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

Integrated step selection analysis: bridging the gap between resource selection and animal movement Methods in Ecology and Evolution 2016, 7, 619–630 doi: 10.1111/2041-210X.12528 Integrated step selection analysis: bridging the gap between resource selection and animal movement 1 2 1,3 1 Tal Avgar *, Jonathan R. Potts ,MarkA.Lewis and Mark S. Boyce 1 2 Department of Biological Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada; School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK; and Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada Summary 1. A resource selection function is a model of the likelihood that an available spatial unit will be used by an ani- mal, 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 consecu- tive observed positions of the animal) with a set of ‘available steps’ randomly sampled from a distribution of observed steps or their characteristics. 2. 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 autocorrela- tions) 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. 3. 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. 4. Our work demonstrates the utility of iSSA as a general, flexible and user-friendly approach for both evaluat- ing a variety of ecological hypotheses, and predicting future ecological patterns. Key-words: conditional logistic regression, dispersal, habitat selection, movement ecology, random walk, redistribution kernel, resource selection, step selection, telemetry, utilisation distribution ing explicit movement behaviours into spatial models of Introduction animal density has led to improved predictive performance (Moorcroft, Lewis & Crabtree 2006; Fordham et al. 2014). Ecology is the scientific study of processes that determine the distribution and abundance of organisms in space and time Deriving predictive space-use models based on the beha- (Elton 1927). Hence, asking how and why living beings vioural process underlying animal movement patterns is of change their spatial position through time is fundamental to particular importance when dealing with altered or novel ecological research (Nathan et al. 2008). Animal movement landscapes that might differ substantially from the landscape links the behavioural ecology of individuals with population used to inform the models. and community level processes (Lima & Zollner 1996). Its Over the past three decades, a great deal of research has been study is consequently vital for understanding basic ecological dedicated to explaining and predicting spatial population dis- processes, as well as for applications in wildlife management tribution patterns based on underlying habitat attributes (often and conservation. termed resources). In that regard, much focus has been given Whether basic or applied, common to many empirical to estimating resource selection functions (Manly et al. 2002)– studies of animal movement is the aspiration to reliably pre- phenomenological models of the relative probability that an dict population density through space and time by modelling available discrete spatial unit (e.g. an encountered patch or the spatiotemporal probability of animal occurrence, also landscape pixel) will be used given its resource type/value (Lele known as the utilisation distribution (Keating & Cherry et al. 2013). Indeed, its intuitive nature and ease of application 2009). Despite much progress in statistical characterisation has made resource selection analysis (RSA) the tool of choice of animal movement and habitat associations, our ability to for many wildlife scientists and managers seeking to use envi- predict utilisation distributions is limited by our understand- ronmental information in conjunction with animal positional ing of the underlying behavioural processes. Indeed, includ- data (Boyce & McDonald 1999; McDonald et al. 2013; Boyce et al. 2015). Whereas much progress has been gained in the application *Correspondence author. E-mail: avgar@ualberta.ca of RSAs to animal positional data, the problem of defining © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. 620 T. Avgar et al. the appropriate spatial domain available to the animal a much sought shift from an individual-based Lagrangian remains as a major concern (Matthiopoulos 2003; Lele et al. perspective to population-level Eulerian models (Turchin 2013; McDonald et al. 2013; Northrup et al. 2013). 1991, 1998). SSAs are thus at an interface between statisti- Weighted distribution approaches deal with this problem by cal (phenomenological) RSAs and mathematical (mechanis- modelling space-use as a function of a movement model and tic) RW models (Potts, Mokross & Lewis 2014b; Potts a selection function, but most weighted distribution models et al. 2014a), models that form the backbone of much of are challenging to implement (but see Johnson, Hooten & the existing body of theory in the field of animal move- Kuhn 2013). Matched case–control logistic regressions ment (Codling, Plank & Benhamou 2008; Benhamou 2014; (CLRs; also known as discrete-choice models) may be con- Fagan & Calabrese 2014). sidered a simplified alternative to the weighted distribution In this paper, we outline a CLR-based approach for approach where each observed location is matched with a simultaneous estimation of the movement and habitat-selec- conditional availability set, limited to some predefined spatial tion components, an approach we name integrated step selec- and/or temporal range (Arthur et al. 1996; McCracken, tion analysis (iSSA; Fig. 1). The iSSA allows the effects of environmental variables on the movement and selection pro- Manly & Heyden 1998; Compton, Rhymer & McCollough 2002; Boyce et al. 2003; Baasch et al. 2010). A major cesses to be distinguished, thus providing a valuable tool for strength of this approach is that maximum-likelihood esti- testing hypotheses (e.g. to test whether animals travel faster mates (MLEs) of the parameters can be efficiently obtained in certain times or through certain habitats), while resulting though commonly used statistical software (often relying on in an empirically parameterised mechanistic movement a Cox Proportional Hazard routine; e.g. function clogit in model (i.e. a mechanistic step selection model; Potts et al. R). One particular type of such conditional RSA is step 2014a), that can be used to translate individual-level observa- selection analysis (SSA), where each ‘used step’ (connecting tions to population-level utilisation distributions across space two consecutive observed positions of the animal) is coupled and time (Potts et al. 2014a; Potts, Mokross & Lewis 2014b; with a set of ‘available steps’ randomly sampled from the Appendix S1). empirical distribution of observed steps or their characteris- The iSSA is related to several recently published works tics (e.g. their length and direction; Fortin et al. 2005; Duch- integrating animal movement and resource selection. Both esne, Fortin & Courbin 2010; Thurfjell, Ciuti & Boyce Forester, Im & Rathouz (2009) and Warton & Aarts 2014). (2013) demonstrated the inclusion of movement variables The definition of availability is challenging, however, even in an RSA and its marked effect on the resulting inference. when using the SSA approach. The problem arises due to the Johnson, Hooten & Kuhn (2013) have shown that animal sequential, rather than simultaneous, estimation of the move- telemetry data can be viewed as a realisation of a non- ment and habitat-selection components of the process. Owing homogenous space–time point process, and MLEs of this to this stepwise procedure, the resulting habitat-selection process can be obtained using a generalised linear model. inference is conditional (on movement), whereas movement is These contributions focused on gaining unbiased resource assumed independent of habitat selection. In reality, the two selection inference while treating the movement process as are tightly linked, with habitat selection and availability nuisance that must be ‘controlled for’. Here, we seek expli- affecting the animal’s movement patterns (Avgar et al. cit inference of this process. State-space models of animal 2013b), and the animal’s movement capacity affecting its movement (reviewed by Jonsen, Myers & Flemming 2003; habitat-use patterns (Rhodes et al. 2005; Avgar et al. 2015). and Patterson et al. 2008) predict the future state (e.g. spa- Failure to adequately account for the movement process may tial position) of the animal given its current state (where consequently lead to biased habitat-selection estimates (Fores- an ‘observation model’ provides the probability