knitr::opts_chunk$set( collapse = FALSE, comment = "#>", dev="png", global.par = TRUE ) library("knitr") library("plotrix") library("trajr")
The trajr
R package is a toolkit for the numerical characterisation and analysis of the trajectories of moving animals. Trajectory analysis is applicable in fields as diverse as optimal foraging theory, migration, and behavioural mimicry (e.g. for verifying similarities in locomotion). A trajectory is simply a record of the path followed by a moving animal. Trajr
operates on trajectories in the form of a series of locations (as x, y coordinates) with times. Trajectories may be obtained by any method which provides this information, including manual tracking, radio telemetry, GPS tracking, and motion tracking from videos.
The goal of this package (and this document) is to aid biological researchers, who may not have extensive experience with R, to analyse trajectories without being handicapped by a limited knowledge of R or programming. However, a basic understanding of R is required.
Within this document, the plots are generated by running the code examples immediately before them. For that reason, most code examples start with a call to TrajGenerate
, which generates a random trajectory to be plotted. These calls would not occur in an analysis of real animal trajectories.
If you use trajr
in your publications, please cite McLean DJ, Skowron Volponi MA. trajr: An R package for characterisation of animal trajectories. Ethology. 2018;00:1–9. https://doi.org/10.1111/eth.12739.
Before using any R package, it is first necessary to install and load it. To install the latest released version of trajr
, simply run:
install.packages("trajr")
To install the latest development version, which is available on GitHub, run:
install.packages("devtools") devtools::install_github("JimMcL/trajr")
Once the package has been installed, you must load it, then it's ready for use:
library("trajr")
Packages only need to be installed once, but they must be loaded in every session before they can be used.
Once the trajr
package has been installed and loaded, you have access to the set of R functions it provides. All trajr
functions (other than the generic functions plot
, lines
and points
) start with the prefix Traj
. Several of the functions provided by trajr
do not provide additional analytic functionality, rather they make it easier to work with multiple trajectories. These convenience functions are prefixed with Trajs
.
trajr
Trajr
works with trajectories, where a trajectory is a simplification of a real path travelled by an animal, and is a set of 2-dimensional spatial coordinates with a third temporal dimension. A trajectory can also be thought of as a series of steps, each comprised of a length (L~i~), a turning angle ($\Delta$~i~), and a time. While trajectories have been modelled in various ways, trajr
works with two theoretical models: correlated random walks in which the turning angle of each step is the direction of the previous step ± some error; and directed walks, or compass-based navigation, in which the angular errors at each step are added to the "ideal" or compass direction. Trajr
generally assumes correlated random walks, but provides some support for directed walks.
As of version 1.5.0, trajr
provides limited functionality for characterising 3-dimensional trajectories. See the section [Three dimensional trajectories] below.
par(mar = c(4, 4, 0.5, 0.5) + 0.1) # Generate and plot a short random trajectory set.seed(4) trj <- TrajGenerate(5, random = TRUE) plot(trj, turning.angles = "random")
par(mar = c(4, 4, 0.5, 0.5) + 0.1) # Generate and plot a short directed trajectory set.seed(1) trj <- TrajGenerate(5, random = FALSE) plot(trj, turning.angles = "directed")
Within trajr
, trajectories are represented by R objects with class Trajectory
. The TrajFromCoords
function is used to create a Trajectory
object from a set of x-y coordinates, with optional times.
par(mar = c(4, 4, 0.5, 0.5) + 0.1) # Define x, y, and time coordinates coords <- data.frame(x = c(1, 1.5, 2, 2.5, 3, 4), y = c(0, 0, 1, 1, 2, 1), times = c(0, 1, 2, 3, 4, 5)) # Create a trajectory from the coordinates trj <- TrajFromCoords(coords) # Plot it plot(trj)
In practice, trajectory coordinates are typically read from a data file such as a CSV file, and then passed in to the TrajFromCoords
function.
coords <- read.csv("mycoords.csv") trj <- TrajFromCoords(coords)
TrajFromCoords
assumes that the first column in coords
contains x values, the second contains y values, and that points were sampled at a constant sampling frequency of 50 points/second. The help page for TrajFromCoords
describes how to alter these defaults - run ?TrajFromCoords
to access the help page. Many functions (including TrajFromCoords
) contain code examples as part of their help information. You can also display and run an example by calling the example
function: i.e. example("TrajFromCoords")
.
