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.

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.

```
# 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.

```
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 stime.

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.

```
trj <- TrajGenerate(10, stepLength = 2)
# 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 step length 1
resampled <- TrajRediscretize(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", "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.

```
# 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.

Function | Description |
---|---|

`TrajRotate` |
Rotates a trajectory |

`TrajTranslate` |
Translates a trajectory |

`TrajReverse` |
Reverses a trajectory |

`TrajGetFPS` |
Returns the frames-per-second of a trajectory |

`TrajGetNCoords` |
Returns the number of coordinates of a trajectory |

`TrajGetUnits` |
Returns the spatial units of a trajectory |

`TrajGetTimeUnits` |
Returns the temporal units of a trajectory |

`TrajStepLengths` |
Returns the lengths of each step within the trajectory |

`TrajLength` |
Returns the total length of the trajectory (or a portion) |

`TrajDistance` |
Returns the straight-line distance from the start to the end of the trajectory (or a portion) |

`TrajDuration` |
Returns the temporal duration of the trajectory (or a portion) |

`TrajMeanVelocity` |
Returns the mean velocity vector of the trajectory (or a portion) |

`TrajAngles` |
Returns the turning angles (radians) of a trajectory |

`TrajMeanVectorOfTurningAngles` |
Returns the mean vector of the turning angles |

`TrajExpectedSquareDisplacement` |
Returns the expected square displacement of a correlated random walk |

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 `TrajDerivatives`

function calculates linear speed and acceleration 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).

```
trj <- TrajGenerate()
# Smooth before calculating derivatives
smoothed <- TrajSmoothSG(trj, 3, 101)
# Calculate speed and acceleration
derivs <- TrajDerivatives(smoothed)
# Plot acceleration and speed
plot(derivs$acceleration ~ derivs$accelerationTimes, type = 'l', col = 'red',
yaxt = 'n',
xlab = 'Time (s)',
ylab = expression(paste('Acceleration (', 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 and acceleration have been obtained, it is simple to calculate values such as mean, maximum, minimum or standard deviation of speed (or acceleration): `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.

```
# 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)
```

```
#> startFrame startTime stopFrame stopTime duration
#> 1 11 0.2337088 27 0.5488033 0.3150945
#> 2 85 1.7073675 94 1.8866781 0.1793107
```

```
# 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).

```
# 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)))
```

```
# Calculate straightness (D/L) for all of the rediscretized trajectories
ds <- sapply(reds, function(rtrjs) sapply(1:n, function(i) TrajStraightness(rtrjs[[i]])))
# Calculate alternate straightness (r) for all of the rediscretized trajectories
rs <- sapply(reds, function(rtrjs) sapply(1:n, function(i) Mod(TrajMeanVectorOfTurningAngles(rtrjs[[i]]))))
# Plot both indices on the same graph
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)
```

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. Sinuosity should be calculated for a trajectory with a constant step length, so trajectories may first require rediscretization. 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.

```
# 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.

```
# Calculate Emax for all of the rediscretized trajectories (from the previous example)
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 Emax vs angular error 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 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.

```
# 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.

```
# Calculate DC and SDDC for all of the original trajectories (from above)
dcs <- sapply(1:n, function(i) mean(TrajDirectionalChange(trjs[[i]])))
sddcs <- sapply(1:n, function(i) sd(TrajDirectionalChange(trjs[[i]])))
# Plot DC vs angular error
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)
```

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.

```
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 <- TrajSinuosity(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)
```

```
#> mean_speed sd_speed sinuosity Emax min_deltaS
#> deltaS 0.061296301 0.009069424 0.9833243 3180.72321 33
#> 2 0.010573501 0.005718966 20.1799859 100.43632 50
#> 3 0.014175447 0.005206138 10.5781944 692.21342 NA
#> 4 0.006245555 0.003005911 6.3442130 67.52479 7
#> 5 0.005344287 0.007936181 60.5565802 14.90601 NA
#> 6 0.019378799 0.005363627 3.3069095 74.33277 35
#> 7 0.014796829 0.002573456 2.4142475 112.93798 NA
#> min_C
#> deltaS 0.899013229
#> 2 0.303049396
#> 3 NA
#> 4 0.930992488
#> 5 NA
#> 6 0.002917922
#> 7 NA
```

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.

```
#---
# 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
# Get rid of NAs in stats because prcomp can't handle them.
# 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))`

```
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 1.000e-10 7.813e-05 2.266e-04 2.266e-04 3.243e-04 1.550e-03
```

`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
```

```
#> time seconds
#> 1 0:00:00:029 0.029
#> 2 0:01:00:216 60.216
#> 3 0:02:01:062 121.062
#> 4 1:00:02:195 3602.195
#> 5 1:06:03:949 3963.949
#> 6 1:42:04:087 6124.087
```

`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.

```
# 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)
```

Processing time can be a problem when operating on large numbers of large trajectories.

- Use an appropriate step length for
*rediscretization*. Too small a step length will result in trajectories with an excessive number of steps and will take a long time to process. - Some steps, such as calculating directional autocorrelations, may be amenable to memoisation.

`Trajr`

is open source, with the source available on Github. If you have any problems or suggestions, email jim_mclean@optusnet.com.au.

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