of observ- ter, Im & Rathouz 2009). ing these states), environmental covariates, and an explicit As we will show here, the benefits of adequately ‘process model’. Once parametrised, the process model can accounting for the movement process may extend beyond be used to generate space-use prediction, but parametrisa- obtaining unbiased habitat-selection estimates. SSAs rely tion is often technically demanding and computationally on a simple depiction of animal movement as a series of intensive (Patterson et al. 2008). More recently, Potts et al. stochastic discrete steps, characterised by specific velocity (2014a) demonstrated the use of a ‘mechanistic step selec- and autocorrelation distributions. This same depiction tion model’ to infer both the drivers and the steady-state underlies the mathematical modelling of animal movement distribution of animal space-use, but the model was framed as a discrete-time random walk (RW), including correlated around one specific functional form of the movement ker- and/or biased RW (Kareiva & Shigesada 1983; Turchin nel, and parameter estimates were obtained using a cus- 1998; Codling, Plank & Benhamou 2008). Indeed, many tom-made likelihood maximisation procedure. Lastly, SSA formulations correspond to a correlated RW process Duchesne, Fortin & Rivest (2015) demonstrated that an with local bias produced by resource selection (BCRW; SSA can be used to obtain unbiased estimates of the direc- Duchesne, Fortin & Rivest 2015). Apart from their com- tional persistence and bias of a BCRW, but did not patibility with the way we often observe animal movement address parametrisation of the step-length distribution. (i.e. in continuous space and at discrete times), many RW The iSSA builds and expands on these contributions. We can be well approximated by diffusion equations, allowing will demonstrate that, by statistically accounting for an explicit © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 Integrated step selection analysis 621 Fig. 1. Step selection analysis workflow. Light grey shading indicates conventional SSA whereas dark grey shading indicates the integrated approach advocated here (iSSA). See Appendix S4 for detailed iSSA guidelines and tips. movement process within an SSA, a complete habitat-depen- Materials and methods dent mechanistic movement model can be parametrised from telemetry data using a standard CLR routine. In the following, INTEGRATED STEP SELECTION ANALYSIS we provide a detailed description of the approach and evaluate In their work on the subject of accounting for movement in resource- its performance (compared with standard RSA and SSA) in selection analysis, Forester, Im & Rathouz (2009) demonstrated that correctly inferring the movement and habitat-selection including a distance function in SSA substantially reduces the bias in processes underlying observed space-use patterns, and in pre- habitat-selection estimates. Mathematically, their argument is based dicting the resulting UD. © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 622 T. Avgar et al. on the habitat-independent movement kernel (the function governing Ψ (e.g. selection for snow-free or flat localities). Eqn 1 is equivalent to movement in the absence of resource selection, or across a homoge- the formulation used (for example) by Rhodes et al. (2005, Eqn 1], neous landscape; Hjermann 2000; Rhodes et al. 2005) belonging to Forester, Im & Rathouz (2009) and Johnson, Hooten & Kuhn (2013, the exponential family, so that it can be accounted for with the logistic Eqn 1) and is a generalised form of a redistribution kernel – awidely formulation of the SSA likelihood function. Here we shall make the used mechanistic model of animal movement and habitat selection (see assumption that, in the absence of habitat selection, step lengths fol- Discussion for recent examples). low either an exponential, half-normal, gamma or log-normal distri- The denominator in Eqn 1 is an integral over the entire spatial bution. Under this assumption, the statistical coefficients associated domain, Ω, serving as a normalisation factor to ensure the resulting with step length, its square, its natural logarithm and/or the square of probability density integrates to one. Whereas in most cases it would its natural logarithm (depending on the assumed distribution), when be impossible to solve this integral analytically, various forms of incorporated as covariates in a standard SSA, serve as statistical esti- numerical (discrete-space) approximations can be used to fit redistri- mators of the parameters of the assumed step-length distribution (see bution kernel functions, such as Eqn 1, to data (see Avgar, Deardon Appendices S2 and S3 for details, and below for an example). Stan- & Fryxell 2013a and the Discussion). Here we focus on a simple like- dard model selection (e.g. likelihood ratio or AIC) then can be used to lihood-based alternative to such numerical methods, one that can be select the best-fit theoretical distribution (out of the four listed above). implemented using common statistical software and is hence accessi- The iSSA approach moreover can be applied to infer directional per- ble to most ecologists. Assuming an exponential form for both Φ sistence and external bias. Assuming that the angular deviations from and Ψ, MLEs for the parameter vectors h and x can be obtained preferred directions (either the previous heading, the target heading or using conditional logistic regression, where observed positions (cases) both) are von Mises distributed (an analogue of the normal distribution are matched with a sample of available positions (controls; Fig. 1 on the circle), the cosine of these angular deviations can be included as and Appendices S2–S4). covariates in an SSA to obtain MLEs of the corresponding von Mises concentration parameters (Duchesne, Fortin & Rivest 2015). Hence, A HYPOTHETICAL EXAMPLE MLEs of iSSA coefficients affiliated with directional deviations and step lengths are directly interpretable as the parameters of distributions Letusassumewehave obtained aset of T spatial positions, sampled at governing the underlying BCRW. a unit temporal interval along an animal’s path, and that we also have We shall make the assumption here that animal space-use beha- maps of two (temporally stationary) spatial covariates, h(x)and y(x). viour is adequately captured by a separable model, involving the pro- We shall now assess the statistical support for the following proposi- duct of two kernels, a movement kernel and a habitat-selection kernel. tions (examples of the sort of hypotheses that could be tested): Formally, we define a discrete-time movement kernel, Φ, which is pro- A The animal is selecting high values of h(x). portional to the probability density of occurrence in any spatial posi- B At the observed temporal scale, and in the absence of variability in h tion, x,at time t, in the absence of habitat selection. The determinants (x), the animal’s movement is directionally persistent (i.e. consecu- of Φ are as follows: the Euclidian distances between x and the preced- tive headings are positively correlated), and the degree of this persis- ing position, x (the step length; l =||x – x ||), the distances t1 t t1 tence varies with y(x) (e.g. the animal moves more directionally between x and x (the previous step length; l =||x – x ||), t1 t2 t1 t1 t2 where y(x) is lower). The resulting turn angles are von Mises dis- the associated step headings, a and a (the directions of movement t t1 tributed with mean 0 (i.e. left and right turns are equally likely) and from x to x and from x to x , respectively), and a vector of t1 t2 t1 a y-dependent concentration parameter. spatial and/or temporal movement predictors at time t and/or at the C At the observed temporal scale, and in the absence of variability in h vicinity of x and/or x , Y(x,x ,t) (e.g. terrain ruggedness, migra- t1 t1 (x)and y(x), the animal’s movement is characterised by gamma dis- tory phase, snow depth, etc.). The effects of these step attributes on Φ tributed step lengths, and the shape of this step-length distribution are controlled by the associated coefficients vector, h. Note that the depends on the time of day (e.g. the animal moves faster during day- effects of spatial attributes here are assumed to operate through local time). biomechanical interactions between the animal and its immediate Note that these propositions are contingent on the temporal gap environment, interactions that determine the rate of displacement (i.e. between observed relocations (i.e. step duration), as well as on the spa- kinesis), not where