Trajectories may need to be scaled before analysis. For instance, when a trajectory is digitised from video, the spatial units will be pixels, and require conversion to more meaningful units such as metres. Trajectories are scaled by calling TrajScale
, specifying the scale factor and the abbreviation of the transformed units. In the case of a trajectory digitised from a video, you can calculate scale as $width^{m}/width^{p}$, where $width^{m}$ is the width of an object in metres, and $width^{p}$ is the width of the same object in pixels, as measured from the video. A commonly used method to determine the scale is to include a measuring tape in the video, then measure the number of pixels in a length of the tape.
# Example 1 coords <- read.csv("H. pahangensis.csv") trj <- TrajFromCoords(coords, spatialUnits = "pixels") # A 10 mm (= .01 m) object had length 47 pixels in the video, scale to metres trj <- TrajScale(trj, .01 / 47, "m") # Example 2 coords <- read.csv("A. argentifasciata.csv") # In this video, a 20 cm (= .2 m) object had length 1800 pixels, scale to metres trj <- TrajScale(trj, .2 / 1800, "m")
Many trajectories will suffer from noise which may bias the results of some analyses. The TrajSmoothSG
function reduces high frequency noise while preserving the shape of the trajectory, by applying a Savitzky-Golay smoothing filter. It requires two arguments (in addition to the trajectory), n
, the filter order, and m
, the filter length (which must be odd). Refer to the help for the function sgolayfilt
(run ?signal::sgolayfilt
) for details of the filter function.
There are some issues which should be considered when deciding whether to smooth and by how much. If duplicate points are omitted from a trajectory (i.e. multiple points representing the same stopped location are not explicitly specified), smoothing is likely to result in a segment spanning the gap which is decreasing in velocity, but not stopped. Similarly, smoothing (or too much smoothing) may eliminate short periods of zero (or near-zero) velocity.
set.seed(3) par(mar = c(4, 4, 0.5, 0.5) + 0.1) trj <- TrajGenerate(200, random = TRUE, angularErrorSd = .25) # Plot original trajectory plot(trj, lwd = 1, lty = 1) # Create a smoothed trajectory, filter order 3, length 31 smoothed <- TrajSmoothSG(trj, p = 3, n = 31) # Plot it in slightly transparent red lines(smoothed, col = "#FF0000A0", lwd = 2) legend("topright", c("Original", "Smoothed"), lwd = c(1, 2), lty = c(1, 1), col = c("black", "red"), inset = 0.01)
trajr
provides functions to resample a trajectory to a fixed step length or a fixed step time.
Some functions require a Trajectory
with a fixed step length. The process of resampling a trajectory to a fixed step length is called rediscretization. The function TrajRediscretize
implements trajectory resampling using the algorithm described by Bovet & Benhamou (1988). There are no clear guidelines for choosing a suitable step length for rediscretization: too small a step length leads to oversampling, leading to high autocorrelation between steps and high variability; too large a step length results in undersampling and loss of information. See Turchin (1998) section 5.2.2 for a discussion. By default, rediscretization discards time (and hence speed) information, however setting simConstantSpeed = TRUE
simulates a constant speed trajectory with (approximately) the same duration (and hence average speed) as the original trajectory.
```r = 2$ to step length $1$._", echo=-1:-2} par(mar = c(4, 4, 0.5, 0.5) + 0.1) set.seed(2) trj <- TrajGenerate(10, stepLength = 2)
plot(trj, lwd = 2) points(trj, draw.start.pt = FALSE, pch = 16, col = "black", cex = 1.2)
resampled <- TrajRediscretize(trj, 1)
lines(resampled, col = "#FF0000A0", lwd = 2) points(resampled, type = 'p', col = "#FF0000A0", pch = 16) legend("topright", c("Original", "Rediscretized"), col = c("black", "red"), lwd = 2, inset = c(0.01, 0.02))
The function `TrajResampleTime` linearly interpolates points along a trajectory to create a new trajectory with fixed step time intervals. ```r par(mar = c(4, 4, 0.5, 0.5) + 0.1) set.seed(5) # Generate trajectory with a point every 2 hours and highly variable speed (which equates to step length) trj <- TrajGenerate(10, stepLength = 1, fps = .5, timeUnits = "hours", linearErrorSd = .8) # Plot original trajectory with dots at trajectory coordinates plot(trj, lwd = 2) points(trj, draw.start.pt = FALSE, pch = 16, col = "black", cex = 1.2) # Resample to 1 hourly steps resampled <- TrajResampleTime(trj, 1) # Plot rediscretized trajectory in red lines(resampled, col = "#FF0000A0", lwd = 2) points(resampled, type = 'p', col = "#FF0000A0", pch = 16) legend("topright", c("Original", "Resampled"), col = c("black", "red"), lwd = 2, inset = c(0.01, 0.02))
This table lists trajectory operations not otherwise addressed by this document. Refer to the help for each function for information.
