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Integrated step selection analysis: bridging the gap between resource selection and animal movement

Tal Avgar, Jonathan R. Potts, Mark A. Lewis, Mark S. Boyce

arXiv:1512.01614v1q-bio.QMq-bio.PE

TL;DR

Resource-selection analysis must define availability, while standard step-selection analysis assumes movement attributes are independent of selection. iSSA jointly estimates movement and resource-selection parameters within a mechanistic movement model, and simulations show good predictive performance across tested scales.

  • Problem

    Resource-selection analysis faces a difficult availability definition, while standard step-selection analysis conditions habitat inference on movement attributes assumed independent of resource selection.

  • Method

    iSSA simultaneously estimates movement and habitat-selection parameters by integrating movement and habitat kernels within step-selection analysis using conditional logistic regression.

  • Results

    iSSA parameters were fully identifiable in the test scenario, and iSSA-based utilization-distribution predictions performed well across different temporal and spatial scales.

  • Takeaways & Limitations

    iSSA provides a general, flexible, and user-friendly approach for evaluating ecological hypotheses and predicting ecological patterns.

  • Takeaways & Limitations

    Conclusions may depend on the mechanistic movement process used in simulations, and iSSA inference is scale dependent on relocation intervals and covariate-map resolution.

Abstract

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A resource selection function is a model of the likelihood that an available spatial unit will be used by an animal, given its resource value. But how do we appropriately define availability? Step-selection analysis deals with this problem at the scale of the observed positional data, by matching each used step (connecting two consecutive observed positions of the animal) with a set of available steps randomly sampled from a distribution of observed steps or their characteristics. Here we present a simple extension to this approach, termed integrated step-selection analysis (iSSA), which relaxes the implicit assumption that observed movement attributes (i.e. velocities and their temporal autocorrelations) are independent of resource selection. Instead, iSSA relies on simultaneously estimating movement and resource-selection parameters, thus allowing simple likelihood-based inference of resource selection within a mechanistic movement model. We provide theoretical underpinning of iSSA, as well as practical guidelines to its implementation. Using computer simulations, we evaluate the inferential and predictive capacity of iSSA compared to currently used methods. Our work demonstrates the utility of iSSA as a general, flexible and user-friendly approach for both evaluating a variety of ecological hypotheses, and predicting future ecological patterns.

Introduction

Animal movement is central to ecology and conservation because it links individual behavior to population patterns, yet predicting space use remains limited by incomplete understanding of movement processes.

  • Animal movement connects individual behavioral ecology with population- and community-level processes.
  • Many studies aim to predict utilization distributions, the spatiotemporal probability of animal occurrence.
  • Step-selection analysis matches each observed used step with randomly sampled available steps to address availability at the scale of positional data.
  • Sequentially estimating movement and habitat selection makes habitat-selection inference conditional on movement while treating movement as independent of selection.
  • iSSA jointly accounts for an explicit movement process within SSA, enabling estimation of a habitat-dependent mechanistic movement model from telemetry data using conditional logistic regression.

Materials and methods

iSSA represents movement and habitat selection as separable kernels, then estimates their parameters jointly through conditional logistic regression using observed and available positions.

  • iSSA models space use as the normalized product of a movement kernel Φ and a habitat-selection kernel Ψ.Φ depends on movement attributes and predictors, whereas Ψ depends on habitat attributes and selection coefficients.
  • Movement predictors include step lengths, successive headings, and spatial or temporal covariates, with effects controlled by movement coefficients.
  • The same environmental variable can affect both movement and habitat selection when included in the movement-predictor vector Y and habitat vector H.For example, snow depth may reduce speed while also influencing selection for snow-free locations.
  • Under exponential kernels, observed positions are matched with available controls and estimated with conditional logistic regression.Observed points receive a response value of 1 and control points receive 0.
  • The method samples control points for each observed point and uses their step attributes to approximate available movement choices.Sampling more controls improves approximation but increases computational costs.
  • The iSSA parameterization includes habitat selection, basal and habitat-dependent directional persistence, and habitat or temporal modifiers of step-length distributions.Turn-angle and step-length coefficients can be interpreted as parameters of distributions governing the underlying movement model.

Results

The simulations supported iSSA formulations that include endpoint habitat and step-length terms, while revealing sensitivity in inferred movement parameters and predictive performance across habitat autocorrelation levels. Movement-capacity underestimation was small at high habitat autocorrelation, but estimability remained scenario-dependent.