the animal ‘wants’ to be (i.e. taxis). Also note that tial resolution of our covariate maps, h(x)and y(x). We thus explicitly the kernel Φ can be non-Markovian and accommodate various types acknowledge that our inference is scale dependent. of velocity autocorrelations (lack of independence between directions We start by sampling, for each (but the first two) of the observed and/or lengths of consecutive steps), including correlated and biased points along a path (x , t = 3, 4, .. ., T), a set of s control points (avail- random walks (if directional biases are known apriori). able spatial positions at time t; x , i = 1, 2, .. ., s), where the probability t;i We further define the habitat-selection function, Ψ, which is propor- of obtaining a sample at some distance, l , from the previous observed t;i tional to the probability density of observing the animal in any spatial 0 0 point (l ¼kx  x k)isgivenbythegammaPDF: t1 t;i t;i position, x,attime t, in the absence of movement constraints. The determinants of Ψ are the habitat attributes in x at t, H(x,t), and their t;i 0 0 b 1 gl jb ; b ¼ l e eqn 2 1 2 t;i t;i corresponding selection coefficients, x. The normalised product of Φ b Cðb Þb 1 2 andΨ yields the probability density of occurrence in x at t,whichis: Here, b and b are initial estimates of the gamma shape and scale 1 2 U½ l ;l ;a ;a ;Yðx;x ;tÞ;hW½ HxðÞ ;t ;x t t1 t t1 t1 fxðÞ jx ;x ¼ : t t1 t2 parameters (respectively) obtained based on the observed step-length U½ l ;l ;a ;a ;Yðx;x ;tÞ;hW½ HxðÞ ;t ;x dx t t1 t t1 t1 distribution (using either the method of moments or maximum likeli- eqn 1 hood). As noted earlier, this estimation is confounded by the process of Note that the same environmental variable (e.g. snow depth or ter- habitat selection, and hence, a method to unravel movement inference rain ruggedness) might be included in both Y and H and hence affect from habitat selection is needed. The iSSA will provide estimates of the both Φ[e.g.decreasedspeedindeepsnoworacrossruggedterrain) and deviations of these initial values from the unobserved habitat-indepen- © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 Integrated step selection analysis 623 dent shape and scale (Appendices S2–S3). Note that these control sets (2012) can be included in an iSSA with the MLEs obtained using also could be sampled randomly within some finite spatial domain (e.g. standard statistical packages. An iSSA thus holds promise as a within the maximal observed displacement distance; Appendices S2 user-friendly yet versatile approach in the movement ecologist’s and S4). Distance weighted sampling is expected to increase inferential toolbox. In Appendix S4, we provide practical guidelines for the efficiency, resulting in a smaller standard error for a given s value, but is application of iSSA. In the next sections, we explore the utility of not a necessity (Forester, Im & Rathouz 2009). In general, any increase this approach using computer simulations. in T and/or s will result in better approximation of the used and/or availability distributions (respectively), and hence better inference SIMULATIONS (together with larger computational costs). Once sampled, control (available) points, x , are assigned a value of t;i Testing the inferential and predictive capacities of any statistical 0, whereas the observed (used or case) points, x , are assigned a value of space-use model is challenging because we are often ignorant of the true 1. The resulting binomial response variable can now be statistically process giving rise to the observed patterns, as well as of the true distri- modelled using conditional (case–control) logistic regression, as the bution of space-use from which these patterns are sampled (Avgar, likelihood of the observed data is exactly proportional to (Gail, Lubin Deardon & Fryxell 2013a; Van Moorter et al. 2013). To deal with this & Rubinstein 1981; Forester, Im & Rathouz 2009; Duchesne, Fortin & challenge, we employ here a simple process-based movement Rivest 2015): exp½b hðx Þþ ½b þ b yðx Þ  cosða  a Þþ b l þðb þ b D Þ lnðl Þ 3 t 4 5 t1 t1 t 6 t 7 8 t t ; eqn 3 0 0 0 0 exp½b hðx Þþ ½b þ b yðx Þcosða  a Þþ b l þðb þ b D Þ lnðl Þ 3 4 5 t1 t1 6 7 8 t t;i t;i t;i t;i i¼0 t¼3 0 0 simulation framework. We provide full details of the simulation where a is the direction of movement from x to x ,and D is an t1 t t;i t;i procedure and its statistical analysis in Appendix S5. indicator variable having the value 1 when t is daytime and 0 otherwise. Note that the summation in the denominator starts at s = 0(rather Fine-scale space-use dynamics were simulated using stochastic than 1) to indicate that the used step is included in the availability set ‘stepping-stone’ movement across a hexagonal grid of cells. Each (x ¼ x ). Also note that it is the inclusion of turn angles that neces- cell, x, is characterised by habitat quality, h(x) with spatial autocor- t;i¼0 sitates the exclusion of the first two positions (x and x ); if no relation set by an autocorrelation range parameter, q (=0, 1, 5, 10 t =1 t =2 velocity autocorrelation is modelled, only the first position is excluded. and 50). For each q value, 1000 trajectories were simulated and Lastly, note that this formulation implies that the degree of directional then rarefied (by retaining every 100th position). Each of these rar- persistence is affected by the value of y at the onset of the step only; in efied trajectories was then separately analysed using RSA and 10 the next section, we provide an example of modelling habitat effects on different (i)SSA formulations, including one or more of the follow- movement along the step. ing covariates (Table 1): habitat values at the end of each step, h (x ), the average habitat value along each step, h(x ,x ), the step Equation 3 is a discrete-choice approximation of Eqn 1 (specifi- t t1 t length, l (=||x – x ||), its natural-log transformation, ln(l ), and cally tailored according to propositions A–C), and we provide its t t1 t t the statistical interactions between l,ln(l)and h(x ,x ). Models full derivation in Appendix S3. In summary, b is the habitat-selec- t t t1 t that included only h(x)and/or h(x ,x ) correspond to traditionally tion coefficient (corresponding to proposition A and estimating the t t1 t used SSA (relying on empirical movement distributions with no only component of the parameter vector x in Eqn 1), b and b 4 5 movement attribute included as covariates; models a, b and c in are the basal (habitat-independent) and y-dependent directional Table 1), whereas models that additionally included l and ln(l)cor- persistence coefficients (corresponding to proposition B and esti- t t respond to iSSA. The predictive capacity of the models was esti- mating two components of the parameter vector h in Eqn 1), and mated based on the agreement between their predicted utilisation b , b and b are the modifiers of the step-length shape and scale 6 7 8 distributions (UD) and the ‘true’ UD, generated by the true under- coefficients (corresponding to proposition C and estimating the remaining components of the parameter vector h). Once maxi- lying movement process. We refer the reader to Appendix S5 for mum-likelihood estimates are obtained, the shape and scale param- further details. eters of the basal step-length distribution can be calculated A separate simulation study was conducted to evaluate the identifia- (Appendix S3), where the shape is given by: [(b + b ) + b .D ], bility and estimability of the iSSA parameters as function of sample size 1 7 8 t and the scale is given by: [1/(b –b )]. Similarly, b can be shown and habitat-selection strength (Appendix S6). 