Step selection analysis with non-linear and random effects in mgcv

Abstract

Step selection analysis is used to jointly describe animal movement patterns and habitat preferences. Recent work has extended this framework to model inter-individual differences, account for unexplained structure in animals' space use, and capture temporally-varying patterns of movement and habitat selection. In this paper, we formulate step selection functions with penalised smooths (similar to generalised additive models) to unify new and existing extensions, and conveniently implement the models in the popular, open-source mgcv R package. We explore non-linear patterns of movement and habitat selection, and use the equivalence between penalised smoothing splines and random effects to implement individual-level and spatial random effects. This framework can also be used to fit varying-coefficient models to account for temporally or spatially-heterogeneous patterns of selection (e.g., resulting from behavioural variation), or any other non-linear interactions between drivers of the animal's movement decisions. We provide a broad overview of how smooth effects can be applied to increase the flexibility and biological realism of step selection analysis. We provide the necessary technical details to understand several key special cases of smooths and their implementation in mgcv, showcase the ecological relevance using two illustrative examples, and provide R code (available at https://github.com/NJKlappstein/smoothSSF) to facilitate the adoption of these methods.

Predicting fine-scale distributions and emergent spatiotemporal patterns from temporally dynamic step selection simulations

“…For NDVI (top row) and canopy cover (second row), the models with one, two and three pairs of harmonics (1p, 2p and 3p) are similar, suggesting that increasing the number of harmonics is unlikely to change the model fit and therefore simulation outputs. For previous space use, the model fit for each number of covariates is different, suggesting that the increasing flexibility of the models attempt to capture more of the complicated memory process, and may benefit from even more harmonics or a different functional form (such as 2D splines - Klappstein et al (2024)). The NDVI selection surface suggests that buffalo have an attraction for intermediate values of NDVI in the middle of the day, and selection against high values in the dawn and dusk periods, possibly to ease transit.…”

“…Unlike the model fits for the four habitat covariates (NDVI, canopy cover, herbaceous vegetation and slope), the estimated coefficients of previous space use density, as indicated by the selection surfaces, did not appear to stabilise with increasing harmonics, suggesting that this process is more complicated and may require more flexibility from the responses. In this case we suggest trying more harmonics to capture the dynamics (at the expense of more computation and the risk of overfitting depending on the amount of data), or using a different functional form of the responses through spline terms or similar (Klappstein et al, 2024). The quadratic term may also not be flexible enough the capture the selection within the range of previous space use density, and 2-dimensional spline surfaces may be more useful.…”

“…These are pairs of sine and cosine terms that have cyclic periods of varying frequency, such as sin(2 πτ/T ) and cos(2 πτ/T ), where τ ∈ T . Harmonics are a natural choice as they are cyclic, which often aligns with temporal changes that animals respond to, such as daily or seasonal cycles (Boyce et al, 2010), although other additive terms such as splines are another obvious choice (Hanks et al, 2015; Klappstein et al, 2024). For a yearly cycle for instance, τ would represent the day of the year, and T would equal 365, although τ does not need to be integer-valued and can be arbitrarily fine.…”

“…to incorporate temporal dynamics into SSFs), and one of the popular approaches is through harmonic terms (Ager et al, 2003; Forester et al, 2009; Tsalyuk et al, 2019; Richter et al, 2020). Harmonics are a natural choice as they are cyclical, which often aligns with temporal changes that animals respond to, such as daily or seasonal cycles (Boyce et al, 2010), although other additive terms, such as splines, are another obvious choice (Hanks et al, 2015; Klappstein et al, 2024). Harmonics are pairs of sine and cosine terms that have cyclic periods of varying frequency, such as sin(2 πτ/T ) and cos(2 πτ/T ), where τ ∈ T .…”

“…Simulation-based models can include temporal dynamics on any scale, and they explicitly incorporate the animal’s movement dynamics. Step selection functions (SSFs) are particularly advantageous as they can be used to simulate trajectories (Signer et al, 2017; Potts & Börger, 2023; Signer et al, 2023), are straightforward to parameterise, and can incorporate temporal dynamics (Ager et al, 2003; Forester et al, 2009; Tsalyuk et al, 2019; Richter et al, 2020; Klappstein et al, 2024). An SSF combines a movement and a external selection kernel, can take a range of forms (Munden et al, 2021; Klappstein et al, 2022; Beumer et al, 2023; Eisaguirre et al, 2024; Pohle et al, 2024), and can accommodate a wide range of covariates including habitat, linear features, distance-to-feature variables, proximity to other animals (Potts et al, 2022; Ellison et al, 2024) and representations of previous space use (Schlägel & Lewis, 2014; Oliveira-Santos et al, 2016; Rheault et al, 2021).…”