kable(data.frame(`Function` = c("`TrajRotate`", "`TrajTranslate`", "`TrajReverse`", "`TrajGetFPS`", "`TrajGetNCoords`", "`TrajGetUnits`", "`TrajGetTimeUnits`", "`TrajStepLengths`", "`TrajLength`", "`TrajDistance`", "`TrajDuration`", "`TrajMeanVelocity`", "`TrajAngles`", "`TrajMeanVectorOfTurningAngles`", "`TrajExpectedSquareDisplacement`", "`TrajFromTrjPoints`"), Description = c("Rotates a trajectory", "Translates a trajectory", "Reverses a trajectory", "Returns the frames-per-second of a trajectory", "Returns the number of coordinates of a trajectory", "Returns the spatial units of a trajectory", "Returns the temporal units of a trajectory", "Returns the lengths of each step within the trajectory", "Returns the total length of the trajectory (or a portion)", "Returns the straight-line distance from the start to the end of the trajectory (or a portion)", "Returns the temporal duration of the trajectory (or a portion)", "Returns the mean velocity vector of the trajectory (or a portion)", "Returns the turning angles (radians) of a trajectory", "Returns the mean vector of the turning angles", "Returns the expected square displacement of a correlated random walk", "Creates a trajectory from a subset of the points in another trajectory")))
Once you have obtained a correctly scaled Trajectory
object, you may call various functions to analyse the trajectory. Broadly speaking, analysis may be divided into measures of speed or acceleration, and measures of straightness or tortuosity.
The functions TrajVelocity
and TrajAcceleration
estimate velocity and acceleration as vectors at each point along a trajectory. Velocity is calculated using first-order finite differences, and acceleration uses second-order finite differences. The TrajDerivatives
function calculates speed and change in speed along a Trajectory
. If the trajectory is noisy, it should be smoothed before the derivatives are calculated (see Smoothing trajectories, but take note of the caveats described there).
set.seed(1) par(mar = c(4, 4.5, 1, 4.5) + 0.1) trj <- TrajGenerate() # Smooth before calculating derivatives smoothed <- TrajSmoothSG(trj, 3, 101) # Calculate speed and acceleration derivs <- TrajDerivatives(smoothed) # Plot change-in-speed and speed plot(derivs$acceleration ~ derivs$accelerationTimes, type = 'l', col = 'red', yaxt = 'n', xlab = 'Time (s)', ylab = expression(paste('Change in speed (', m/s^2, ')'))) axis(side = 2, col = "red") lines(derivs$speed ~ derivs$speedTimes, col = 'blue') axis(side = 4, col = "blue") mtext('Speed (m/s)', side = 4, line = 3) abline(h = 0, col = 'lightGrey')
Once the trajectory speed has been obtained, it is simple to calculate values such as mean, maximum, minimum or standard deviation of speed: mean(derivs$speed)
, max(derivs$speed)
, min(derivs$speed)
, or sd(derivs$speed)
. Similarly, calculations excluding periods when speed drops below (or above) some value are also straightforward, for example: mean(derivs$speed[deriv$speed > STOPPED_SPEED])
. In addition, the function TrajSpeedIntervals
allows you to determine the time intervals within the trajectory that satisfy a speed constraint, either slower than some speed, faster than a speed, or both. This may be useful, for example, when testing how often, or for how long, a flying insect hovers, or a running insect stops, i.e. when its speed drops below some threshold. The object returned by a call to TrajSpeedIntervals
can be plotted.
set.seed(1) par(mar = c(4, 4, 0.5, 0.5) + 0.1) # Simulate a flying bee trj <- TrajGenerate(100, angularErrorSd = 0.3, stepLength = 0.001) # Smooth the trajectory trj <- TrajSmoothSG(trj, 3, 51) # Calculate hovering intervals intervals <- TrajSpeedIntervals(trj, slowerThan = 2) print(intervals) # Plot speed over time with hovering intervals highlighted plot(intervals)
Various methods to measure the straightness, or conversely, tortuosity, of trajectories are available within trajr
. The simplest is $D/L$, where $D$ is the distance from the start to the end of the trajectory, and $L$ is the length of the trajectory. This straightness index is calculated by the function TrajStraightness
, and is a number ranging from 0 to 1, where 1 indicates a straight line. Benhamou (2004) considers the straightness index to be a reliable measure of the efficiency of a directed walk, but inapplicable to random trajectories. Batschelet (1981) considers this straightness index to be an approximation of r, which is the length of the mean vector of turning angles after rediscretizing to a constant step length. r can be calculated by calling Mod(TrajMeanVectorOfTurningAngles(trj))
, assuming trj
is a Trajectory with constant step length.