2 6 4 to be an unbiased estimator of the concentration parameter of the (habitat-independent) von Mises turn angles distribution (Duch- Results esne, Fortin & Rivest 2015). Including movement attributes as covariates in SSA, which we PARAMETERISATION termed here iSSA, thus allows simple likelihood-based estimation of explicit ecological hypotheses within a framework of a mechanistic All models converged in a timely manner and the convergence habitat-mediated movement model. Such hypotheses might include, time for the most complex model (model j in Table 1) was in addition to those mentioned thus far, long- and short-term target approximately 1 CPU sec. Of the 10 (i)SSA formulations speci- prioritisation (Duchesne, Fortin & Rivest 2015), barrier crossing fied in Table 1, AIC ranking indicated support for only four and avoidance behaviour (Beyer et al. 2015), and interactions with (d, f, h and j), all of which include the habitat value at the step’s conspecifics and intraspecifics (Latombe, Fortin & Parrott 2014; endpoint (with coefficient b ) and the step length and its natu- Potts, Mokross & Lewis 2014b; Potts et al. 2014a). In fact, many of 3 the facets of the generic approach developed by Langrock et al. ral logarithm (with coefficients b and b ) as covariates. Hence, 5 6 © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 624 T. Avgar et al. Table 1. The 11 different models fitted here and their relative performance ranking at five different levels of habitat spatial autocorrelation (with 1000 realisations at each level). To enable AIC comparison, RSA’s were run with only those positions included in the SSA (i.e. excluding the first position) Covariates % Scord as best (based on AIC) R (x x ) R (x x ) t, t1 t, t1 Model R (x ) R (x x ) l ln (l ) l ln (l ) q = 0 q = 1 q = 5 q = 10 q = 50 t t, t1 t t t t RSA b 00 0 0 0 0 0 203 457447 RSA SSA a b 00 0 0 0 0 0 0 0 0 b 0 b 0 0 0 0 00 00 0 c b b 0 0 0 0 00 00 0 3 4 iSSA d b 0 b b 00 0 0 0 1111 3 5 6 e 0 b b b 0 0 00 00 0 4 5 6 f b b b b 0010444139240284 3 4 5 6 g 00 b b b b 00 00 0 5 6 7 8 h b 0 b b b b 662 2831075 23 3 5 6 7 8 i 0 b b b b b 00 00 0 4 5 6 7 8 j b b b b b b 234 673551 233235 3 4 5 6 7 8 Bolded numbers mark the best performing model at each level of spatial autocorrelation. iSSAs better explain our simulated data than traditionally used and j), b wasclosertozero(Fig. 3).Interestingly,whenonly SSAs (excluding step length as a covariate), but only as long as the habitat at the end of the step and the habitat along the step an endpoint effect (i.e. selection for/against the habitat value at were included in the model (i.e. model c;acommonly used the end of the step) is included. In fact, models that excluded SSA formulation), and at low q values (=0, 1), b was negative, the habitat value at the step’s endpoint (models b, e, g and i) indicating ‘selection against’ high-quality steps. In fact, this had AIC scores that were typically two orders of magnitude reflects the low probability of observing a ‘used’ step that tra- larger than those including it. In comparison to RSA, iSSA verses high-quality habitat but does not end there. formulations had unequivocal AIC support at low habitat spa- As explained above (and in Appendices S2 and S3), iSSA tial autocorrelation levels, but only partial support at high coefficients affiliated with the step length (b ) and its natural autocorrelation levels (Table 1). logarithm (b ), when combined with the estimated shape and Estimated habitat-selection strengths, as indicated by our scale values of the observed step-length distribution (b and b ; 1 2 RSA and SSA coefficient estimates (b and b respectively), on which sampling was conditioned; Appendices S3 and S5), RSA 3, were appreciably larger than the true habitat-selection strength could be used to infer the shape and scale of the ‘habitat-inde- (x = 1), and more so for RSA estimates than for SSA (Fig. 2). pendent’ step-length distribution [i.e. assuming h(x ,x ) = 0]. t t+1 Note that this in itself does not mean these estimates are ‘bi- Under most imaginable scenarios, we would expect this basal ased’ but rather reflects the inherent difference between the movement kernel to be wider (i.e. with larger mean) than the intensity of the true process and that of the emerging pattern, observed one, as animals tend to linger in preferred habitats at the scale of observation (see further discussion below). These and hence display more restricted movements compared to the estimates showed little sensitivity to the level of habitat spatial basal expectation. Indeed, the mean of these inferred distribu- ðb þb Þ 1 6 autocorrelation, although a substantial increase in variance is tions (the product of their shape and scale: ) corre- ðb b Þ 2 5 observed in the RSA case (Fig. 2a). As found before by Fores- sponds exactly to the observed mean, as long as no other ter, Im & Rathouz (2009), the strength of SSA-inferred habitat covariates are included in the analysis (model x in Fig. 4). selection is larger when step lengths are included as a covariate Once other covariates are included in the model (and hence in the analysis (iSSA), but this effect is fairly weak and dimin- habitat selection is at least partially accounted for), inferred ishesasthe habitat’sspatial autocorrelation increases mean step-length values were significantly higher from the (Fig. 2b). observed values, showed little sensitivity to model structure, Overall, SSA-inferred habitat selections were substantially but increased with q (as do the observed mean step lengths). less variable than RSA-based estimates and showed little sensi- One exception is model g, which strongly underestimated the tivity to the inclusion or exclusion of other covariates in the mean step length at moderate-high q valuesasitdoesnot model fit (Fig. 2). This is not the case, however, for the effect include any main habitat effects. of the mean habitat value along the step (b ), which varied sub- Even at high q values, inferred mean step length slightly but stantially with both the level of habitat spatial autocorrelation consistently underestimates the true habitat-independent step- length distribution (calculated by simulating the process based and the inclusion of an endpoint effect (b ). Where b was not 3 3 included in the model fit (models b, e and i in Table 1), b on Eqn S51with x = 0; Fig. 4). This bias is a result of an increased with q,whereas where b was included (models c, f iSSA’s limited capacity to account for the full movement © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 Integrated step selection analysis 625 (a) (b) Fig. 2. Statistically inferred habitat-selection coefficient estimates for RSA (a) and SSA (b; letters along the x-axis refer to the SSA formulations listed in Table 1), for five levels of habitat spatial autocorrelation, q. Each box-and-whiskers is based on 1000 independent estimates, where the cen- tral mark is the median, the edges of the box are the 25th and 75th percentiles, the whiskers extend to the most extreme data points not considered outliers (i.e. within approximately 99% coverage if the data are normally distributed), and outliers are plotted individually. Horizontal dashed lines represent the true habitat-selection intensity, x = 1. See Appendix S5 for further details. Fig. 3. Statistically inferred effects of the mean habitat along the step. The dashed line represents no effect. Other details are as in Fig 2. process as it unfolds in between observations. The animal does of the underlying movement process, the animal’s true move- not actually travel along the straight lines that we term ‘steps’ ment capacity is never fully manifested in the observed reloca- and, even if it would, the mean habitat value along the step tion pattern and is thus always underestimated. Note, does not exactly correspond to its probability to travel farther. however, that this bias is negligibly small where the spatial As long as the scale of the observation is coarser than the scale autocorrelation of habitats is high (q > 1; Fig. 4). © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 626 T. Avgar et al. Fig. 4. Mean of gamma step-length distributions (displacement in spatial units per Dt; Appendix S5) inferred based on the different iSSA formula- tions (see Table 1). Model x is a null model, including only the step length and its natural logarithm (with no habitat effects), added here to demon- strate that the conditional logistic regression produces unbiased MLEs. The dotted lines correspond to the observed mean step length across all 1000 realisations at each of the five levels of habitat spatial autocorrelation. The dashed line corresponds to the ‘true’ habitat-free mean step length, calcu- lated by simulating the process using Eqn S5.1 but with x = 0. Other details are as in Fig 2. Finally, despite apparent support for iSSA formulations predictions were robust to these scale mismatches, with G KLD including interaction between the step length and habitat qual- scores of ~098 and ~096, for the q = 50 landscape (with ity along the step (Table 1; models h and j), the estimated val- parameter estimates based on q = 0 data) and the q = 0 ues of the interaction coefficients, b and b , mostly overlapped landscape (with parameter estimates based on q = 50 data), 7 8 zero (Appendix S7). Generally speaking, the mean habitat respectively. Similarly, all step selection models including an value along the step has a weak negative effect on both the endpoint effect performed well, with G scores ranging KLD shape (through b ) and the scale (through its