Saturday, February 4, 2012

Implementing Circles example

This week I reimplemented part of Conic Sections 1 model from NetLogo. In the model turtles seek to to be in target distance from center.

My code takes only one center point, so only circles can be obtained. Apart from turtle location plot given in NetLogo implementation I added:
1. plot showing maximal difference between turtle distance and target distance;
2. decreasing turtle step size.
Here is the plot showing final simulation state, but it is also nice to watch the simulation run:

Below is the code generating the simulation:
```# n: number of turtles
# p.x, p.y: location of center
# range: turtles have random position from [0,range]
#        and will move in random angle a
# step: how fast turtles move
# target: target distance from center
# time: simulation time
init <- function(n, p.x, p.y, range, step, target, time) {
sim <- list(
turtles = data.frame(x = runif(n, max = range),
y = runif(n, max = range),
a = runif(n, max = 2 * pi)),
p.x = p.x, p.y = p.y, step = step, target = target,
time = time, max.dist = rep(NA, time))

# Calculate turtle distance from center
sim\$turtles\$dist <- sqrt((sim\$turtles\$x - p.x) ^ 2 +
(sim\$turtles\$y - p.y) ^ 2)
return(sim)
}

step <- function(sim) {
x <- sim\$turtles\$x
y <- sim\$turtles\$y

# Remember last distance and save current distance
o.dist <- sim\$turtles\$dist
n.dist <- sqrt((x - sim\$p.x) ^ 2 + (y - sim\$p.y) ^ 2)
sim\$turtles\$dist <- n.dist

# For turtles that are too far and are moving out
# or too close and are moving in randomly change direction
w.dist <- ((n.dist < o.dist) & (n.dist < sim\$target)) |
((n.dist > o.dist) & (n.dist > sim\$target))
sim\$turtles\$a[w.dist] <- runif(sum(w.dist), max = 2 * pi)

sim\$turtles\$x <- x + sin(sim\$turtles\$a) * sim\$step
sim\$turtles\$y <- y + cos(sim\$turtles\$a) * sim\$step
return(sim)
}

do.plot <- function(sim) {
rng <- quantile(c(sim\$turtles\$x, sim\$turtles\$y),
c(0.05, 0.95))
rng <- round(rng, -1) + c(-10, 10)
par(mai = rep(0.5, 4), mfrow = c(1, 2))
plot(sim\$turtles\$x, sim\$turtles\$y, pch = ".",
xlim = rng, ylim = rng, xlab = "", ylab = "",
main = "Turtle location")
points(sim\$p.x, sim\$p.y, col = "red", pch = 20, cex = 2)
plot(sim\$max.dist, type = "l",
ylim = c(0, max(sim\$max.dist, na.rm = TRUE) + 5),
xlab = "", ylab = "", main = "Max difference from target")
}

run <- function(sim) {
for (i in 1:sim\$time) {
sim <- step(sim)
sim\$step <- sim\$step * 127 / 128
sim\$max.dist[i] <- max(sim\$turtles\$dist) - sim\$target
do.plot(sim)
}
}

sim <- init(4096, 128, 128, 256, 2, 128, 512)
set.seed(0)
run(sim)
```

1. Very fun to run and watch!

You are decreasing the step size at a constant rate: the new step is (127/128) of the previous. For some initial conditions this might be too slow or too fast. I wonder if you can tie the rate to the number of simulations? For example, if you use
r <- (1e-2)**(1/sim\$time)
...
sim\$step <- sim\$step * r
then the step is getting very small at about the same time that the simulation ends.

2. Another option is to use adaptive step size:

sim\$step <- (max(sim\$turtles\$dist) - sim\$target) / 16

Interestingly for small divisors (for given parameters treshold seems to be around 10) the process diverges due to randomness of angle.