The data of step selection functions

Where does an animal go when it moves? Part 4 turned the fishers' tracks into regular steps of 10 minutes. In this part you will ask what those steps chose. For every step a fisher took, you will draw ten steps it could have taken from the same place, take the land cover and the elevation at the end of all of them, and compare: did the real steps end in forest more often than the possible ones? This comparison of used and available steps is the step selection function (SSF), the most used method for relating animal movement to the landscape. Terraflux builds its data, as R's amt does, to the same numbers; the model itself, a conditional logistic regression, is fitted in R or Python, and this page shows how and what it says.

The idea#

Used and available, step by step#

A resource selection function compares the places an animal used with places drawn at random from an area, usually its home range. That raises the question of what was available: a fisher at one end of its range could not reach the other end in 10 minutes. Step selection functions answer it with the animal's own movement. From the start of every observed step, the steps the animal could have taken are drawn from the lengths and turns it actually makes; each observed step and its random alternatives form a stratum, and the comparison is made only within strata (Fortin et al., 2005; Thurfjell et al., 2014).

Where the random steps come from#

amt draws random steps in three moves (Signer et al., 2019):

1. fit a gamma distribution to the step lengths and a von Mises distribution to the turn angles of each track, the tentative distributions of its movement;
2. from the start of every observed step with a turn angle, draw n_control lengths and turns from them;
3. turn each drawn angle from the direction of the previous step, and walk the drawn length.

Integrated step selection#

The tentative distributions are fitted to the observed steps, which were themselves shaped by the landscape. Avgar et al. (2016) showed that adding the step's length and turn to the model, as covariates, corrects them: with \( \log(\text{sl}) \) and \( \cos(\text{ta}) \) among the covariates, the shape of the gamma distribution becomes \( k + \beta_{\log \text{sl}} \) and the concentration of the von Mises distribution \( \kappa + \beta_{\cos \text{ta}} \) (\( \text{sl} \) itself would correct the gamma's scale). The model then estimates habitat selection and movement together: the integrated step selection analysis (iSSA), amt's fit_issf.

When to use it#

• How animals use a landscape while they move: selection of forest, avoidance of roads or of open land, attraction to water, at the scale of a step.
• Selection that changes with time or state: by day and by night, by season, with the distance to a predator (Fortin et al., 2005), through interactions with the covariates.
• Movement that depends on the landscape: faster or straighter in some covers (interactions of \( \log(\text{sl}) \) or \( \cos(\text{ta}) \) with a covariate at the start of the step).
• Resistance surfaces from data: the coefficients of an SSF are one way of calibrating the costs of circuit connectivity or grid distances.

An SSF needs regular steps (part 4) at an interval at which the animal could plausibly choose among the random steps. With fixes days apart, the "available" steps are no longer local, and a resource selection function over the home range is the simpler choice.

What you need#

• The fishers' steps of part 4: resampled every 10 minutes with a tolerance of 1 minute, in bursts of 3 locations or more, projected (move.aeqd_crs). move.random_steps refuses steps in longitude and latitude: it draws lengths and angles on a plane.
• The landscape under the steps: ESA WorldCover 2021 (land cover, 10 m) and the Copernicus DEM (30 m), which data.download fetches for the fishers' area. The rasters can stay in longitude and latitude: move.extract_covariates takes the steps to each raster's coordinate system.
• To fit the model: R with the package survival (or amt), or Python with a conditional logistic regression (statsmodels' ConditionalLogit, or the few lines of numpy that made this page's numbers).

Build the model#

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 = "random"
op = "move.random_steps"
inputs = { steps = "steps.steps" }
params = { n_control = 10, seed = 1 }

[[nodes]]
id = "dem"
op = "data.download"
params = { source = "copernicus_dem", dem_resolution = 30, extent = [-74.0, 42.65, -73.33, 42.9], cellsize = 0.0003 }

[[nodes]]
id = "cover"
op = "data.download"
params = { url = "https://esa-worldcover.s3.eu-central-1.amazonaws.com/v200/2021/map/ESA_WorldCover_10m_2021_v200_N42W075_Map.tif", extent = [-74.0, 42.65, -73.33, 42.9], cellsize = 0.0003, resampling = "mode" }

[[nodes]]
id = "forest"
op = "raster.calc"
inputs = { cover = "cover.raster" }
params = { expression = "1 if cover == 10 else 0" }

[[nodes]]
id = "covariates"
op = "move.extract_covariates"
inputs = { tracks = "random.steps", forest = "forest.map", elevation = "dem.raster" }

[[nodes]]
id = "write"
op = "vector.write"
inputs = { vector = "covariates.tracks" }
params = { path = "random-steps.gpkg" }

The distributions on their own#

move.random_steps fits the distributions it needs, but they can be fitted, inspected and drawn from as nodes of their own:

[[nodes]]
id = "sl"
op = "move.fit_distribution"
inputs = { values = "steps.steps" }
params = { name = "gamma" }

[[nodes]]
id = "ta"
op = "move.fit_distribution"
inputs = { values = "steps.steps" }
params = { name = "vonmises" }