inverse relation- from ~094 (model f)to ~098 (model h)for the q = 0land- ship with b ) of the step-length distribution – long steps are less scape (with parameter estimates based on q = 50 data), and likely through high-quality habitats. G scores ranging from ~083 (model j)to ~097 (model a) KLD for the q = 50 landscape (with parameter estimates based on q = 0 data). Overall, iSSA-based predictive capacity PREDICTIVE CAPACITIES remained mostly unaltered by mismatches between the data’s At approximate steady state, RSA-based UD predictions are landscape structure and the structure of the landscape on slightly more accurate and precise than SSA-based predictions which projections are made. As can be expected, step selection models predict transient (Fig. 5 and Appendix Table S8). The RSA’s predictive capac- ity increases with q (while its precision dramatically decreases), UDs better than the inherently stationary RSA (except when whereas the opposite is true for SSA predictions, where the q = 50; Appendix Table S8). In comparison with steady-state minimum KLD value (Kullback-Leibler Divergence; see predictions, complex iSSA-based predictions perform better Appendix S5) is reached when q = 0 (Appendix Table S8). than simpler ones (Appendix Table S8). As for the steady-state KLD values coarsely mirror the AIC ranking of the different predictions, all iSSA formulations including an endpoint effect SSA formulations in distinguishing those that include an end- performed well in predicting transient UDs, with G scores KLD point effect (b ), but the best performing formulations based ranging from ~89% (model h when q = 0) to ~098 (model d on KLD are simpler than the ones selected based on AIC when q = 1). G scores for RSA-based predictions showed KLD (Tables 1 and S8). That said, all iSSA formulations including substantial sensitivity to the level of spatial autocorrelation, an endpoint effect performed well overall, with G scores (a ranging from ~069 (q = 0) to ~097 (q = 50). KLD measure of goodness of fit; Appendix S5) ranging from ~084 (model f when q = 50) to ~098 (model d when q = 0). For ref- ISSA IDENTIFIABILITY AND ESTIMABILITY erence, the G scores for RSA-based predictions ranged KLD from ~098 (q = 0) to ~099 (q = 50). Results are presented in detail in Appendix S6. In short, our To test the sensitivity of the models’ predictions to the analysis revealed that, under the test scenario, all iSSA sampling scale (see Appendix S9 for relating q to the sam- parameters are fully identifiable, that estimates are unbiased pling scale), we generated predicted UDs, both SSA-based in relation to the true values of the kernel generating func- and RSA-based, across a highly autocorrelated landscape tions, andthatan increasein samplesizebeyond ~400 (q = 50) using parameter estimates obtained from samples of observed positions does not seem to substantially enhance a random landscape (q = 0), and vice versa. RSA-based precision (and hence estimability). That said, our results also © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 Integrated step selection analysis 627 Fig. 5. Log–log plots of the true UDs vs the predicted UDs. Each dot represents the utilisa- tion probability of a single spatial cell. Black dots correspond to the median parameter esti- mates, whereas grey dots correspond to the 25 and 975 percentiles of the estimated parame- ters distribution. Black diagonal lines repre- sent a perfect 1:1 mapping – dots appearing above these lines are spatial cells where the true UD value exceeded the predicted UD value (under-predictions), whereas dots appearing below these lines represent over- predictions. Note that iSSA results are pre- sented for the simplest iSSA including an end- point effect, formulation d in Table 1. indicate that inferential accuracy of movement related param- to the true UD when the time-scale is long (and hence eters may be highly variable, leading to compromised preci- approaches the steady-state limit). sion (with up to 1000% departure from the true value; Two caveats are in place here. First, in our analysis the defi- Appendix S6) even at a fairly largesamplesize. This maybe nition of the availability set for the RSA was exact (i.e. the particularly true given the inherent trade-off between sam- entire domain), a situation that seldom occurs in empirical pling extent and frequency (Fieberg 2007). Estimability of cer- studies where availability is unknown. This is not the case for tain parameters, under certain scenarios, may thus be weak iSSA where the availability set always can be adequately andmustbeevaluatedonacase-by-case basis. defined (but is conditional on the temporal resolution of the positional data). Secondly, the high variability characterising the RSA coefficient estimates, and its resulting predictions Discussion (Figs 2 and 5) indicate substantial risk of erroneous inference. This may be particularly true when sample size is smaller than The ideas, simulations and results presented above are aimed the relatively large sample used here, resulting in data that are at providing a comprehensive assessment of using integrated not adequate unbiased samples of the steady-state UD, which step selection analysis, iSSA, with emphasis on its predictive is likely the case in most empirical studies. The more mechanis- capacity. The iSSA allows simultaneous inference of habitat- tic nature of the iSSA makes it less sensitive to stochastic differ- dependent movement and habitat selection and is hence a ences between specific realisations of the space-use process powerful tool for both evaluating ecological hypotheses and (e.g. due to differences in landscape configuration) and thus predicting ecological patterns. We have shown that iSSA- leads to more precise inference. Hence, even if the sole objec- based habitat-selection inference is relatively insensitive to tive of a given study is to predict the long-term (steady-state) model structure and landscape configuration, and that iSSA- utilisation distribution, the more complicated iSSA-based pre- based UD predictions perform well across different temporal dictions mightbemorereliablethanthose based onRSA. and spatial scales (we discuss the connection between the Moreover, in many real-world ecological scenarios, a steady temporal resolution of the data and the habitat spatial auto- state is never reached, and consequently, the static RSA-based correlation in Appendix S9). On the other hand, our results approach is less appropriate than the dynamical iSSA. We thus indicate that movement and habitat selection may not be com- conclude that iSSA should be the method of choice whenever: pletely separable once observations are collected at a coarser (i) RSA availability cannot be properly defined, (ii) predicting temporal resolution than the underlying behavioural process. across a landscape different from the landscape used for Consequently, stationary RSA-based predictions, whereas parametrisation, (iii) the data used for parametrisation are not much simpler to obtain, provide slight but consistent better fit © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 628 T. Avgar et al. an adequate sample of the true steady-state UD or (iv) predict- et al. 2005; Getz & Saltz 2008; Avgar, Deardon & Fryxell ing transient space-use dynamics. 2013a; Potts et al. 2014a; Beyer et al. 2015). Unlike the Many movement and selection processes could be consid- iSSA, however, fitting these kernel-based models to empiri- ered plausible, and the particular details of the mechanistic cal data relies on complex, and often specifically tailored model used to simulate space-use data might substantially alter likelihood maximisation algorithms (namely discrete-space our conclusions. Our aim here was to use the simplest, and approximations of the integral in Eqn 1). The statistical hence most general, mechanistic process imaginable, leading machineryused iniSSA, basedonobtaining asmall setof us to choose a stepping-stone movement process as our pat- random samples from an inclusive availability domain, is tern-generating process. Interestingly, this simple process, gov- accessible to most ecologists because it relies on software erned by only two parameters (Eqn S5.1), gave rise to complex that is already used (Thurfjell, Ciuti & Boyce 2014). patterns once rarefied. In particular, the emerging step-length Through the addition of appropriate covariates and interac- distributions fit remarkably well with a gamma distribution, tion terms, iSSA can moreover address many of the ques- with shape and scale that reflect the underlying landscape tions that were the