Within this section, most of the figures show the appropriate statistic plotted for multiple randomly generated correlated random walks which vary in the standard deviations of angular errors, $\sigma_{\Delta}$ (higher values of $\sigma_{\Delta}$ produce more tortuous trajectories).
set.seed(1) # Generate some trajectories for use in examples n <- 100 # Random magnitude of angular errors angularErrorSd <- runif(n, 0, 2) # Generate some trajectories with varying angular errors trjs <- lapply(1:n, function(i) TrajGenerate(500, stepLength = 2, angularErrorSd = angularErrorSd[i])) # Rediscretize each trajectory to a range of step sizes stepSizes <- c(1, 2, 10) reds <- lapply(stepSizes, function(ss) lapply(1:n, function(i) TrajRediscretize(trjs[[i]], ss)))
```r$, plotted for several different step sizes (linear axes)._", fig.height=4, fig.width=6, echo=-1} par(mar = c(4, 4, 0.5, 0.5) + 0.1)
ds <- sapply(reds, function(rtrjs) sapply(1:n, function(i) TrajStraightness(rtrjs[[i]])))
rs <- sapply(reds, function(rtrjs) sapply(1:n, function(i) Mod(TrajMeanVectorOfTurningAngles(rtrjs[[i]]))))
plot(rep(angularErrorSd, 3), rs, pch = 16, cex = .8, col = c(rep('red', n), rep('blue', n), rep('darkgreen', n)), xlab = expression(sigma[Delta]), ylab = "Straightness", ylim = range(c(ds, rs))) points(rep(angularErrorSd, 3), ds, pch = 3, cex = .8, col = c(rep('red', n), rep('blue', n), rep('darkgreen', n)))
legend("bottomleft", c(expression(italic(r)), "D/L", paste("Step length", stepSizes)), pch = c(16, 3, 16, 16), col = c("black", "black", "red", "blue", "darkgreen"), inset = 0.01)
#### Sinuosity The sinuosity index defined by Benhamou (2004) may be an appropriate measure of the tortuosity of a random search path. Sinuosity is a function of the mean cosine of turning angles, and is a corrected form of the original sinuosity index defined by Bovet & Benhamou (1988). The function `TrajSinuosity2` calculates the corrected form while `TrajSinuosity` calculates the original index. In general, `TrajSinuosity2` should be used rather than `TrajSinuosity`. The uncorrected form of sinuosity (`TrajSinuosity1`) should be calculated for a trajectory with a constant step length, so trajectories may first require [rediscretization](#resampling-trajectories). In the absence of a biologically meaningful step size, rediscretizing to the mean step length of the trajectory will produce a trajectory with approximately the same shape and number of steps as the original. ```r$, plotted for several different step sizes (linear axes)._", fig.height=4, fig.width=6, echo=-1} par(mar = c(4, 4, 0.5, 0.5) + 0.1) # Calculate sinuosity for all of the rediscretized trajectories sins <- sapply(reds, function(rtrjs) sapply(1:n, function(i) TrajSinuosity2(rtrjs[[i]]))) # Plot sinuosity vs angular error plot(rep(angularErrorSd, 3), sins, pch = 16, cex = .8, col = c(rep('red', n), rep('blue', n), rep('darkgreen', n)), xlab = expression(sigma["angular error"]), ylab = "Sinuosity") legend("bottomright", paste("Step length", stepSizes), pch = 16, col = c("red", "blue", "darkgreen"), inset = 0.01)
$E^{a}{max}$ is a dimensionless estimate of the maximum expected displacement of a trajectory. Larger $E^{a}{max}$ values (approaching infinity) represent straighter paths (Cheung, Zhang, Stricker, & Srinivasan, 2007). $E^{b}{max}$ is $E^{a}{max}$ multiplied by the mean step length, so gives the maximum possible displacement in spatial units. TrajEmax
returns the value of $E^{a}{max}$ for a trajectory, or $E^{b}{max}$ if the argument eMaxB
is TRUE
. TrajEmax
differentiates between random and directed walks; see ?TrajEmax
for details.
