Regular steps and track summaries

Part 2 showed that every measure of a track depends on its sampling interval, and that the fishers were sampled at three different ones, irregularly. In this part you will make the intervals regular without inventing any location, as amt does: by keeping the fixes that fall on a regular grid of times and cutting the track into bursts where they do not. From the bursts you will build steps, the unit of amt's analyses, label each step as day, night, dawn or dusk from the position of the sun, follow each animal's net squared displacement over the weeks, and summarise each track in a few numbers. At the end you will have the fishers' steps at a common 10-minute interval, ready for the step selection functions of part 5, and a table of their tracks that you can report.

The idea#

Resampling into bursts#

An irregular track can be made regular in two ways. Part 3 placed locations at regular times by interpolation, inventing them. Resampling keeps only real fixes: it walks along the track and keeps a fix when it falls at the chosen interval after the last one kept, within a tolerance. When no fix is close enough, the regular sequence is broken, and a new one, a new burst, begins at the next fix.

Key idea: amt's track_resample

With a rate \( r \) and a tolerance \( \tau \), from the first location of a track:

the next location kept is the first one at least \( r - \tau \) after the last one kept;
if it is at most \( r + \tau \) after it, it belongs to the same burst; otherwise it starts a new one;
the locations in between are dropped.

Every step within a burst then lasts \( r \pm \tau \). A step between bursts is not a step at all: amt never builds one. A burst of one or two locations gives no turn angle, so bursts are usually kept only from three locations on (move.filter_min_n_burst).

Signer et al. (2019) made this the first step of amt's workflow, because the step selection functions of part 5 compare steps by their lengths and turns, which must all have the same duration.

Steps#

A step in amt is a line from one location to the next of the same burst, with its start (x1_, y1_), end (x2_, y2_), length (sl_), direction (direction_p), turn angle from the step before (ta_), start and end times (t1_, t2_) and duration (dt_). It differs from move2's measures of part 2 in two ways: the step is a feature of its own (a line), not a column of a location; and its direction is the angle of mathematics, anticlockwise from east, in radians, not the azimuth clockwise from north. The turn angle is the same quantity in both, but its sign follows the convention: positive to the left in amt (anticlockwise), positive to the right in move2.

Day and night#

Most animals are not equally active around the clock. amt labels each location or step by the sun: day between sunrise and sunset, night between sunset and sunrise, and, if asked, dawn and dusk, the civil twilights when the sun is less than 6° below the horizon. The times of the sun come from the date and the location, computed as the R package suncalc computes them (sunrise and sunset when the sun's upper edge is at -0.833°, refraction included).

Summaries of a track#

Key idea: summarising a track

For a track of \( n \) locations \( p_1, \dots, p_n \) with step lengths \( l_i \) and turn angles \( \theta_i \):

cum_dist, the distance travelled: \( \sum l_i \).
tot_dist, the net distance: \( |p_n - p_1| \), from the first location to the last.
straightness (Batschelet, 1981): \( \text{tot\_dist} / \text{cum\_dist} \), 1 for a straight line, near 0 for an animal that comes back to where it started.
msd, the mean squared distance of the locations from their centroid, a measure of the spread of the locations (the square of the radius of gyration).
sinuosity (Benhamou, 2004): \( S = 2 \left[ p \left( \frac{1 + c}{1 - c} + b^2 \right) \right]^{-1/2} \), where \( p \) is the mean step length, \( c \) the mean cosine of the turn angles and \( b \) the coefficient of variation of the step lengths. It estimates the tortuosity of a random search path, in units of 1 / √length; higher is more tortuous.
tac, the turn angle correlation (Dray et al., 2010): the mean squared difference between consecutive turns on the unit circle, \( \frac{1}{n} \sum (\cos\theta_{i+1} - \cos\theta_i)^2 + (\sin\theta_{i+1} - \sin\theta_i)^2 \). It is 0 when every turn is the same, 2 when consecutive turns are independent and uniform, and up to 4 when they alternate.
NSD, the net squared displacement of each location: \( |p_i - p_1|^2 \), the squared distance from the first location (Bunnefeld et al., 2011).