focus of other kernel-based approaches, such as home-range behaviour (Rhodes et al. 2005), mem- structure. Note that this is a purely phenomenological descrip- tion of the movement kernel, as the true underlying process ory-use (Avgar, Deardon & Fryxell 2013a; Merkle, Fortin had a fixed, habitat-independent movement parameter & Morales 2014; Avgar et al. 2015; Schl€agel & Lewis 2015), (Appendix S5). Ideally, a truly mechanistic approach will habitat-dependent habitat selection (Potts et al. 2014a) and involve maximising the likelihood over all possible paths the barrier effects (Beyer et al. 2015). Hence, iSSA allows ecolo- animal might have taken between two observed locations, and gists to tackle complicated questions using simple tools. hence allowing inference of the true underlying process (Mat- To conclude, our work complements several recent contri- thiopoulos 2003). In most cases, however, this approach is for- butions advocating the use of movement covariates within step biddingly computationally expensive. We showed that the selection analysis (Forester, Im & Rathouz 2009; Johnson, approximation based on samples of straight-line movements Hooten & Kuhn 2013; Warton & Aarts 2013; Duchesne, For- between observed positions, which is the underlying assump- tin & Rivest 2015). We believe a convincing body of theoretical tion of any SSA, performs well over a range of conditions. An evidence now indicates the suitability of integrated step selec- iSSA thus provides a reasonable compromise between compu- tion analysis as a general, flexible and user-friendly approach tationally intensive mechanistic models and the purely phe- for both evaluating ecological hypotheses and predicting nomenological RSA. future ecological patterns. Our work highlights the importance According to Barnett & Moorcroft (2008), the steady-state of including endpoint effects in the analysis together with some UD should scale linearly with the underlying habitat-selection caveats regarding the interpretation of SSA results, specifically function Ψ (Eqn 1) when informed movement capacity when dealing with the effects of the habitat along the step. We exceeds the scale of spatial variation in Ψ, but should scale with also recommend careful consideration of parameter estimabil- the square ofΨ if informed movement capacity is much shorter ity, particularly with regard to the movement components of than the scale of habitat variation. In the particular case of the the model, which may be prone to strong cross-correlations (as exponential habitat-selection function used here (Eqn S5.1), discussed in Appendix S6). Based on our current experience in we would thus expect the following loglinear relationship: ln applying iSSA to empirical data (T. Avgar, work in progress) [UD(x)] = a + b∙x∙h(x), where a is a scaling parameter [the we have provided practical guidelines in Appendix S4. Addi- utilisation probability where h(x) = 0], and b (1 ≤ b ≤ 2) is tional theoretical work is needed to investigate the effects of some increasing function of the habitat spatial autocorrelation, the underlying movement process on iSSA performance, as q. Our results, emerging from a very different model than the well as to come up with computationally efficient iSSA-based continuous-space continuous-time analytical approximation simulations to enable rapid generation of predicted utilisation of Barnett & Moorcroft (2008), corroborate this expectation. distribution (as discussed in Appendix S1). Most importantly, The slope of the loglinear regression model described above the utility of iSSA must now be evaluated by applying it to real increases from b  14to b  2as q increases from 0 to 50 data sets, and using it to solve real ecological problems. (Appendix S10). RSA-based coefficient estimates, b , clo- RSA sely mirror this pattern, increasing from ~16to ~2as q Acknowledgements increases (Fig. 2a). Hence, as can be expected from a phe- TA gratefully acknowledges supported by the Killam and Banting Postdoctoral nomenological model, RSA-based inference reflects the Fellowships. MAL gratefully acknowledges NSERC Discovery and Accelerator steady-state UD rather than the underlying habitat-selection Grants and a Canada Research Chair. MSB thanks NSERC and the Alberta Conservation Association for funding. The authors thank L. Broitman for process. designing Fig. 1 and C. Prokopenko for helpful editorial comments, and Dr. Recent years have seen a proliferation of sophisticated Geert Aarts, Dr. John Fieberg and an anonymous reviewer, for their instructive modelling approaches aimed at mechanistically capturing comments and suggestions. animal space-use behaviours. Many of these models share the theoretical underpinning of iSSA (as formulated in Data accessibility Eqn 1), relying on a depiction of animal space-use as All data used in this paper were simulated as described in the online Supporting emerging from the product of a resource-selection process Information. and a selection-independent movement kernel (e.g. Rhodes © 2015 The Authors. 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Jonsen, I.D., Myers, R.A. & Flemming, J.M. (2003) Meta-analysis of animal Received 18 August 2015; accepted 4 December 2015 movement using state-space models. Ecology, 84, 3055–3063. Handling Editor: Luca Borg € er Kareiva, P.M. & Shigesada, N. (1983) Analyzing insect movement as a correlated random-walk. Oecologia, 56, 234–238. © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 630 T. Avgar et al. Supporting Information Appendix S6. Evaluating the iSSA parameter identifiability and estima- Additional Supporting Information may be found in the online version bility. of this article. Appendix S7. b7and b8. Appendix S1. From step selection to utilisation distribution. Appendix S8. Predictive performance. Appendix S2. Inferring step-length distributions. Appendix S9. Interpreting q. Appendix S3. Deriving an iSSA likelihood function. Appendix S10. Habitat selection and utilisation distribution. Appendix S4. iSSA practical user guide. Appendix S11. Appendices reference list. Appendix S5. Simulation experiments. © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Methods in Ecology and Evolution Wiley

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

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Methods in Ecology and Evolution © 2016 British Ecological Society
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Methods in Ecology and Evolution 2016, 7, 619–630 doi: 10.1111/2041-210X.12528 Integrated step selection analysis: bridging the gap between resource selection and animal movement 1 2 1,3 1 Tal Avgar *, Jonathan R. Potts ,MarkA.Lewis and Mark S. Boyce 1 2 Department of Biological Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada; School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK; and Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada Summary 1. A resource selection function is a model of the likelihood that an available spatial unit will be used by an ani- mal, 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 consecu- tive observed positions of the animal) with a set of ‘available steps’ randomly sampled from a distribution of observed steps or their characteristics. 2. 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 autocorrela- tions) 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. 3. 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. 