BRB {adehabitatHR} R Documentation

## Utilization Distribution of an Animal Based on Biased Random Bridges

### Description

This function estimates the utilization distribution of one/several animals using a biased random bridge approach (Benhamou and Cornelis 2010, Benhamou 2011). This function also allows the decomposition of the utilization distribution into (i) an intensity distribution reflecting the average time spent by the animal in the habitat patches, and (ii) a recursion distribution reflecting the number of visits of the animal in the habitat patches (Benhamou and Riotte-Lambert, 2012).

### Usage

```BRB(ltr, D, Tmax, Lmin, hmin, type=c("UD","ID", "RD"), radius = NULL,
maxt = NULL, filtershort = TRUE, habitat = NULL, activity = NULL,
grid = 200, b=FALSE, same4all=FALSE, extent=0.5, tau = NULL,
boundary=NULL)

BRB.D(ltr, Tmax = NULL, Lmin = NULL, habitat = NULL, activity = NULL)

BRB.likD(ltr, Dr=c(0.1,100),
Tmax = NULL, Lmin = NULL,
habitat = NULL, activity = NULL)

```

### Arguments

 `ltr` an object of class `ltraj`. `D` a number corresponding to the diffusion parameter (in squared "units" per second, where "units" denote the units of the relocation coordinates) used for the estimation. Alternatively this parameter may be an object of class `DBRB` returned by the function `BRB.D`. `Dr` a vector of length two giving the lower and upper limits of the diffusion coefficient, within which the maximum likelihood could be found. `Tmax` the maximum duration (in seconds) allowed for a step built by successive relocations. All steps characterized by a duration dt greater than `Tmax` are not taken into account in the calculations. `Lmin` the minimum distance (in units of the coordinates) between successive relocations, defining intensive use or resting (See details). The distance should be specified in units of the relocation coordinates (i.e. in metres if they are specified in metres). `hmin` The minimum smoothing parameter (in units of the relocations coordinates), applied to all recorded relocations. See details for a description of this parameter. `type` The type of distribution expected by the user: `"UD"` returns the utilization distribution using the approach described by Benhamou and Cornelis (2010). `"ID"` returns the intensity distribution described in Benhamou and Riotte-Lambert (2012), i.e. a distribution reflecting the average time spent in habitat patches. `"RD"` returns the recursion distribution described in Benhamou and Riotte-Lambert (2012), i.e. a distribution reflecting the number of visits in habitat patches. `radius` If `type = "ID"` or `"RD"`, the radius of the patches (in units of the relocation coordinates) used in the calculation of the residence time or the number of visits. If `NULL`, the radius is set to `3*hmin`. `maxt` If `type = "ID"` or `"RD"`, maximum time threshold (in seconds) that the animal is allowed to spend outside the patch before that we consider that the animal actually left the patch (see `?residenceTime`). `filtershort` logical indicating the behaviour of the function when the length of a step is lower than `Lmin` (see details). It must be set to TRUE if track segments shorter than Lmin are assumed to correspond to resting periods, which thereby will be filtered out systematically, or to FALSE to take these short track segments into account when not associated to resting (animals active without moving more than Lmin). They then will be given a null diffusion coefficient. `habitat` optionally, an object of class `SpatialPixelsDataFrame` with one column describing the habitat type on the area. `activity` optionally, a character indicating the name of the variable in the infolocs component of `ltr` indicating the proportion of time between relocation i and relocation i+1 during which the animal was active (Users of adehabitatLT prior to version 0.3 should read the section Warning below). `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 `SpatialPixels`, or a list of objects of class `SpatialPixels` with as many elements as there are bursts in `ltr`. `b` logical specifying how relocation and movement variances are combined. If `TRUE`, the relocation variance progressively merges with the movement component; if `FALSE`, the relocation variance has a constant weight (see Benhamou, 2011). `same4all` logical. if `TRUE`, the same grid is used for the estimation of all bursts. If `FALSE`, one grid is used per burst. `extent` a value indicating the extent of the grid used for the estimation (the extent of the grid on the abscissa is equal to `(min(xy[,1]) + extent * diff(range(xy[,1])))`). `tau` interpolation time (tau, in seconds). Defaults to `tmin/10`, where `tmin` is the minimum duration of a step in `ltr`. `boundary` If, not `NULL`, an object inheriting the class `SpatialLines` defining a barrier that cannot be crossed by the animals. There are constraints on the shape of the barrier that depend on the smoothing parameter (***see details***)

### Details

The function `BRB` uses the biased random bridge approach to estimate the Utilization Distribution of an animal with serial autocorrelation of the relocations. This approach is similar to the Brownian bridge approach (see `?kernelbb`), with several noticeable improvements. Actually, the Brownian bridge approach supposes that the animal movement is random and purely diffusive between two successive relocations: it is supposed that the animal moves in a purely random fashion from the starting relocation and reaches the next relocation randomly. The BRB approach goes further by adding an advection component (i.e., a "drift") to the purely diffusive movement: is is supposed that the animal movement is governed by a drift component (a general tendency to move in the direction of the next relocation) and a diffusion component (tendency to move in other directions than the direction of the drift).

