In JimMcL/trajr: Animal Trajectory Analysis

knitr::opts_chunk$set(echo = FALSE)
library(trajr)

The problem

Trajectories in trajr consist of locations sampled at discrete times. We generally make the simplifying assumption that the animal moves in a straight line at a constant velocity between locations (Fig. 1).

x <- c(1, 1, 1, 1, 1)
y <- 1:5 * 8
times <- cumsum(c(0, 8, 4, 2, 1))
trj <- TrajFromCoords(data.frame(x, y, times), timeCol = 3)

displ <- cumsum(Mod(trj$displacement))
speed <- TrajDerivatives(trj)$speed

plot(NULL, xlim = range(times), ylim = range(displ), xlab = "Time", ylab = "Distance", main = "Distance travelled")
lines(times, displ, type = "b")

In fact, a trajectory like this would have a discontinuous first derivative (Fig. 2, black lines). Consequently, acceleration would be infinite at the jumps between steps, and 0 along each step. Since this is not a useful way to model acceleration, we instead assume that acceleration occurs for some amount of time. The method implemented in TrajDerivatives (Fig. 2, red lines) assumes the acceleration occurs from the start until the end of the step. The method implemented in TrajAcceleration (Fig. 2, blue lines) assumes the acceleration occurs from half-way through the first step in a pair, until half-way through the second step.

More sophisticated techniques exist to estimate continuous displacement and speed from trajectories, but they are not implemented by trajr. For one such technique and a comparison with others, see:

Noonan, M. J., Fleming, C. H., Akre, T. S., Drescher-Lehman, J., Gurarie, E., Harrison, A.-L., . . . Calabrese, J. M. (2019). Scale-insensitive estimation of speed and distance traveled from animal tracking data. Movement Ecology, 7(1), 35. doi:10.1186/s40462-019-0177-1

plot(NULL, xlim = range(times), ylim = range(c(0, speed)), xlab = "Time", ylab = "Speed", main = "Approximating acceleration")
for (i in seq_along(speed)) {
  segments(times[i], speed[i], times[i + 1], speed[i], lwd = 2)
}

# Two ways to calculate acceleration
lines(tail(times, -1), speed, type = "b", col = "red", lwd = 2)
lines(head(times, -1) + diff(times) / 2, speed, type = "b", col = "blue", lwd = 2)
legend("topleft", c("Constant speed", "TrajDerivatives", "TrajAcceleration"), lwd = 2, col = c("black", "red", "blue"))