====== Brownian Motion ======

The function ''brownian.motion()'' has illustrated the phenomenon of [[wp>random_walk|Random Walk]] on the 2D plane. For a point ''(x, y)'', its next location is ''(x + rnorm(1), y + rnorm(1))''.

===== R code =====

<code r>
# create an HTML animation page of Brownian Motion
# store the old option to restore it later
oopt = ani.options(interval = 0.05, nmax = 100, ani.dev = png, ani.type = "png",
    title = "Demonstration of Brownian Motion",
    description = "Random walk on the 2D plane: for each point (x, y), 
    x = x + rnorm(1) and y = y + rnorm(1).")
ani.start()
opar = par(mar = c(3, 3, 1, 0.5), mgp = c(2, .5, 0), tcl = -0.3,
    cex.axis = 0.8, cex.lab = 0.8, cex.main = 1)
brownian.motion(pch = 21, cex = 5, col = "red", bg = "yellow")
par(opar)
ani.stop()
ani.options(oopt)
</code>

===== Animation =====

The animation is like this:

<ani bm http://i288.photobucket.com/albums/ll181/xieyihui/brownian_motion/ png 100 480 0.05|Demonstration of Brownian Motion>
Random walk on the 2D plane: for each point (x, y), x = x + rnorm(1) and y = y + rnorm(1).
</ani>

===== Other resources =====

  * I find this Java applet is much more interesting than my function: [[http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/brownian/brownian.html|Einstein's Explanation of Brownian Motion]]

===== Reminder to lattice users =====

If you use the ''lattice'' package, don't forget to ''print()'' your plots, otherwise graphics devices like ''png()''
 will not be able to record your images. Here is an example:

<code r>
library(animation)
library(lattice)
bm.mat = data.frame(x = rnorm(10), y = rnorm(10))
oopt = ani.options(interval = 0.05, nmax = 100, 
    title = "Demonstration of Brownian Motion",
    description = "Random walk on the 2D plane: for each point (x, y), 
    x = x + rnorm(1) and y = y + rnorm(1).")
## Brownian Motion in lattice
for (i in 1:ani.options("nmax")) {
    print(xyplot(y ~ x, bm.mat, xlim = c(-10, 10), ylim = c(-10, 10), pch = 19))
    bm.mat = bm.mat + data.frame(x = rnorm(10), y = rnorm(10))
}
ani.stop() 
ani.options(oopt)
</code>