The BRB approach is based on the biased random walk model. This model is the following: at a given time t, the speed of the animal is drawn from a probability density function (pdf) and the angle between the step and the east direction is drawn from a circular pdf with given mean angle and concentration parameters. A biased random walk occurs when this angular distribution is not uniform (i.e. there is a preferred direction of movement). Now, consider two successive relocations r1 = (x1, y1) and r2 = (x2, y2) collected respectively at times t1 and t2. The aim of the Biased Random Bridges approach is to estimate the pdf that the animal is located at a given place r = (x,y) at time ti (with t1 < ti < t2), given that it is located at r1,r2 at times t1,t2, and given that the animal moves according a biased random walk with an advection component determined by r1 and r2.

Benhamou (2011) proposed an approximation for this pdf, noting that it can be approximated by a circular bivariate normal distribution with mean location corresponding to (x1 + pi*(x2-x1), y1 + pi*(y2-y1)), where pi = (ti-t1)/(t2-t1). The variance-covariance matrix of this distribution is diagonal, with both diagonal elements corresponding to the diffusion coefficient D. This coefficient D can be estimated using the plug-in method, using the function `BRB.D` (for details, see Benhamou, 2011). Note that the diffusion parameter `D` can be estimated for each habitat type is a habitat map is available. Note that the function `BRB.likD` can be used alternatively to estimate the diffusion coefficient using the maximum likelihood method.

An important aspect of the BRB approach is that the drift component is allowed to change in direction and strength from one step to the other, but should remain constant during each of them. For this reason, it is required to set an upper time threshold `Tmax`. Steps characterized by a longer duration are not taken into account into the estimation of the pdf. This upper threshold should be based on biological grounds.

As for the Brownian bridge approach, this conditional pdf based on biased random walks takes an infinite value at times ti = t1 and ti = t2 (because, at these times, the relocation of the animal is known exactly). Benhamou proposed to circumvent this drawback by considering that the true relocation of the animal at times t1 and t2 is not known exactly. He noted: "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. This is controlled by the parameter `b` of the function. In both cases, the minimum uncertainty over the relocation of an animal is observed for ti = t1 or t2. This minimum standard deviation corresponds to the parameter `hmin`. According to Benhamou and Cornelis, "`hmin` 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, `hmin` 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, hmin should be large enough to encompass potential locations occurring in the same habitat patch as the recorded location".

Practically, the BRB approach can be carried out with the help of the movement-based kernel density estimation (MKDE) developed by Benhamou and Cornelis (2010). This method consists in dividing each step i into Ti/tau intervals, where Ti is the duration of the step (in seconds) and `tau` is the interpolation time (in seconds). A kernel density estimation is then used to estimate the required pdf, with a smoothing parameter varying with each interpolated location ri and corresponding to: hi^2 = hmin^2 + 4pi(1-pi)(hmax^2 - hmin^2)Ti/Tmax. In this equation, `hmax^2` corresponds to `hmin^2+ D*Tmax/2` if `b` is FALSE and to `D*Tmax/2` otherwise. Note that this smoothing parameter may be a function of the habitat type where the interpolated relocation occurs if the diffusion parameters are available for each habitat types.

The special case where a given step covers a distance lower than `Lmin` merits further details. When the parameter `filtershort = TRUE`, it is always assumed that the animal was resting at this time, and this step is filtered out before the estimation. When the parameter `filtershort = FALSE`, this assumption is not made. In this case, the behaviour of the function depends on the availability of a variable measuring the activity of the animal (when the name of the variable containing the ai in the `infolocs` component is passed as the parameter `activity`; see `?infolocs` for additionnal information on this component). If the animal was active during the step, the smoothing parameter hi is set to `hmin` for this step. This procedure allows to give more weight to the immediate surroundings of this relocation (indicating an intensive use of these immediate surroundings). If the animal was inactive, then the animal was resting and the step is filtered out before the estimation. Note however that activity value may sometimes be relatively high while the animal is resting, e.g. if disturbed by flies, possibly requiring manual correction of activity values based on the distance moved between relocations).

If no activity variable is available and `filtershort = FALSE`, it is always suppose that the animal was active between the two relocations, the step is not filtered out and the smoothing parameter hi is set to `hmin` for this step.

The parameter `boundary` allows to define a barrier that cannot be crossed by the animals. When this parameter is set, the method described by Benhamou and Cornelis (2010) for correcting boundary biases is used. The boundary can possibly be defined by several nonconnected lines, each one being built by several connected segments. Note that there are constraints on these segments (not all kinds of boundary can be defined): (i) each segment length should at least be equal to `3*h` (the size of "internal lane" according to the terminology of Benhamou and Cornelis), (ii) the angle between two line segments should be greater that `pi/2` or lower that `-pi/2`. The UD of all the pixels located within a band defined by the boundary and with a width equal to `6*h` ("external lane") is set to zero.

