# simm.mba: Simulation of an Arithmetic Brownian Motion In adehabitat: Analysis of Habitat Selection by Animals

## Description

This function simulates an Arithmetic Brownian Motion.

## Usage

 ```1 2``` ```simm.mba(date = 1:100, x0 = c(0, 0), mu = c(0, 0), sigma = diag(2), 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 `mu` a vector of length 2 describing the drift of the movement `sigma` a 2*2 positive definite matrix `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

The arithmetic Brownian motion (Brillinger et al. 2002) can be described by the stochastic differential equation:

dz = mu * dt + Sigma dB2(t)

Coordinates of the animal at time t are contained in the vector `z(t)`. `dz = c(dx, dy)` is the increment of the movement during dt. `dB2(t)` is a bivariate brownian Motion (see `?simm.brown`). The vector `mu` measures the drift of the motion. The matrix `Sigma` controls for perturbations due to the random noise modeled by the Brownian motion. It can also be used to take into account a potential correlation between the components dx and dy of the animal moves during dt (see Examples).

## 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

Brillinger, D.R., Preisler, H.K., Ager, A.A. Kie, J.G. & Stewart, B.S. (2002) Employing stochastic differential equations to model wildlife motion. Bulletin of the Brazilian Mathematical Society 33: 385–408.

`simm.brown`, `ltraj`, `simm.crw`, `simm.mou`
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```set.seed(253) u <- simm.mba(1:1000, sigma = diag(c(4,4)), burst = "Brownian motion") v <- simm.mba(1:1000, sigma = matrix(c(2,-0.8,-0.8,2), ncol = 2), burst = "cov(x,y) > 0") w <- simm.mba(1:1000, mu = c(0.1,0), burst = "drift > 0") x <- simm.mba(1:1000, mu = c(0.1,0), sigma = matrix(c(2, -0.8, -0.8, 2), ncol=2), burst = "Drift and cov(x,y) > 0") z <- c(u, v, w, x) plot(z, addpoints = FALSE, perani = FALSE) ```