# simm.brown: Simulate a Bivariate Brownian Motion In adehabitat: Analysis of Habitat Selection by Animals

## Description

This function simulates a Bivariate Brownian Motion.

## Usage

 `1` ```simm.brown(date = 1:100, x0 = c(0, 0), h = 1, id = "A1", burst = id) ```

## Arguments

 `date` a vector indicating the date (in seconds) at which relocations should be simulated. This vector can be of class `POSIXct` `x0` a vector of length 2 containing the coordinates of the startpoint of the trajectory `h` Scaling parameter for the brownian motion (larger values give smaller dispersion) `id` a character string indicating the identity of the simulated animal (see `help(ltraj)`) `burst` a character string indicating the identity of the simulated burst (see `help(ltraj)`)

## Details

A bivariate Brownian motion can be described by a vector `B2(t) = (Bx(t), By(t))`, where `Bx` and `By` are unidimensional Brownian motions. Let `F(t)` the set of all possible realisations of the process `(B2(s), 0 < s < t)`. `F(t)` therefore corresponds to the known information at time `t`. The properties of the bivariate Brownian motion are therefore the following: (i) `B2(0)= c(0,0)` (no uncertainty at time `t = 0`); (ii) `B2(t) - B2(s)` is independent of `F(s)` (the next increment does not depend on the present or past location); (iii) `B2(t) - B2(s)` follows a bivariate normal distribution with mean `c(0,0)` and with variance equal to `(t-s)`.

Note that for a given parameter `h`, the process ```1/h * B2( t * h^2 )``` is a Brownian motion. The function `simm.brown` simulates the process `B2(t * h^2)`. Note that the function `hbrown` allows the estimation of this scaling factor from data.

## Value

An object of class `ltraj`

## Author(s)

Clement Calenge [email protected]
Stephane Dray [email protected]
Manuela Royer [email protected]
Daniel Chessel [email protected]

## References

~put references to the literature/web site here ~

`ltraj`, `hbrown`
 ```1 2 3 4``` ```plot(simm.brown(1:1000), addpoints = FALSE) ## Note the difference in dispersion: plot(simm.brown(1:1000, h = 4), addpoints = FALSE) ```