Benhamou and Riotte-Lambert (2012) showed that the space use at any given location, as estimated by this approach, can be seen as the product between the mean residence time per visit times the number of visits of the location. They proposed an approach allowing the decomposition of the UD into two components: (i) the intensity distribution reflecting this average residence time and (ii) a recursion distribution reflecting the number of visits. This function allows to estimate these two components, by setting the argument `type` to `"ID"` and `"RD"` respectively.

Note that all the methods available to deal with objects of class `estUDm` are available to deal with the results of the function `BRB` (see `?kernelUD`).

### Value

`BRB` returns an object of class `estUDm` when the UD is estimated for several animals, and `estUD` when only one animal is studied.

`BRB.D` and `BRB.likD` returns a list of class `DBRB`, with one component per burst containing a data frame with the diffusion parameters.

### Warning

Users of the version 0.2 of adehabitatHR should be careful that there was a slight inconsistency in the package design: whereas all the parameters characterizing the steps in an object of class `"ltraj"` (e.g. `dist, dx, dy`) describe the step between relocations i and i+1, it was expected for `BRB` that the activity described the proportion of activity time between relocation i-1 and i. This inconsistency has now been corrected since version 0.3.

### Author(s)

Clement Calenge clement.calenge@ofb.gouv.fr, based on a C translation of the Pascal source code of the program provided by Simon Benhamou.

### References

Benhamou, S. (2011) Dynamic approach to space and habitat use based on biased random bridges PLOS One, 6, 1–8.

Benhamou, S. and Cornelis, D. (2010) Incorporating Movement Behavior and Barriers to Improve Biological Relevance of Kernel Home Range Space Use Estimates. Journal of Wildlife Management, 74, 1353–1360.

Benhamou, S. and Riotte-Lambert, L. (2012) Beyond the Utilization Distribution: Identifying home range areas that are intensively exploited or repeatedly visited. Ecological Modelling, 227, 112–116.

### See Also

`kernelbb` for the Brownian bridge kernel estimation, `kernelUD` and `estUD-class` for additional information about objects of class `estUDm` and `estUD`, `infolocs` for additional information about the `infolocs` component, `as.ltraj` for additional information about the class `ltraj`.

### Examples

```
## Example dataset used by Benhamou (2011)
data(buffalo)

## The trajectory:
buffalo\$traj

## The habitat map:
buffalo\$habitat

## Show the dataset
plot(buffalo\$traj, spixdf = buffalo\$habitat)

## Estimate the diffusion component for each habitat type
## Using the plug-in method
vv <- BRB.D(buffalo\$traj, Tmax = 180*60, Lmin = 50,
habitat = buffalo\$habitat, activ = "act")

vv

## Note that the values are given here as m^2/s, whereas
## they are given as m^2/min in Benhamou (2011). The
## values in m^2 per min are:
vv[][,2]*60

## Approximately the same values, with slight differences due to
## differences in the way the program of Benhamou (2011) and the present
## one deal with the relocations occurring on the boundary between two
## different habitat types
## Note that an alternative estimation of the Diffusion coefficient
## could be found using maximum likelihood
vv2 <- BRB.likD(buffalo\$traj, Tmax = 180*60, Lmin = 50,
habitat = buffalo\$habitat, activ = "act")
vv2
vv[][,2]*60

## Estimation of the UD with the same parameters as those chosen by
## Benhamou (2011)
ud <- BRB(buffalo\$traj, D = vv, Tmax = 180*60, tau = 300, Lmin = 50, hmin=100,
habitat = buffalo\$habitat, activity = "act", grid = 50, b=0,
same4all=FALSE, extent=0.5)
ud

## Show the UD.
image(ud)

## Not run:
## Example of the decomposition of the UD into a recursion distribution
## and a intensity distribution (Benhamou and Riotte-Lambert 2012).
##
## 1. Intensity Distribution using the same parameters as Benhamou and
## Riotte-Lambert (2012)

id <- BRB(buffalo\$traj, D = 440/60, Tmax = 3*3600, Lmin = 50, type = "ID",
hmin=100, radius = 300, maxt = 2*3600, activity="act", filtershort=FALSE,
grid = 200, extent=0.1)

rd <- BRB(buffalo\$traj, D = 440/60, Tmax = 3*3600, Lmin = 50, type = "RD",
hmin=100, radius = 300, maxt = 2*3600, activity="act", filtershort=FALSE,
grid = 200, extent=0.1)

ud <- BRB(buffalo\$traj, D = 440/60, Tmax = 3*3600, Lmin = 50,
hmin=100, radius = 300, maxt = 2*3600, activity="act", filtershort=FALSE,
grid = 200, extent=0.1)

par(mfrow = c(2,2), mar=c(0,0,2,0))
image(getvolumeUD(id))
title("ID")
image(getvolumeUD(rd))
title("RD")
image(getvolumeUD(ud))
title("UD")

## End(Not run)

```

[Package adehabitatHR version 0.4.19 Index]