  • Parameterization: Four of ten formulations received AIC support, all including endpoint habitat, step-length, and log step-length covariates.Models without endpoint habitat had AIC scores typically two orders of magnitude larger.
  • Habitat selection: iSSA habitat-selection estimates were larger than SSA estimates when step length was included, although the effect weakened as habitat autocorrelation increased.RSA estimates showed substantially greater variance, while SSA estimates were less variable.
  • Movement parameters: The inferred mean step length matched the observed mean only when no other covariates were included; adding covariates generally produced higher estimates.Inferred means still slightly underestimated the habitat-independent distribution because iSSA cannot fully represent movement between observations.
  • Movement parameters: Movement-capacity underestimation was negligibly small when habitat spatial autocorrelation exceeded 1.Coarser observation scales otherwise prevent the full underlying movement process from appearing in relocation patterns.
  • Predictive capacities: RSA predictions were slightly more accurate and precise than SSA predictions at approximate steady state, whereas SSA prediction was best at zero habitat autocorrelation.Predictive performance patterns differed with habitat autocorrelation, and KLD broadly mirrored AIC distinctions involving endpoint effects.
  • Estimability: Beyond approximately 400 observed positions, larger samples did not substantially improve precision, while movement-parameter departures reached 1000% under some scenarios.Parameter estimability therefore requires case-by-case evaluation, especially given the trade-off between sampling extent and frequency.

Discussion

iSSA jointly models habitat selection and movement, supporting inference and utilisation-distribution prediction across scales while revealing limits at coarse temporal resolution and under mechanistic-model uncertainty.

  • iSSA simultaneously infers habitat-dependent movement and habitat selection, supporting ecological hypothesis evaluation and prediction of ecological patterns.
  • iSSA habitat-selection inference is relatively insensitive to model structure and landscape configuration, while its utilisation-distribution predictions perform well across temporal and spatial scales.
  • At coarse temporal resolution, movement and habitat selection may not be completely separable, limiting interpretation relative to the underlying behavioural process.
  • When the time scale is long and approaches steady state, stationary RSA predictions fit the true utilisation distribution slightly but consistently better than iSSA predictions.
  • The study’s conclusions may depend on the mechanistic movement process used in simulations, which was deliberately chosen to be simple and general.
  • iSSA uses accessible conditional-logistic-regression tools and can address questions involving home-range behaviour, memory, habitat-dependent selection, and barriers through covariates and interactions.

Appendix S1 – from step selection to utilisation distribution

Step-selection models describe individual movement in discrete time, while redistribution kernels and integral equations connect these processes to transient and steady-state utilisation distributions.

  • Step-selection models mechanistically depict individual animal movement in discrete time.
  • Diffusion, partial-differential, and integral-equation approximations shift from individual movement trajectories to population-level utilisation distributions.
  • The redistribution kernel R(x’|x) determines how the utilisation distribution changes across a time interval τ within the spatial domain Ω.
  • The steady-state utilisation distribution is the limiting distribution reached when the time-evolving distribution no longer changes.
  • The steady-state integral equation is usually not exactly solvable, so numerical, approximate, or Monte Carlo methods are required.

Appendix S2 – inferring step-length distributions

Conditional logistic regression can estimate parameters of several step-length distributions from observed steps and matched control points, with sampling choices affecting efficiency and habitat-selection confounding movement estimates.

  • Conditional logistic regression estimates parameters for exponential, half-normal, gamma, and log-normal step-length distributions from positive step-length data.
  • Observed step lengths are represented by lt = ||xt − xt-1||, while control points provide alternative step lengths relative to the previous used location.
  • Control points are sampled within a maximum distance from the previous location and analyzed with matched case-control logistic regression.
  • For gamma and log-normal models, the regression coefficients map to distribution parameters through the likelihood formulations for step lengths.
  • Sampling controls under the assumed step-length distribution reduces estimation error for a given number of controls.
  • For exponential step lengths, the regression estimate has expectation zero when no other process is modeled; otherwise, λ = λ1 + β1.

Appendix S3 – deriving an iSSA likelihood function

The iSSA likelihood combines habitat selection, directional persistence or bias, and step-length terms into a movement probability, then approximates its normalization using sampled available steps.

  • The conditional movement density f(xt|xt-1,xt-2) is constructed from habitat selection, directional persistence or bias, and step-length components.
  • Habitat selection enters as exponential selection for high values of h(x), with resource value evaluated at the candidate endpoint.
  • The likelihood includes angular deviations from movement directions and step length through cosine, l, and ln(l) terms with covariate-dependent coefficients.
  • The denominator integral normalizes the movement density so that it integrates to one over the spatial domain.
  • Because direct integration is computationally intensive, iSSA samples directions uniformly and distances from a gamma availability distribution.
  • Substituting the gamma density into the discrete-choice approximation yields the conditional probability used in the main-text iSSA likelihood.

Appendix S4 – iSSA practical guide

The practical guide recommends consistent positional sampling, flexible step-length modeling, and conditional logistic regression while emphasizing identifiability constraints.