Straightness suits directed movement (a migration, a dispersal); it is near 0 for any animal living in a home range, whose last location is near its first. Sinuosity suits searching movement within an area. NSD over time tells movement patterns apart: it stays within a band for a resident animal, rises and levels off for one that disperses and settles, rises and falls back for a migrant that returns.

When to use it#

Resample before any analysis that compares steps: step selection functions (part 5), the distributions of step lengths and turns, comparisons of animals or seasons. Choose the rate from the collars' schedule (the histogram of time lags of part 2): at least the scheduled interval, a multiple of it to compare animals on different schedules.
Time of day to compare activity, or as a variable of a step selection function (do animals use roads at night and avoid them by day?).
NSD to classify individuals as resident, dispersing, migrating or nomadic.
Summaries for a table of the tracks in a paper, and to compare individuals at a common interval.

What you need#

The cleaned tracks of part 2, projected on the azimuthal equidistant projection of part 3 (move.aeqd_crs), so that steps have lengths in metres and the random steps of part 5 can be drawn. move.steps also works in longitude and latitude (lengths in metres on the sphere, lonlat = true), but the random steps need projected tracks.

Build the model#

The projected tracks (node 5) resampled every 10 minutes with a tolerance of 1 minute (node 6), the bursts of at least 3 locations (node 7), their steps (node 8) and the time of day at the end of each step (node 9).

Model file (TOML)

format = "terraflux/1"

[[nodes]]
id = "read"
op = "move.read_tracks"
params = { path = "fishers.csv.gz" }

[[nodes]]
id = "located"
op = "vector.filter"
inputs = { vector = "read.tracks" }
params = { expression = "not isnull(geometry.x)" }

[[nodes]]
id = "visible"
op = "move.filter_movebank_visible"
inputs = { tracks = "located.vector" }

[[nodes]]
id = "unique"
op = "move.filter_unique"
inputs = { tracks = "visible.tracks" }

[[nodes]]
id = "aeqd"
op = "move.aeqd_crs"
inputs = { tracks = "unique.tracks" }

[[nodes]]
id = "resample"
op = "move.resample"
inputs = { tracks = "aeqd.tracks" }
params = { rate = "10min", tolerance = "1min" }

[[nodes]]
id = "bursts"
op = "move.filter_min_n_burst"
inputs = { tracks = "resample.tracks" }
params = { min_n = 3 }

[[nodes]]
id = "steps"
op = "move.steps"
inputs = { tracks = "bursts.tracks" }

[[nodes]]
id = "tod"
op = "move.time_of_day"
inputs = { tracks = "steps.steps" }
params = { include_crepuscule = true, where = "end" }

Resample with move.resample (library subject Animal movement), connected to aeqd.tracks. rate and tolerance take seconds (600) or a text with a unit (10min, 2h, 30s). The node keeps the locations of the regular sequence and adds the column burst_, the number of the burst within each track (1, 2...). start (0 by default) is the location each track starts from, counted from 0. Ten minutes suits both the 10-minute collars and the 2-minute ones, of which it keeps every fifth fix.
Keep the bursts of 3 locations or more with move.filter_min_n_burst (min_n = 3). A burst of two locations is one step with no turn angle; most analyses drop them.
Build the steps with move.steps: one line per step within each burst, with the columns described above. keep_cols copies the tracks' other columns onto the steps: those of the location the step starts at (start), ends at (end), or both (both, with the suffixes _start and _end). Steps never join two bursts.
Label the time of day with move.time_of_day. On steps, where chooses the end (the default, as amt) or the start of each step; include_crepuscule = true adds dawn and dusk to day and night. The node adds the column tod (or the name in column).

The summaries and the NSD read the tracks, not the steps:

The net squared displacement of every location (node 10), and the summaries of each track at 10 minutes (node 11).

Model file (TOML)

[[nodes]]
id = "nsd"
op = "move.nsd"
inputs = { tracks = "aeqd.tracks" }

[[nodes]]
id = "summary"
op = "move.track_summary"
inputs = { tracks = "bursts.tracks" }

move.nsd adds the column nsd to every location, in the squared units of the coordinates: square metres here, square degrees on tracks in longitude and latitude (as amt), which is one more reason to project first. move.track_summary gives one line per track (the track's line of move.track_lines, with its number of locations and its constant columns) with the columns cum_dist, tot_dist, straightness, msd, sinuosity and tac.