```r$ (logarithmic axes)._", fig.height=4, fig.width=6, echo=-1} par(mar = c(4, 4, 1.5, 0.5) + 0.1)
emaxs <- sapply(reds, function(rtrjs) sapply(1:n, function(i) TrajEmax(rtrjs[[i]]))) emaxs[emaxs < 0] <- NA # Avoid warnings when plotting on log axes
plot(rep(angularErrorSd, 3), emaxs, log = 'xy', pch = 16, cex = .8, col = c(rep('red', n), rep('blue', n), rep('darkgreen', n)), xlab = expression(sigma["angular error"]), ylab = expression(E[max]))
legend("bottomleft", c("Step length 1", "Step length 2", "Step length 10"), pch = 16, col = c("red", "blue", "darkgreen"), inset = 0.01)
#### Fractal dimension Fractal dimension has been considered a promising measure of straightness or tortuosity, varying between 1 (straight line movement) and 2 (Brownian motion). However, several studies have found it inappropriate for use with animal trajectories, as animal trajectories are not fractal curves, and the fractal dimension of a non-fractal curve depends critically on the range of step sizes used to calculate it (Nams, 2006; Turchin, 1996). Nonetheless, fractal dimension continues to be used, and the `TrajFractalDimension` function, by default, calculates fractal dimension using the modified dividers method to account for truncation error (Nams, 2006). See `?TrajFractalDimension` and `?TrajFractalDimensionValues` for more information. ```r$._", fig.width=6, fig.height=4, echo=-1} par(mar = c(4, 4, 0.5, 0.5) + 0.1) # Use the same range of step sizes for all trajectories stepSizes <- TrajLogSequence(.1, 7, 10) # Fractal dimension is a slow calculation, so just plot a subset # of trajectories from the previous example fn <- n / 4 use <- sample.int(fn, n = length(angularErrorSd)) fangularErrorSd <- angularErrorSd[use] # Calculate fractal dimension for all of the rediscretized trajectories d <- sapply(reds, function(rtrjs) sapply(use, function(i) { TrajFractalDimension(rtrjs[[i]], stepSizes) })) # Plot fractal dimension vs angular error plot(rep(fangularErrorSd, 3), d, pch = 16, cex = .8, col = c(rep('red', fn), rep('blue', fn), rep('darkgreen', fn)), xlab = expression(sigma["angular error"]), ylab = "Fractal dimension") legend("topleft", c("Step length 1", "Step length 2", "Step length 10"), pch = 16, col = c("red", "blue", "darkgreen"), inset = 0.01)
Directional change is defined as the change in direction over time, and has been used to assess locomotor mimicry in butterflies (Kitamura & Imafuku, 2015). Directional change is defined for each pair of steps, so a trajectory (or a portion thereof) may be characterised by the mean (DC
) and standard deviation (SDDC
) of all directional changes. DC
may be used as an index of nonlinearity, and SDDC
as a measure of irregularity. The directional changes for a trajectory are calculated by the function TrajDirectionalChange
, so SD
may be calculated as mean(TrajDirectionalChange(trj))
, and SDDC
as sd(TrajDirectionalChange(trj))
. Note that directional change cannot be calculated on rediscretized trajectories, as rediscretizing discards time information.
