD between three locations (which
corresponds to the next potential cluster) and (ii) the minimum of the
NNJD between a cluster "c" and the closest location.  If (i) is
smaller that (ii), another cluster is defined with these three
locations.  If (ii) is smaller than (i), the cluster "c" gains a new
location.  If this new location belong to another cluster, the two
cluster fuses.  The process stop when all relocations are assigned to
the same cluster.\\

At each step of the clustering process, the proportion of all
relocations which are assigned to a cluster is computed (so that
the home range can be defined to enclose a given proportion of the
relocations at hand, i.e. to an uncomplete process).  At a given
step, the home range is defined as the set of minimum convex
polygons enclosing the relocations in the clusters.\\


\subsubsection{The function \texttt{clusthr}}


Take as example the \texttt{puechabonsp} dataset:

<<>>=
uu <- clusthr(puechabonsp$relocs[,1])
class(uu)
@

The class \texttt{MCHu} is returned by all the home range estimation
methods based on clustering methods (LoCoH and single linkage).
The object \texttt{uu} is basically a list of
\texttt{SpatialPolygonsDataFrame} objects (one per animal).  We can
have a display of the home ranges by typing:

<<label=kkkssssm,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(uu)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<kkkssssm>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<kkkssssm>>
<<zfig>>
@
\end{center}

The information stored in each \texttt{SpatialPolygonsDataFrame}
object is the following (for example, for the third animal):

<<>>=
as.data.frame(uu[[3]])
@

From this data frame, it is possible to extract a home range
corresponding to a particular percentage of relocations enclosed in
the home range.  For example, the smallest home range enclosing 90\% of
the relocations corresponds to the row 26 of this data frame:

<<label=ksqkskq,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(uu[[3]][26,])
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<ksqkskq>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<ksqkskq>>
<<zfig>>
@
\end{center}



\subsubsection{Rasterizing an object of class \texttt{MCHu}}

It is possible to rasterize the estimated home ranges using the
function \texttt{MCHu.rast}.  For example, using the \texttt{map}
component of the \texttt{puechabonsp} dataset:

<<label=sldlsl,echo=FALSE,fig=FALSE,eval=FALSE>>=
uur <- MCHu.rast(uu, puechabonsp$map, percent=90)
image(uur, 3)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<sldlsl>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<sldlsl>>
<<zfig>>
@
\end{center}


\subsubsection{Computing the home-range size}

We already noticed that the object of class \texttt{MCHu} returned by
the function \texttt{clusthr} contains the area of each home
range.  For example, for the first animal:

<<>>=
head(as.data.frame(uu[[1]]))
@

and as usual, the units of these areas are controlled by the
parameters \texttt{unin} and \texttt{unout} of the function
\texttt{clusthr}.  I also included a function \texttt{MCHu2hrsize},
which generates an object of class \texttt{hrsize}, and makes easier
the exploration of the relationship between the home range size and
the percentage of relocations included in the home range:

<<label=slqllqq,echo=FALSE,fig=FALSE,eval=FALSE>>=
ii <- MCHu2hrsize(uu, percent=seq(50, 100, by=5))
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<slqllqq>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<slqllqq>>
<<zfig>>
@
\end{center}

Here, the choice of 90\% seems a good choice for the
estimation, as the area-percentage relationship seems to reach a rough
asymptote (no steep fall in the decrease below this value).



\subsection{The LoCoH methods}

The LoCoH (Local Convex Hulls) family of methods has been proposed by
Getz et al. (2007) for locating Utilization Distributions from
relocation data.  Three possible algorithms can be used: Fixed k
LoCoH, Fixed r LoCoH, and Adaptive LoCoH.  The three algorithms are
implemented in \texttt{adehabitatHR}.  All the algorithms work by constructing
a small convex hull for each relocation and its neighbours, and then
incrementally merging the hulls together from smallest to largest into
isopleths.  The 10\% isopleth contains 10\% of the points and
represents a higher utilization than the 100\% isopleth that contains
all the points (as for the single linkage method).  I describe the
three methods below (this description is also on the help page of the
functions \texttt{LoCoH.k}, \texttt{LoCoH.r} and \texttt{LoCoH.a}):\\

\begin{itemize}
\item \textbf{Fixed k LoCoH}: Also known as k-NNCH, Fixed k LoCoH is
  described in Getz and Willmers (2004).  The convex hull for each
  point is constructed from the (k-1) nearest neighbors to that point.
  Hulls are merged together from smallest to largest based on the area
  of the hull.  This method is implemented in the function
  \texttt{LoCoH.k};\\

\item \textbf{Fixed r LoCoH}: In this case, hulls are created from all
  points within $r$ distance of the root point.  When merging hulls,
  the hulls are primarily sorted by the value of k generated for each
  hull (the number of points contained in the hull), and secondly by
  the area of the hull.  This method is implemented in the function
  \texttt{LoCoH.r};\\

\item \textbf{Adaptive LoCoH}:  Here, hulls are created out of the
  maximum nearest neighbors such that the sum of the distances from
  the nearest neighbors is less than or equal to d.  Use the same hull
  sorting as Fixed r LoCoH.  This method is implemented in the
  function \texttt{LoCoH.a};\\
\end{itemize}

All of these algorithms can take a significant amount of time.\\

\noindent \textit{Note}: these functions rely on the availability of
the packages rgeos and gpclib, which may not be available for all OS
(especially MacOS).