4. Our work demonstrates the utility of iSSA as a general, flexible and user-friendly approach for both evaluat- ing a variety of ecological hypotheses, and predicting future ecological patterns. Key-words: conditional logistic regression, dispersal, habitat selection, movement ecology, random walk, redistribution kernel, resource selection, step selection, telemetry, utilisation distribution ing explicit movement behaviours into spatial models of Introduction animal density has led to improved predictive performance (Moorcroft, Lewis & Crabtree 2006; Fordham et al. 2014). Ecology is the scientific study of processes that determine the distribution and abundance of organisms in space and time Deriving predictive space-use models based on the beha- (Elton 1927). Hence, asking how and why living beings vioural process underlying animal movement patterns is of change their spatial position through time is fundamental to particular importance when dealing with altered or novel ecological research (Nathan et al. 2008). Animal movement landscapes that might differ substantially from the landscape links the behavioural ecology of individuals with population used to inform the models. and community level processes (Lima & Zollner 1996). Its Over the past three decades, a great deal of research has been study is consequently vital for understanding basic ecological dedicated to explaining and predicting spatial population dis- processes, as well as for applications in wildlife management tribution patterns based on underlying habitat attributes (often and conservation. termed resources). In that regard, much focus has been given Whether basic or applied, common to many empirical to estimating resource selection functions (Manly et al. 2002)– studies of animal movement is the aspiration to reliably pre- phenomenological models of the relative probability that an dict population density through space and time by modelling available discrete spatial unit (e.g. an encountered patch or the spatiotemporal probability of animal occurrence, also landscape pixel) will be used given its resource type/value (Lele known as the utilisation distribution (Keating & Cherry et al. 2013). Indeed, its intuitive nature and ease of application 2009). Despite much progress in statistical characterisation has made resource selection analysis (RSA) the tool of choice of animal movement and habitat associations, our ability to for many wildlife scientists and managers seeking to use envi- predict utilisation distributions is limited by our understand- ronmental information in conjunction with animal positional ing of the underlying behavioural processes. Indeed, includ- data (Boyce & McDonald 1999; McDonald et al. 2013; Boyce et al. 2015). Whereas much progress has been gained in the application *Correspondence author. E-mail: avgar@ualberta.ca of RSAs to animal positional data, the problem of defining © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. 620 T. Avgar et al. the appropriate spatial domain available to the animal a much sought shift from an individual-based Lagrangian remains as a major concern (Matthiopoulos 2003; Lele et al. perspective to population-level Eulerian models (Turchin 2013; McDonald et al. 2013; Northrup et al. 2013). 1991, 1998). SSAs are thus at an interface between statisti- Weighted distribution approaches deal with this problem by cal (phenomenological) RSAs and mathematical (mechanis- modelling space-use as a function of a movement model and tic) RW models (Potts, Mokross & Lewis 2014b; Potts a selection function, but most weighted distribution models et al. 2014a), models that form the backbone of much of are challenging to implement (but see Johnson, Hooten & the existing body of theory in the field of animal move- Kuhn 2013). Matched case–control logistic regressions ment (Codling, Plank & Benhamou 2008; Benhamou 2014; (CLRs; also known as discrete-choice models) may be con- Fagan & Calabrese 2014). sidered a simplified alternative to the weighted distribution In this paper, we outline a CLR-based approach for approach where each observed location is matched with a simultaneous estimation of the movement and habitat-selec- conditional availability set, limited to some predefined spatial tion components, an approach we name integrated step selec- and/or temporal range (Arthur et al. 1996; McCracken, tion analysis (iSSA; Fig. 1). The iSSA allows the effects of environmental variables on the movement and selection pro- Manly & Heyden 1998; Compton, Rhymer & McCollough 2002; Boyce et al. 2003; Baasch et al. 2010). A major cesses to be distinguished, thus providing a valuable tool for strength of this approach is that maximum-likelihood esti- testing hypotheses (e.g. to test whether animals travel faster mates (MLEs) of the parameters can be efficiently obtained in certain times or through certain habitats), while resulting though commonly used statistical software (often relying on in an empirically parameterised mechanistic movement a Cox Proportional Hazard routine; e.g. function clogit in model (i.e. a mechanistic step selection model; Potts et al. R). One particular type of such conditional RSA is step 2014a), that can be used to translate individual-level observa- selection analysis (SSA), where each ‘used step’ (connecting tions to population-level utilisation distributions across space two consecutive observed positions of the animal) is coupled and time (Potts et al. 2014a; Potts, Mokross & Lewis 2014b; with a set of ‘available steps’ randomly sampled from the Appendix S1). empirical distribution of observed steps or their characteris- The iSSA is related to several recently published works tics (e.g. their length and direction; Fortin et al. 2005; Duch- integrating animal movement and resource selection. Both esne, Fortin & Courbin 2010; Thurfjell, Ciuti & Boyce Forester, Im & Rathouz (2009) and Warton & Aarts 2014). (2013) demonstrated the inclusion of movement variables The definition of availability is challenging, however, even in an RSA and its marked effect on the resulting inference. when using the SSA approach. The problem arises due to the Johnson, Hooten & Kuhn (2013) have shown that animal sequential, rather than simultaneous, estimation of the move- telemetry data can be viewed as a realisation of a non- ment and habitat-selection components of the process. Owing homogenous space–time point process, and MLEs of this to this stepwise procedure, the resulting habitat-selection process can be obtained using a generalised linear model. inference is conditional (on movement), whereas movement is These contributions focused on gaining unbiased resource assumed independent of habitat selection. In reality, the two selection inference while treating the movement process as are tightly linked, with habitat selection and availability nuisance that must be ‘controlled for’. Here, we seek expli- affecting the animal’s movement patterns (Avgar et al. cit inference of this process. State-space models of animal 2013b), and the animal’s movement capacity affecting its movement (reviewed by Jonsen, Myers & Flemming 2003; habitat-use patterns (Rhodes et al. 2005; Avgar et al. 2015). and Patterson et al. 2008) predict the future state (e.g. spa- Failure to adequately account for the movement process may tial position) of the animal given its current state (where consequently lead to biased habitat-selection estimates (Fores- an ‘observation model’ provides the probability of observ- ter, Im & Rathouz 2009). ing these states), environmental covariates, and an explicit As we will show here, the benefits of adequately ‘process model’. Once parametrised, the process model can accounting for the movement process may extend beyond be used to generate space-use prediction, but parametrisa- obtaining unbiased habitat-selection estimates. SSAs rely tion is often technically demanding and computationally on a simple depiction of animal movement as a series of intensive (Patterson et al. 2008). More recently, Potts et al. stochastic discrete steps, characterised by specific velocity (2014a) demonstrated the use of a ‘mechanistic step selec- and autocorrelation distributions. This same depiction tion model’ to infer both the drivers and the steady-state underlies the mathematical modelling of animal movement distribution of animal space-use, but the model was framed as a discrete-time random walk (RW), including correlated around one specific functional form of the movement ker- and/or biased RW (Kareiva & Shigesada 1983; Turchin nel, and parameter estimates were obtained using a cus- 1998; Codling, Plank & Benhamou 2008). Indeed, many tom-made likelihood maximisation procedure. Lastly, SSA formulations correspond to a correlated RW process Duchesne, Fortin & Rivest (2015) demonstrated that an with local bias produced by resource selection (BCRW; SSA can be used to obtain unbiased estimates of the direc- Duchesne, Fortin & Rivest 2015). Apart from their com- tional persistence and bias of a BCRW, but did not patibility with the way we often observe animal movement address parametrisation of the step-length distribution. (i.e. in continuous space and at discrete times), many RW The iSSA builds and expands on these contributions. We can be well approximated by diffusion equations, allowing will demonstrate that, by statistically accounting for an explicit © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 Integrated step selection analysis 621 Fig. 1. Step selection analysis workflow. Light grey shading indicates conventional SSA whereas dark grey shading indicates the integrated approach advocated here (iSSA). See Appendix S4 for detailed iSSA guidelines and tips. movement process within an SSA, a complete habitat-depen- Materials and methods dent mechanistic movement model can