  • Data collection: Collect positional data at a constant fix rate to maximize iSSA usefulness.Higher fix rates, corresponding to shorter time steps and step lengths, are expected to improve reliability.
  • Step-length distribution: Use the gamma distribution for step lengths when no distribution is justified a priori.The gamma distribution is flexible and includes the exponential distribution as a special case.
  • Step-length distribution: Estimate theoretical step-length parameters from observed step lengths rather than treating the observed distribution as habitat-independent.Observed step lengths can be used with either the method of moments or maximum likelihood.
  • Covariates: Independent effects of purely temporal or step-start covariates are statistically unidentifiable within clusters.Such effects become identifiable when they interact with other variables, such as step length and season.
  • Model fitting: Fit iSSA separately for each individual when sample sizes are sufficient.The guide recommends this instead of a mixed-effects approach to support straightforward evaluation of inter- and intra-individual effects.

Appendix S5 – simulation experiments

The simulations generate habitat-dependent movement on autocorrelated landscapes, then compare RSA, SSA, and iSSA formulations for inference and utilization-distribution prediction.

  • Simulation design: The simulated domain contains 11,600 hexagonal cells representing 10,000 spatial units on a torus.Each cell has a continuous, normally distributed habitat-quality value, h(x).
  • Simulation design: Movement selects among the current cell and six adjacent cells using habitat attraction or repulsion and basal movement cost.Attractiveness is an exponential function of habitat quality h(x), selection intensity ω, and movement cost μ.
  • Simulation design: The redistribution kernel is normalized by summing attractiveness across valid landscape cells.The indicator function identifies valid cells, and the denominator makes the kernel sum to one.
  • Model formulations: The full iSSA formulation includes endpoint habitat, mean habitat along the step, step length, log step length, and their interactions.Models using only habitat covariates represent traditional SSA, whereas adding step-length terms defines iSSA.
  • Prediction: Utilization distributions are compared using steady-state and transient simulations, including GKLD relative to a null expectation.GKLD values closer to 1 indicate better focal-model performance relative to the null, whereas values below approximately 0.368 indicate worse performance.

Appendix S6 – Evaluating the iSSA parameter identifiability and estimability

The identifiability experiments assess iSSA parameter recovery across trajectory lengths using simulated movement, habitat selection, and repeated conditional-logistic fits.

  • Simulation design: Three trajectories of 10^6 steps were simulated across an intermediate-autocorrelation landscape to evaluate parameter identifiability and estimability.The trajectories were generated by stochastic sampling from a redistribution kernel.
  • Simulation design: The simulated habitat-selection coefficient took values ω = 0, 1, or 2 across the three trajectories.The scenario represents movement that may slow in higher-quality habitat while selecting high-quality locations.
  • Fitting procedure: Available points were sampled from gamma-distributed step lengths, and used versus available points were analyzed with conditional logistic regression.The gamma shape and scale parameters were estimated from observed step lengths along each trajectory.
  • Fitting procedure: Fits using segments of 100–1,000 observed points therefore ranged from 99 to 999 steps.The constrained-sample analysis repeated fitting 1,000 times using different trajectory segments.
  • Results: Full-trajectory estimates were unbiased, and shorter-segment estimates were also unbiased on average.All model fits converged successfully; identifiability was assessed by examining whether the likelihood maximum was unique and by repeated fits.

Appendix S7 - β7 and β8

The appendix evaluates interactions between mean habitat along a step and step length or its logarithm.

  • Interaction estimates: The reported interactions are between mean habitat along the step and step length or the natural logarithm of step length.The dashed line represents no effect.

Appendix S8 – predictive performance

The analysis compares 11 models using Kullback–Leibler divergence from true steady-state and transient utilisation distributions across five habitat-autocorrelation levels. RSA projections performed better for steady-state utilisation distributions, while the best SSA results were identified within each autocorrelation level.

  • 11 models were evaluated using Kullback–Leibler divergence from true steady-state and transient utilisation distributions at five habitat spatial-autocorrelation levels.
  • RSA projections showed superior performance in predicting steady-state utilisation distributions.
  • The lowest-performing-divergence SSA formulation at each habitat-autocorrelation level was bolded for comparison with corresponding RSA values.

Appendix S9 – interpreting ρ

The study evaluates scale sensitivity in movement inference by varying temporal and spatial resolutions of simulated positional data. This follows prior work showing that parameter estimates can change when data are coarsened.

  • Movement-behaviour inference is scale-dependent, varying with the temporal and spatial resolution of observed positional time series.
  • The evaluation varies a single parameter to assess sensitivity to temporal and spatial resolution.

Appendix S10 – habitat selection and utilisation distribution

The simulated stepping-stone process produced utilisation distributions whose log values were plotted against habitat quality across spatial-autocorrelation levels. The relationship did not approach a unit slope because movement scale could not exceed habitat-variation scale.

  • Log utilisation distributions were plotted against underlying habitat values for four spatial-autocorrelation levels; the ρ = 50 level was omitted because it was indistinguishable from ρ = 10.
  • The discrete simulation allowed movement scale to equal, but not exceed, the scale of habitat variation.
  • Consequently, the observed log-linear slope did not approach 1 and was minimised at an intermediate value.
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