\subsection{Example with the Fixed k LoCoH method}

We use the fixed k LoCoH method to estimate the home range of the wild
boars monitored at Puechabon, for example with $k = 5$:

<<eval=FALSE>>=
res <- LoCoH.k(puechabonsp$relocs[,1], 10)
@

The object returned by the function is of the class \texttt{MCHu}, as
the objects returned by the function \texttt{clusthr} (see previous
section).  Consequently, all the functions developed for this class of
objects can be used (including rasterization with \texttt{MCHu.rast},
computation of home-range size with \texttt{MCHu2hrsize}, etc.).  A
visual display of the home ranges can be obtained with:

<<eval=FALSE, fig=TRUE>>=
plot(res)
@

Note that the relationship between the home range size and the number
of chosen neighbours can be investigated thanks to the function
\texttt{LoCoH.k.area}:

<<eval=FALSE>>=
LoCoH.k.area(puechabonsp$relocs[,1], krange=seq(4, 15, length=5),
             percent=100)
@

Just copy and paste the above code (it is not executed in this
report).  An asymptote seems to be reached at about 10 neighbours,
which justifies the choice of 10 neighbours for an home range
estimated with 100\% of the relocations (a more stable home range
size).


\subsection{The characteristic hull method}

Downs and Horner (2009) developed an interesting method for the home
range estimation. This method relies on the calculation of the
Delaunay triangulation of the set of relocations. Then, the triangles
are ordered according to their area (and not their perimeter). The
smallest triangles correspond to the areas the most intensively used
by the animals. It is then possible to derive the home range estimated
for a given percentage level.\\

For example, consider the \texttt{puechabonsp} dataset.

<<>>=
res <- CharHull(puechabonsp$relocs[,1])
class(res)
@

The object returned by the function is also of the class \texttt{MCHu}, as
the objects returned by the functions \texttt{clusthr} and
\texttt{LoCoH.*} (see previous sections).  Consequently, all the
functions developed for this class of objects can be used (including
rasterization with \texttt{MCHu.rast}, computation of home-range size
with \texttt{MCHu2hrsize}, extraction of the home-range contour with
\texttt{getverticeshr} etc.).  A visual display of the home ranges can
be obtained with:

<<label=plotcharhul,echo=FALSE,fig=FALSE,eval=FALSE>>=
plot(res)
@

<<results=tex,echo=TRUE,fig=FALSE,eval=FALSE>>=
<<plotcharhul>>
@

\begin{center}
<<results=tex,echo=FALSE>>=
<<afig>>
<<plotcharhul>>
<<zfig>>
@
\end{center}




\section{Conclusion}

I included in the package \texttt{adehabitatHR} several functions allowing
the estimation of the home range and utilization distribution of
animals using relocation data.  These concepts are extremely useful for
the study of animal space use, but they consider the use of space from
a purely spatial point of view.  In other words, time is not taken into
account...\\

The package \texttt{adehabitatLT} represents an attempt to shift our point
of view on relocation data toward a more dynamic point of view.
Indeed, with the increasing use of GPS collars, the time dimension of
the monitoring becomes more and more important in studies of animals
space use.  This package contains several functions developed for the
analysis of trajectories.  The class \texttt{ltraj}, which is at the
very core of this package, has been developped to make possible the
analysis of animals trajectories.\\

Another important aspect in the study of animals space use is the
study of their relationships with the environment (i.e., habitat
selection).  The package \texttt{adehabitatHS} proposes several methods
allowing to explore this aspect of animals space use.  Several
functions are implemented, relying on the study of the distribution of
the time spent by the animals in the multidimensional space defined by
the environmental variables.  In addition to common methods in the
analysis of habitat selection (compositional analysis of habitat use,
selection ratios, etc.), the package \texttt{adehabitatHS} proposes several
exploratory tools useful to identify patterns in habitat
selection.  The study of habitat selection by animals is possible
through the interaction with the other brother packages (of the suite
\texttt{adehabitat}), but also thanks to te tools provided by the
package \texttt{sp}, which allow to handle raster and vector maps of
environmental variables.  Note that the package \texttt{adehabitatMA}
provides additionnal tools which can be useful in the
particular field of habitat selection study.



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\end{description}

\end{document}

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