```r$._", fig.height=4, fig.width=6, echo=-1} par(mar = c(4, 4, 0.5, 0.5) + 0.1)
dcs <- sapply(1:n, function(i) mean(TrajDirectionalChange(trjs[[i]]))) sddcs <- sapply(1:n, function(i) sd(TrajDirectionalChange(trjs[[i]])))
plot(angularErrorSd, dcs, pch = 16, cex = .8, col = "blue", xlab = expression(sigma["angular error"]), ylab = "DC") points(angularErrorSd, sddcs, pch = 16, cex = .8, col = 'red') legend("bottomright", c("DC", "SDDC"), pch = 16, col = c("blue", "red"), inset = 0.01)
#### Direction autocorrelation The direction autocorrelation function was devised to detect and quantify regularities within trajectories, in the context of locomotor mimicry by ant-mimics (Shamble, Hoy, Cohen, & Beatus, 2017). It detects wave-like periodicities, and provides an indication of their wavelength and amplitude, so can be used to detect, for example, the winding movements of trail-following ants. The function $C(\Delta s)$ is applied to a rediscretized trajectory, and calculates the differences in step angles at all steps separated by $\Delta$, for a range of $\Delta s$. The position of the first local minimum in $C(\Delta s)$ (i.e. the 2-dimensional position $(\Delta s, c(\Delta s))$) may be used to characterise the periodicity within a trajectory. The function is calculated by `TrajDirectionAutocorrelations`, and the result may be plotted. The position of the first local minimum (or maximum) is calculated by `TrajDAFindFirstMinimum` (or `TrajDAFindFirstMaximum`). Some trajectories will not have a first local minimum (or maximum), which indicates that they do not contain regular changes in direction. ```r set.seed(1) par(mar = c(4, 4, 0.5, 0.5) + 0.1) trj <- TrajGenerate(1000) corr <- TrajDirectionAutocorrelations(trj) plot(corr)
As trajectory studies are likely to involve multiple trajectories per individual, per species, or trajectories for multiple species, trajr
provides some functions to simplify the manipulation of multiple trajectories. TrajsBuild
assumes that you have a set of trajectory files (such as CSV files), each of which may have a scale and a frames-per-second value. The function reads each of the files specified in a list of file names, optionally using a custom function, then passes the result to TrajFromCoords
, which assumes that the first column is x
, the second is y
, and there is no time column. If these assumptions are incorrect, the csvStruct
argument can be specified to identify the x
, y
and time
columns (see the example below). TrajsBuild
optionally scales the trajectories with the specified scale (by calling TrajScale
), and optionally smooths them with the specified smoothing parameters (by calling TrajSmoothSG
). The value returned is a list of Trajectory
objects.
tracks <- as.data.frame(rbind( c("3527.csv", "Zodariid2 sp1", "spider", "red"), c("3530.csv", "Daerlac nigricans", "mimic bug", "blue"), c("3534.csv", "Daerlac nigricans", "mimic bug", "blue"), c("3537.csv", "M. erythrocephala", "mimic spider", "blue"), c("3542.csv", "Polyrhachis sp1", "ant", "black"), c("3543.csv", "Polyrhachis sp1", "ant", "black"), c("3548.csv", "Crematogaster sp1", "ant", "black") ), stringsAsFactors = FALSE) colnames(tracks) <- c("filename", "species", "category", "col") # If your list is read from a CSV file which contains empty lines, # remove them like this: tracks <- na.omit(tracks) # Order of columns in the CSV files is unknown so identify them by name csvStruct <- list(x = "x", y = "y", time = "Time") trjs <- TrajsBuild(tracks$filename, scale = .220 / 720, spatialUnits = "m", timeUnits = "s", csvStruct = csvStruct, rootDir = "..")
TrajsMergeStats
simplifies the construction of a data frame of values, with one row per trajectory. To use it, you need a list of trajectories (which you will probably obtain from a call to TrajsBuild
), and a function which calculates statistics for a single trajectory. The following example demonstrates how you might write and use such a function.
# Define a function which calculates some statistics # of interest for a single trajectory characteriseTrajectory <- function(trj) { # Measures of speed derivs <- TrajDerivatives(trj) mean_speed <- mean(derivs$speed) sd_speed <- sd(derivs$speed) # Measures of straightness sinuosity <- TrajSinuosity2(trj) resampled <- TrajRediscretize(trj, .001) Emax <- TrajEmax(resampled) # Periodicity corr <- TrajDirectionAutocorrelations(resampled, 60) first_min <- TrajDAFindFirstMinimum(corr) # Return a list with all of the statistics for this trajectory list(mean_speed = mean_speed, sd_speed = sd_speed, sinuosity = sinuosity, Emax = Emax, min_deltaS = first_min[1], min_C = first_min[2] ) } # Calculate all stats for trajectories in the list # which was built in the previous example stats <- TrajsMergeStats(trjs, characteriseTrajectory) print(stats)
Principal components analysis (PCA) is a commonly used statistical technique which can be used to visualise similarities and differences between groups of trajectories. In R, the function prcomp
is usually used to perform a PCA. Prcomp
will not process data containing NA
s (in R, the value NA
is used to indicate a 'N
ot A
vailable', or missing, value). However, NA
s can be potentially informative, especially when using the first local minimum of the direction autocorrelation function - in this case, an NA
indicates that there was no periodicity detected in the trajectory. To avoid discarding this information, Trajr
provides a utility function, TrajsStatsReplaceNAs
. It is used to replace NA
values with some other value, and optionally adds an additional column to the data which flags trajectories which originally contained NA
values. It operates on a single column at a time.