[[nodes]]
id = "sl_draws"
op = "move.random_numbers"
inputs = { distribution = "sl.distribution" }
params = { n = 20000, seed = 1 }

[[nodes]]
id = "ta_draws"
op = "move.random_numbers"
inputs = { distribution = "ta.distribution" }
params = { n = 20000, seed = 1 }

[[nodes]]
id = "slow"
op = "move.distribution"
params = { name = "gamma", shape = 0.8, scale = 100 }

[[nodes]]
id = "straight"
op = "move.distribution"
params = { name = "vonmises", kappa = 0.5 }

[[nodes]]
id = "given"
op = "move.random_steps"
inputs = { steps = "steps.steps", sl_distr = "slow.distribution", ta_distr = "straight.distribution" }
params = { n_control = 10, seed = 1 }

move.fit_distribution fits gamma, exp, lnorm or hnorm to the step lengths (sl_) and vonmises or unif to the turn angles (ta_), by amt's methods (fit_distr), to the same numbers; column chooses another column, and a table works as well as steps. move.distribution makes a distribution from parameters you give (amt's make_distribution), move.random_numbers draws from either. Connected to sl_distr and ta_distr, they replace the per-track fits of move.random_steps: one distribution for all the animals, or a distribution taken from another study.

Read the results#

The tentative distributions#

Enlarge

Enlarge

A gamma distribution with a shape under 1, as here, has its mode at 0: most steps short, a long tail of travel, which fits the lengths well. The turn angles are harder. The fishers either go on (0°) or turn back (±180°), and a von Mises distribution centred on 0 has a single peak: fitted to all the turns, it is almost uniform. Fitted per track, as move.random_steps does, it is more concentrated:

The steps and their strata#

The 7,480 steps of part 4 become 6,026 strata of 11 steps each, 66,286 steps in all: the first step of each burst has no turn angle (no previous step to turn from), and bursts of fewer than 3 steps are left out. The observed steps end in tree cover more often than the random ones (96.4 % against 91.0 %); their elevations are the same within a metre.

Enlarge

A single stratum shows the comparison the model makes. M4 started this step in tree cover, near a wetland and open water. Of its ten random alternatives, two end in the wetland; the observed step, and eight of the random ones, end in forest. Strata like this one, where the forest varies, carry the information about forest selection; a stratum entirely in forest says nothing about it. Here 38 % of the strata have both forest and other covers among their ends.

The conditional logistic regression#

Terraflux prepares the data; it does not fit the model yet. In R, from the file of node 14:

library(sf)
library(survival)
d <- st_drop_geometry(st_read("output/random-steps.gpkg"))
# step_id_ restarts in every track: the stratum is the animal and the step
d$stratum <- paste(d$individual.local.identifier, d$step_id_)
m <- clogit(case_ ~ forest + I(elevation / 100) + log(pmax(sl_, 1)) + cos(ta_) +
              strata(stratum), data = d)
summary(m)

With amt, the same model is fit_issf(d, case_ ~ forest + I(elevation / 100) + log(pmax(sl_, 1)) + cos(ta_) + strata(stratum)). R's sf reads the column individual-local-identifier as individual.local.identifier. The lengths are clipped at 1 m before the logarithm, because random steps can be drawn far shorter (1,622 of them are under a metre). The numbers below come from a short conditional logistic regression in numpy (Newton-Raphson on the likelihood above), the same to every digit as R's clogit on these data:

Forest. \( \exp(1.221) = 3.4 \): of two steps equally available, equal in length and turn and elevation, one ending in tree cover and one not, the fishers were 3.4 times as likely to take the one into forest (95 % interval 2.9 to 4.0). Fishers are known as forest carnivores; the SSF measures their preference at the scale of a 10-minute step.

Elevation. \( 0.22 \pm 0.13 \) per 100 m: no clear selection. The ends of a stratum's steps lie within a few hundred metres of each other, and the elevation among them varies by 18 m in the median stratum; on average the observed and random ends differ by half a metre. An SSF sees only the contrast within strata, so a covariate must vary at the scale of a step to be estimated well.

Movement. The coefficient of \( \log(\text{sl}) \) is small (0.027): the gamma distributions fitted to each track describe the fishers' step lengths well. That of \( \cos(\text{ta}) \) is clearly negative (−0.236): the observed steps go straight on less often than the random steps drawn from the von Mises distributions, which are concentrated around 0° (see the warning above). By Avgar et al.'s (2016) correction, the fishers' \( \kappa \) is about 0.24 lower than the tentative ones (0.27 to 0.44), between 0.03 and 0.2: at 10 minutes their turns have little preferred direction around 0°, as the two-peaked histogram shows. The correction is strictly one per animal, each with its own tentative distribution; one model for all of them, as here, gives their average.