Read the results#

Bursts#

Resampling M4's day from part 1 shows what the method keeps. Of 248 locations, 76 fall on the 10-minute grid; they form 25 bursts, many of them short, because the collar's fixes drifted around its schedule and failed at times. The long bursts are in the night, when the fisher travelled and the collar fixed well. Resampling did not invent a single location: every coloured point is a fix.

How much survives depends on the rate:

rate	tolerance	locations	bursts	steps	animals	M4's median step (m)
2 min	20s	30,179	14,504	14,686	5	28.4
10 min	1min	12,833	4,783	7,480	7	76.8
30 min	3min	5,838	2,096	3,577	8	166.6
1 h	6min	3,678	694	2,877	8	80.7
2 h	12min	1,991	457	1,466	8	144.9

The fishers resampled at five rates, each with a tolerance of a tenth of it: the locations kept, the bursts, the steps in bursts of 3 locations or more, the animals that have any, and M4's median step length.

At 2 minutes only the five animals on that schedule have steps; at 10 minutes M1, fixed every 15, has none (its fixes are never 9 to 11 minutes apart); from 30 minutes on all eight have. The rate is therefore also a choice of which animals enter the analysis. And of which parts of a track: M4's median step does not grow steadily with the rate (28, 77, 167, 81 and 145 m), because the steps that fit each grid come from different hours: 9 % of its 30-minute steps start between 07:00 and 17:00 local time, 31 % of its hourly ones, when it rested and its collar fixed sparsely. A resampled track keeps the periods that happen to fit the grid, so check that they are not a biased sample (all night, or all from one week).

Step lengths and turns#

The step lengths at 10 minutes — The length of the 7,480 steps of 10 minutes of seven fishers. Most are short (half are under 59 m), a long tail reaches over a kilometre.

The turn angles at 10 minutes — The turn angles of the same steps (ta_, in degrees). Peaks at 0° (straight on) and at ±180° (reversals).

Steps of a regular 10 minutes have the shape amt's analyses expect: lengths skewed to the right, with many short steps (resting, lingering) and a long tail of travel; turns concentrated at 0° and at ±180°. Part 5 fits distributions to exactly these two histograms: a gamma distribution to the lengths, a von Mises distribution to the turns.

Night owls#

The mean step length by hour of the day — The mean length of the 10-minute steps by the local hour at which they start (UTC - 5), coloured by whether most of the hour's steps end at night or by day.

time of day	steps	share (%)	median step (m)	mean step (m)
day	1,553	20.8	22.6	55.8
dusk	46	0.6	24.7	62
night	5,661	75.7	85	123.6
dawn	220	2.9	65.3	117.5

The 10-minute steps by the time of day at their end (move.time_of_day with include_crepuscule): how many, and their lengths.

The fishers are nocturnal in these data: their steps average about 120 m per 10 minutes at night and 25 to 30 m around midday, and three quarters of the 10-minute steps end at night. The share of night steps is not the share of time: the collars obtained more regular fixes at night, so night steps are over-represented, and a model that pools them must take that into account (part 5 uses time of day as a variable). Dawn and dusk are short in winter at this latitude, about half an hour each, and hold few steps.

Net squared displacement#

None of the eight NSD curves rises and levels off: every fisher went out and came back within a band, the pattern of a resident animal in a home range. The bands differ by an order of magnitude: F3 stays within 2.7 km² (about 1.6 km from its first location), M2 goes up to 70 km² (8 km) late in its track, and M1 makes regular trips to about 30 km² (5.5 km) and back, every few days. The fishers' breeding season, in March and April, falls within the tracking of most of these animals, and long trips by males in search of females are a likely reason for such excursions; the tracks alone cannot tell.

The summaries#

animal	locations	cum_dist (km)	tot_dist (km)	straightness	msd (km²)	sinuosity	tac
F1	1,300	114.2	0.72	0.006	0.95	0.106	1.763
F2	575	109.6	1.41	0.013	1.29	0.096	1.806
F3	300	41.7	0.38	0.009	0.21	0.112	1.944
M2	1,520	173.8	4.46	0.026	3.94	0.089	1.795
M3	385	89.7	0.89	0.01	1.18	0.077	1.832
M4	1,801	225.5	1.46	0.006	1.81	0.111	1.88
M5	2,760	371.1	1.3	0.004	1.5	0.107	1.757

The summaries of each track (move.track_summary) on the locations resampled every 10 minutes, in bursts of 3 or more. M1 has none at this rate. Distances in km, msd in km², sinuosity in 1/√m.