par(mar = c(4, 4, 0.5, 0.5) + 0.1) #--- # Custom PCA plotting function customPcaPlot <- function(x, xlabs, xcols, choices = 1L:2L, ycol = "#ff2222aa", ...) { # Draw points pts <- t(t(x$x[, choices])) plot(pts, type = "p", xlim = extendrange(pts[, 1L]), ylim = extendrange(pts[, 2L]), asp = 1, xlab = "PC1", ylab = "PC2", pch = 16, col = xcols, ...) text(pts, labels = xlabs, pos = 1, ...) # Draw arrows axs <- t(t(x$rotation[, choices])) * 3.5 text(axs, labels = dimnames(axs)[[1L]], col = ycol, ...) arrows(0, 0, axs[, 1L] * .8, axs[, 2L] * .8, length = .1, col = ycol) } # --- # Here we are operating on the statistics from the previous example # The PCA function, prcomp, can't handle NAs, so replace NAs with a "neutral" value # and create a new column that flags which trajectories had an NA value. # First fix min_deltaS, and add an extra column which flags non-periodic trajectories pcaStats <- TrajsStatsReplaceNAs(stats, "min_deltaS", replacementValue = 2 * max(stats$min_deltaS, na.rm = TRUE), flagColumn = "no_first_min") # Also get rid of NAs in min_C - no need to add another column since it would duplicate no_first_min pcaStats <- TrajsStatsReplaceNAs(pcaStats, "min_C", replacementValue = 2 * max(stats$min_C, na.rm = TRUE)) # Perform the PCA PCA <- prcomp(pcaStats, scale. = TRUE) # Plot it using custom plotting function. Could just call biplot instead customPcaPlot(PCA, tracks$category, tracks$col, cex = .8) legend("bottomleft", c("Spider", "Mimic", "Ant"), pch = 16, col = c('red', 'blue', 'black'), inset = c(0.01, .02))
The plot above shows that the trajectory of the non-mimetic spider (red dot) seems quite different from the trajectories of ant mimicking bugs, ant-mimicking spiders (blue dots) and ants (black dots). The arrows indicate the importance and direction of the parameters that separate the trajectories. From the plot, it seems that the non-mimetic spider varies mainly in speed, (it is faster and more variable) and E~max~ (it walks in a straighter path). While these trajectories have been digitised from videos of walking animals, clearly this example does not contain enough data to be meaningful.
The function TrajsStepLengths
returns a vector containing all of the step lengths in a list of trajectories.
summary(TrajsStepLengths(trjs))
TrajConvertTime
is a convenience function that converts a delimited time string to a numeric value. This may be useful if your data contains times which are formatted as strings with millisecond accuracy. In general, the base R function strptime
should be used to convert strings to times, but it does not handle milliseconds.
t <- data.frame(time = c("0:00:00:029", "0:01:00:216", "0:02:01:062", "1:00:02:195", "1:06:03:949", "1:42:04:087"), stringsAsFactors = FALSE) t$seconds <- TrajConvertTime(t$time) t
Trajr
allows you to generate random trajectories with the TrajGenerate
function. Random trajectories may be used in simulation studies, or simply for experimenting with trajectory analysis, and it has been used to generate the majority of trajectories in this document. By default, TrajGenerate
returns a correlated random walk, however, by specifying different arguments, a variety of trajectory types can be created.
set.seed(1) par(mfrow = c(2, 2), mar = c(4, 4, 0.5, 0.5) + 0.1) # Correlated random walk trj <- TrajGenerate(n = 200) # Plot it plot(trj) mtext("a)", 3, -1.3, adj = .05) # Directed walk trj <- TrajGenerate(n = 20, random = FALSE) plot(trj) mtext("b)", 3, -1.3, adj = .05) # Brownian motion trj <- TrajGenerate(n = 500, angularErrorDist = function(n) stats::runif(n, -pi, pi)) plot(trj) mtext("c)", 3, -1.3, adj = .05) # Levy walk - path lengths follow a Cauchy distribution trj <- TrajGenerate(linearErrorDist = stats::rcauchy) plot(trj) mtext("d)", 3, -1.3, adj = .05)
trajr
provides limited assistance for building more sophisticated heterogeneous trajectories. Functions exist for splitting (TrajSplit
) and merging (TrajMerge
) trajectories, for detecting when trajectories cross polygon boundaries (TrajInPolygon
) and splitting a trajectory at a polygon crossing (TrajSplitAtFirstCrossing
).
