Understanding step selection analysis through numerical integration

Abstract

Step selection functions (SSFs) are flexible statistical models used to jointly describe animals' movement and habitat preferences. The popularity of SSFs has grown rapidly, and various extensions have been developed to increase their utility, including the ability to use multiple statistical distributions to describe movement constraints, interactions to allow movements to depend on local environmental features, and random effects and latent states to account for within‐ and among‐individual variability. Although the SSF is a relatively simple statistical model, its presentation has not been consistent in the literature, leading to confusion about model flexibility and interpretation. We believe that part of the confusion has arisen from the conflation of the SSF model with the methods used for statistical inference, and in particular, parameter estimation. Notably, conditional logistic regression (CLR) can be used to fit SSFs in exponential form, and this model fitting approach is often presented interchangeably with the actual model (the SSF itself). However, reliance on CLR reduces model flexibility, and suggests a misleading interpretation of step selection analysis as being equivalent to a case–control study. In this review, we explicitly distinguish between model formulation and inference technique, presenting a coherent framework to fit SSFs based on numerical integration and maximum likelihood estimation. We provide an overview of common numerical integration techniques (including Monte Carlo integration, importance sampling and quadrature), and explain how they relate to popular methods used in step selection analyses. This general framework unifies different model fitting techniques for SSFs, and opens the way for improved inferential methods. In this approach, it is straightforward to model movement with distributions outside the exponential family, and to apply different SSF model formulations to the same data set and compare them with AIC. By separating the model formulation from the inference technique, we hope to clarify many important concepts in step selection analysis.

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

“…Samples from the estimated, temporally dynamic, step length distribution are shown with the black lines, with 0p, 1p, 2p and 3p referring to the number of pairs of harmonic terms included in the model. The movement parameters were updated using the estimated coefficients and the tentative Gamma and von Mises distributions that were fitted to the observed data and that random steps were sampled from prior to model fitting (Avgar et al, 2016; Fieberg et al, 2021; Michelot et al, 2024). The model with a single pair of harmonics is restricted to a single period, and fails to capture the two peaks of movement in the observed data.…”

“…The stability of the shape parameter suggests that the scale parameter is ‘absorbing’ the temporal variation, resulting in pathological results, although we also observed similar behaviour when using the exponential distribution, where only a single parameter is estimated. It is likely that further increasing the number of random steps will increase the accuracy of the numerical integration (Michelot et al, 2024), although even with 100 random steps we observed the pathological behaviour. For a high number of parameters and a large dataset increasing the number of steps beyond this can be computationally prohibitive, particularly when fitting models hierarchically.…”

“…The external selection term describes the association between the animal and any covariates that represent the surrounding landscape (Fortin et al, 2005; Thurfjell et al, 2014). We use the term ‘external selection’ rather than ‘habitat selection’ as this model component can include other factors such as conspecifics, predator or prey density, intensity of previous space use, or any other covariates that can be represented spatially (Oliveira-Santos et al, 2016; Potts, Börger, et al, 2022; Potts & Börger, 2023; Michelot et al, 2024). To denote the SSF, p is the likelihood of observing an animal at location s t given its last two observed locations s t− 1 , s t− 2 with locations indexed by discrete time t and within a spatial domain S .…”

“…The parameter vector α defines the functional relationship between θ and τ . The ω component is the external selection kernel, and the term in the denominator ensures that the probability density, p ( · ), integrates to 1, which in practice is typically approximated through numerical integration, where we sample a set of proposed next steps given what is ‘available’ to the animal as determined by the movement kernel (Avgar et al, 2016; Michelot et al, 2024). Within the external selection kernel, β ( τ ; α ) = ( β 1 ( τ ; α 1 ) ,…, β n ( τ ; α n )) is a vector of coefficients that may also have a functional relationship with τ defined by parameter vectors α = ( α 1 ,…, α n ), where the i th coefficient, β i ( τ ; α i ) is parameterised by parameter vector α i and corresponds to the i th spatial covariate in X .…”

“…Step selection functions (SSFs) are statistical models commonly applied to telemetry data to infer movement behaviour and habitat selection (Fortin et al, 2005; Potts et al, 2014; Thurfjell et al, 2014; Avgar et al, 2016; Northrup et al, 2022; Potts & Börger, 2023; Michelot et al, 2024). The SSF combines two terms, one of which describes the movement behaviour of the animal, and the other describes how animals select areas of the landscape (Forester et al, 2009; Potts et al, 2014; Arce Guillen et al, 2023).…”