TrajExpectedSquareDisplacement: Trajectory expected square displacement

Description

Calculates the expected square displacement for a trajectory assuming it is a correlated random walk, using the formula in Kareiva & Shigesada, (1983).

Usage

TrajExpectedSquareDisplacement(
  trj,
  n = nrow(trj),
  eqn1 = TRUE,
  compass.direction = NULL
)

Arguments

trj: A Trajectory.
n: Number of steps to calculate.
eqn1: If TRUE, calculate using equation 1, otherwise using equation 2. Equation 2 applies when the mean of turning angles is 0, i.e.turns are unbiased.
compass.direction: If not NULL, step angles are calculated relative to this angle (in radians), otherwise they are calculated relative to the previous step angle.

Details

Note that Cheung, Zhang, Stricker, and Srinivasan (2007) define an alternative formulation for expected maximum displacement, Emax (see TrajEmax).

References

Cheung, A., Zhang, S., Stricker, C., & Srinivasan, M. V. (2007). Animal navigation: the difficulty of moving in a straight line. Biological Cybernetics, 97(1), 47-61. doi:10.1007/s00422-007-0158-0

Kareiva, P. M., & Shigesada, N. (1983). Analyzing insect movement as a correlated random walk. Oecologia, 56(2), 234-238. doi:10.1007/bf00379695

Examples

Run this code

set.seed(1)
# A random walk
trj <- TrajGenerate(200)
smoothed <- TrajSmoothSG(trj)

# Calculate actual squared displacement at all points along the trajectory
sd2 <- sapply(2:nrow(smoothed), function(n) TrajDistance(smoothed, 1, n) ^ 2)
# Calculate expected squared displacement
ed2_1 <- sapply(2:nrow(smoothed), function(n) TrajExpectedSquareDisplacement(smoothed, n, TRUE))
ed2_2 <- sapply(2:nrow(smoothed), function(n) TrajExpectedSquareDisplacement(smoothed, n, FALSE))

# Plot expected against actual. According to Kareiva & Shigesada, (1983), if actual
# (approximately) matches expected, the trajectory is probably a correlated random walk
par(mar = c(5, 5, 0.1, 0.1) + .1)
plot(2:nrow(smoothed), sd2, type = 'l', pch = 16, cex = .2, lwd = 2,
     xlab = 'Number of consecutive moves',
     ylab = expression('Squared displacement, ' * R[n]^2))
lines(2:nrow(smoothed), ed2_1, col = "grey", lwd = 2)
lines(2:nrow(smoothed), ed2_2, col = "pink", lwd = 2)

legend("bottomright",
       c(expression("Actual displacement"^2),
         expression("Expected displacement"^2 * " (eqn 1)"),
         expression("Expected displacement"^2 * " (eqn 2)")),
       col = c('black', 'grey', 'pink'), lwd = 2,
       inset = c(0.01, 0.02))