The cumulative distances (42 to 371 km) mostly reflect how long and how densely each animal was followed: compare animals by distance per day at a common rate (part 2), not by these totals. The straightness is near 0 for all, as for any resident animal; it says nothing here. The msd, the spread of each animal's locations, ranges from 0.2 km² (F3) to 3.9 km² (M2), in line with the NSD curves. The sinuosity (0.08 to 0.11) and tac (1.76 to 1.94, below the 2 of independent turns) are similar across animals: at 10 minutes the fishers' paths are equally tortuous, with slightly correlated turns.

To report: the summaries, the rate and tolerance they were computed at, and the number of locations and steps they rest on.

Settings that change the answer#

rate and tolerance of move.resample: the table of rates above. A wider tolerance keeps more fixes and longer bursts, but the steps' durations vary more; amt's examples use a tenth of the rate or less.
min_n of move.filter_min_n_burst: 3 keeps every burst with a turn; a higher value keeps only long bursts (fewer, more regular steps).
start of move.resample: the location the grid starts from. A different start keeps different fixes; with a short tolerance the results change little.
include_crepuscule and where of move.time_of_day: with only day and night, the twilight steps join them; the time at the end or at the start of a step can differ by one label at the boundaries.
The track summarised: summaries of the raw tracks (every fix) differ from those at 10 minutes (more steps, more tortuous). Give move.track_summary the resampled tracks for a fair comparison.

Common pitfalls#

Steps across gaps. Steps built on a track that was not resampled join locations hours apart. Resample, or at least split the track into bursts, before move.steps.
Animals that vanish. An animal whose schedule does not fit the rate has no bursts and silently drops out (M1 at 10 minutes). Check the number of animals after move.filter_min_n_burst.
NSD in squared degrees. On tracks in longitude and latitude, move.nsd gives squared degrees, as amt does. Project first.
Directions from two conventions. direction_p of move.steps is anticlockwise from east (amt); move.azimuth is clockwise from north (move2). In longitude and latitude, amt reads a compass bearing as if it were an angle from east; Terraflux converts it (π/2 minus the bearing), so its directions are right where amt's are rotated.
Times in UTC. The time of day is computed from UTC times and the location, so it is right whatever the local time zone; but a histogram by hour (as above) must convert to local time to be read.

Going further#

The data of step selection functions: the steps of this part with random steps and the landscape under them.
The operators' reference: move.resample, move.filter_min_n_burst, move.steps, move.time_of_day, move.nsd, move.track_summary.

References#

Batschelet, E. (1981). Circular statistics in biology. Academic Press.
Benhamou, S. (2004). How to reliably estimate the tortuosity of an animal's path: straightness, sinuosity, or fractal dimension? Journal of Theoretical Biology, 229(2), 209-220. https://doi.org/10.1016/j.jtbi.2004.03.016
Bunnefeld, N., Börger, L., van Moorter, B., Rolandsen, C. M., Dettki, H., Solberg, E. J., & Ericsson, G. (2011). A model-driven approach to quantify migration patterns: individual, regional and yearly differences. Journal of Animal Ecology, 80(2), 466-476. https://doi.org/10.1111/j.1365-2656.2010.01776.x
Calenge, C., Dray, S., & Royer-Carenzi, M. (2009). The concept of animals' trajectories from a data analysis perspective. Ecological Informatics, 4(1), 34-41. https://doi.org/10.1016/j.ecoinf.2008.10.002
Dray, S., Royer-Carenzi, M., & Calenge, C. (2010). The exploratory analysis of autocorrelation in animal-movement studies. Ecological Research, 25(3), 673-681. https://doi.org/10.1007/s11284-010-0701-7
Signer, J., Fieberg, J., & Avgar, T. (2019). Animal movement tools (amt): R package for managing tracking data and conducting habitat selection analyses. Ecology and Evolution, 9(2), 880-890. https://doi.org/10.1002/ece3.4823