How many random steps#

With 2 random steps per stratum, the estimate is higher and less precise; from 5 on it settles at 1.20 to 1.22, and its interval narrows little beyond 10. More random steps reduce the noise of the sampling of availability, not the noise of the animals' choices, which is fixed by the number of observed steps. The usual advice is to check, as here, that the estimates no longer move (Thurfjell et al., 2014; Fieberg et al., 2021); the cost of more steps is only computing time (move.random_steps and move.extract_covariates take seconds for the 66,286 steps here, 300,000 at 50 controls).

Forest by day and by night#

The time of day is the same for all the steps of a stratum, so it enters the model only through an interaction. Label every step with move.time_of_day, connected to covariates.tracks, and add forest × night:

By day, the coefficient of forest is 1.33; at night it is 0.12 lower, but with a standard error of 0.23: the data show no difference. Three quarters of the strata are at night (4,726 of 6,026), when the fishers move, and the day's 1,300 strata give the day's coefficient its wider interval.

Settings that change the answer#

• The interval of the steps (part 4): an SSF at 10 minutes and one at 2 hours describe choices at different scales, and their coefficients differ; compare only SSFs at the same interval.
• n_control of move.random_steps: the chart above. Check that the estimates are stable.
• The tentative distributions: per track (the default), pooled, or given (move.distribution into sl_distr, ta_distr). Include \( \log(\text{sl}) \) and \( \cos(\text{ta}) \) in the model: without movement terms, the habitat coefficients can be biased by the way availability was sampled (Forester et al., 2009; Avgar et al., 2016).
• seed: another seed draws other random steps, and the coefficients move a little. Report the seed, and check that the conclusions do not depend on it.
• where of move.extract_covariates: the end of the step (where the animal chose to go) is the usual choice; the start serves for covariates that change how it moves (interactions with \( \log(\text{sl}) \) or \( \cos(\text{ta}) \)).
• The covariates' resolution: the land cover here is resampled to cells of about 25 by 33 m (cellsize = 0.0003 degrees), the scale of the steps. Coarser cells blur the contrasts within strata.

Common pitfalls#

• Strata across animals. step_id_ restarts in every track. A stratum of step_id_ alone joins up to seven animals' steps in one stratum, and the model is wrong without any error: here it gives forest 1.06 and elevation −0.03, instead of 1.22 and 0.22. Build the stratum from the animal and the step.
• Pooling animals. One model for all the fishers treats their steps as one animal's. Animals differ in what they select; fit one model per animal and summarise them, or a mixed conditional logistic regression with random slopes per animal (Muff et al., 2020).
• Unprojected steps. move.random_steps refuses steps in longitude and latitude. Project with move.aeqd_crs first (part 3).
• Steps off the map. A random step that ends outside a raster, or on its no data, gets no value, and R drops its row. Download the landscape over the tracks' extent with a margin of a few long steps.
• Categorical rasters resampled as numbers. A land cover resampled by bilinear or cubic interpolation gets classes that do not exist (the mean of forest and water). Use resampling = "mode" or "near" for categories.
• A von Mises centred on 0. amt's tentative turns are centred on 0 even for animals that reverse more often than they go on (the warning above): include \( \cos(\text{ta}) \) in the model.
• Covariates that do not vary within strata. Time of day, season, the animal's sex: only their interactions can be estimated.

Going further#

• Regular steps and track summaries: the steps this part starts from.
• Species distribution models: where a species can live, from occurrences instead of tracks.
• Circuit connectivity: resistance surfaces, which SSF coefficients can calibrate.
• The operators' reference: move.fit_distribution, move.distribution, move.random_numbers, move.random_steps, move.extract_covariates.

References#

• Avgar, T., Potts, J. R., Lewis, M. A., & Boyce, M. S. (2016). Integrated step selection analysis: bridging the gap between resource selection and animal movement. Methods in Ecology and Evolution, 7(5), 619-630. https://doi.org/10.1111/2041-210X.12528
• Fieberg, J., Signer, J., Smith, B., & Avgar, T. (2021). A 'How to' guide for interpreting parameters in habitat-selection analyses. Journal of Animal Ecology, 90(5), 1027-1043. https://doi.org/10.1111/1365-2656.13441
• Forester, J. D., Im, H. K., & Rathouz, P. J. (2009). Accounting for animal movement in estimation of resource selection functions: sampling and data analysis. Ecology, 90(12), 3554-3565. https://doi.org/10.1890/08-0874.1
• Fortin, D., Beyer, H. L., Boyce, M. S., Smith, D. W., Duchesne, T., & Mao, J. S. (2005). Wolves influence elk movements: behavior shapes a trophic cascade in Yellowstone National Park. Ecology, 86(5), 1320-1330. https://doi.org/10.1890/04-0953
• Muff, S., Signer, J., & Fieberg, J. (2020). Accounting for individual-specific variation in habitat-selection studies: efficient estimation of mixed-effects models using Bayesian or frequentist computation. Journal of Animal Ecology, 89(1), 80-92. https://doi.org/10.1111/1365-2656.13087
• 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
• Thurfjell, H., Ciuti, S., & Boyce, M. S. (2014). Applications of step-selection functions in ecology and conservation. Movement Ecology, 2, 4. https://doi.org/10.1186/2051-3933-2-4