As of version 1.5.0, trajr
provides limited functionality for characterising 3-dimensional trajectories. All functions that operate on 3-dimensional trajectories are prefixed with Traj3D
. A 3D trajectory created by Traj3DFromCoords
is a 2D trajectory with an additional z
column, which allows additional functionality. You can call any of the 2D trajr
functions with a 3D trajectory, however, they will simply ignore the 3rd dimension, so that is probably not what you want. Any trajr
functions that manipulate or return metadata will work as expected, e.g. TrajGetUnits
, TrajGetNCoords
, TrajGetFPS
, TrajGetTimeUnits
. Similarly TrajDuration
works correctly as it does not use spatial data.
Functionality for 3D trajectory characterisation is limited to a small set of functions. Each of the 3D functions is equivalent to a 2D function with the same name but prefix Traj
.
kable(data.frame(`Function` = c("`Traj3DFromCoords`", "`Traj3DLength`", "`Traj3DDistance`", "`Traj3DStraightness`", "`Traj3DRediscretize`", "`Traj3DResampleTime`", "`Traj3DSmoothSG`", "`Traj3DStepLengths`"), Description = c("Creates a 3D Trajectory Object", "Return the length along a 3D trajectory", "Returns the distance between the start and end points of a 3D trajectory", "Returns the straightness of a 3D trajectory (distance / length)", "Resamples a 3D trajectory to a constant step length", "Resample a 3D trajectory to a constant time interval", "Smooths a 3D trajectory using a Savitzky-Golay filter", "Returns a vector of step lengths of a 3D trajectory" )))
Plotting a 3D trajectory using plot
will simply plot the x and y dimensions, ignoring the z dimension. If you wish to plot a 3D representation of a trajectory, use a package such as rgl
.
library(rgl) t3 <- Traj3DFromCoords(...) # Plot the 3D line plot3d(t3$x, t3$y, t3$z, type = 'l', lwd = 2, xlim = range(t3$x)) # Mark the start of the trajectory plot3d(t3$x[1], t3$y[1], t3$z[1], add = T, type = 's', size = 0.8)
While trajr
was implemented with processing speed in mind, processing time can be a problem when operating on large numbers of large trajectories.
Trajr
is open source, with the source available on Github. If you have any problems or suggestions, email jim_mclean@optusnet.com.au.
Batschelet, E. (1981). Circular statistics in biology. ACADEMIC PRESS, 111 FIFTH AVE., NEW YORK, NY 10003, 1981, 388.
Benhamou, S. (2004). How to reliably estimate the tortuosity of an animal's path. Journal of Theoretical Biology, 229(2), 209-220. \doi{10.1016/j.jtbi.2004.03.016}
Benhamou, S. (2006). Detecting an orientation component in animal paths when the preferred direction is individual-dependent. Ecology, 87(2), 518-528. \doi{10.1890/05-0495}
Bovet, P., & Benhamou, S. (1988). Spatial analysis of animals' movements using a correlated random walk model. Journal of Theoretical Biology, 131(4), 419-433. \doi{10.1016/S0022-5193(88)80038-9}
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}
Kitamura, T., & Imafuku, M. (2015). Behavioural mimicry in flight path of Batesian intraspecific polymorphic butterfly Papilio polytes. Proceedings of the Royal Society B: Biological Sciences, 282(1809). \doi{10.1098/rspb.2015.0483}
Nams, V. O. (2006). Improving Accuracy and Precision in Estimating Fractal Dimension of Animal movement paths. Acta Biotheoretica, 54(1), 1-11. \doi{10.1007/s10441-006-5954-8}
Shamble, P. S., Hoy, R. R., Cohen, I., & Beatus, T. (2017). Walking like an ant: a quantitative and experimental approach to understanding locomotor mimicry in the jumping spider Myrmarachne formicaria. Proceedings of the Royal Society B: Biological Sciences, 284(1858). \doi{10.1098/rspb.2017.0308}
Turchin, P. (1996). Fractal Analyses of Animal Movement: A Critique. Ecology, 77(7), 2086-2090. \doi{10.2307/2265702}
Turchin, P. (1998). Quantitative analysis of movement. Sunderland (Mass.): Sinauer assoc.
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.