be parametrised from telemetry data using a standard CLR routine. In the following, INTEGRATED STEP SELECTION ANALYSIS we provide a detailed description of the approach and evaluate In their work on the subject of accounting for movement in resource- its performance (compared with standard RSA and SSA) in selection analysis, Forester, Im & Rathouz (2009) demonstrated that correctly inferring the movement and habitat-selection including a distance function in SSA substantially reduces the bias in processes underlying observed space-use patterns, and in pre- habitat-selection estimates. Mathematically, their argument is based dicting the resulting UD. © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 622 T. Avgar et al. on the habitat-independent movement kernel (the function governing Ψ (e.g. selection for snow-free or flat localities). Eqn 1 is equivalent to movement in the absence of resource selection, or across a homoge- the formulation used (for example) by Rhodes et al. (2005, Eqn 1], neous landscape; Hjermann 2000; Rhodes et al. 2005) belonging to Forester, Im & Rathouz (2009) and Johnson, Hooten & Kuhn (2013, the exponential family, so that it can be accounted for with the logistic Eqn 1) and is a generalised form of a redistribution kernel – awidely formulation of the SSA likelihood function. Here we shall make the used mechanistic model of animal movement and habitat selection (see assumption that, in the absence of habitat selection, step lengths fol- Discussion for recent examples). low either an exponential, half-normal, gamma or log-normal distri- The denominator in Eqn 1 is an integral over the entire spatial bution. Under this assumption, the statistical coefficients associated domain, Ω, serving as a normalisation factor to ensure the resulting with step length, its square, its natural logarithm and/or the square of probability density integrates to one. Whereas in most cases it would its natural logarithm (depending on the assumed distribution), when be impossible to solve this integral analytically, various forms of incorporated as covariates in a standard SSA, serve as statistical esti- numerical (discrete-space) approximations can be used to fit redistri- mators of the parameters of the assumed step-length distribution (see bution kernel functions, such as Eqn 1, to data (see Avgar, Deardon Appendices S2 and S3 for details, and below for an example). Stan- & Fryxell 2013a and the Discussion). Here we focus on a simple like- dard model selection (e.g. likelihood ratio or AIC) then can be used to lihood-based alternative to such numerical methods, one that can be select the best-fit theoretical distribution (out of the four listed above). implemented using common statistical software and is hence accessi- The iSSA approach moreover can be applied to infer directional per- ble to most ecologists. Assuming an exponential form for both Φ sistence and external bias. Assuming that the angular deviations from and Ψ, MLEs for the parameter vectors h and x can be obtained preferred directions (either the previous heading, the target heading or using conditional logistic regression, where observed positions (cases) both) are von Mises distributed (an analogue of the normal distribution are matched with a sample of available positions (controls; Fig. 1 on the circle), the cosine of these angular deviations can be included as and Appendices S2–S4). covariates in an SSA to obtain MLEs of the corresponding von Mises concentration parameters (Duchesne, Fortin & Rivest 2015). Hence, A HYPOTHETICAL EXAMPLE MLEs of iSSA coefficients affiliated with directional deviations and step lengths are directly interpretable as the parameters of distributions Letusassumewehave obtained aset of T spatial positions, sampled at governing the underlying BCRW. a unit temporal interval along an animal’s path, and that we also have We shall make the assumption here that animal space-use beha- maps of two (temporally stationary) spatial covariates, h(x)and y(x). viour is adequately captured by a separable model, involving the pro- We shall now assess the statistical support for the following proposi- duct of two kernels, a movement kernel and a habitat-selection kernel. tions (examples of the sort of hypotheses that could be tested): Formally, we define a discrete-time movement kernel, Φ, which is pro- A The animal is selecting high values of h(x). portional to the probability density of occurrence in any spatial posi- B At the observed temporal scale, and in the absence of variability in h tion, x,at time t, in the absence of habitat selection. The determinants (x), the animal’s movement is directionally persistent (i.e. consecu- of Φ are as follows: the Euclidian distances between x and the preced- tive headings are positively correlated), and the degree of this persis- ing position, x (the step length; l =||x – x ||), the distances t1 t t1 tence varies with y(x) (e.g. the animal moves more directionally between x and x (the previous step length; l =||x – x ||), t1 t2 t1 t1 t2 where y(x) is lower). The resulting turn angles are von Mises dis- the associated step headings, a and a (the directions of movement t t1 tributed with mean 0 (i.e. left and right turns are equally likely) and from x to x and from x to x , respectively), and a vector of t1 t2 t1 a y-dependent concentration parameter. spatial and/or temporal movement predictors at time t and/or at the C At the observed temporal scale, and in the absence of variability in h vicinity of x and/or x , Y(x,x ,t) (e.g. terrain ruggedness, migra- t1 t1 (x)and y(x), the animal’s movement is characterised by gamma dis- tory phase, snow depth, etc.). The effects of these step attributes on Φ tributed step lengths, and the shape of this step-length distribution are controlled by the associated coefficients vector, h. Note that the depends on the time of day (e.g. the animal moves faster during day- effects of spatial attributes here are assumed to operate through local time). biomechanical interactions between the animal and its immediate Note that these propositions are contingent on the temporal gap environment, interactions that determine the rate of displacement (i.e. between observed relocations (i.e. step duration), as well as on the spa- kinesis), not where the animal ‘wants’ to be (i.e. taxis). Also note that tial resolution of our covariate maps, h(x)and y(x). We thus explicitly the kernel Φ can be non-Markovian and accommodate various types acknowledge that our inference is scale dependent. of velocity autocorrelations (lack of independence between directions We start by sampling, for each (but the first two) of the observed and/or lengths of consecutive steps), including correlated and biased points along a path (x , t = 3, 4, .. ., T), a set of s control points (avail- random walks (if directional biases are known apriori). able spatial positions at time t; x , i = 1, 2, .. ., s), where the probability t;i We further define the habitat-selection function, Ψ, which is propor- of obtaining a sample at some distance, l , from the previous observed t;i tional to the probability density of observing the animal in any spatial 0 0 point (l ¼kx  x k)isgivenbythegammaPDF: t1 t;i t;i position, x,attime t, in the absence of movement constraints. The determinants of Ψ are the habitat attributes in x at t, H(x,t), and their t;i 0 0 b 1 gl jb ; b ¼ l e eqn 2 1 2 t;i t;i corresponding selection coefficients, x. The normalised product of Φ b Cðb Þb 1 2 andΨ yields the probability density of occurrence in x at t,whichis: Here, b and b are initial estimates of the gamma shape and scale 1 2 U½ l ;l ;a ;a ;Yðx;x ;tÞ;hW½ HxðÞ ;t ;x t t1 t t1 t1 fxðÞ jx ;x ¼ : t t1 t2 parameters (respectively) obtained based on the observed step-length U½ l ;l ;a ;a ;Yðx;x ;tÞ;hW½ HxðÞ ;t ;x dx t t1 t t1 t1 distribution (using either the method of moments or maximum likeli- eqn 1 hood). As noted earlier, this estimation is confounded by the process of Note that the same environmental variable (e.g. snow depth or ter- habitat selection, and hence, a method to unravel movement inference rain ruggedness) might be included in both Y and H and hence affect from habitat selection is needed. The iSSA will provide estimates of the both Φ[e.g.decreasedspeedindeepsnoworacrossruggedterrain) and deviations of these initial values from the unobserved habitat-indepen- © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 Integrated step selection analysis 623 dent shape and scale (Appendices S2–S3). Note that these control sets (2012) can be included in an iSSA with the MLEs obtained using also could be sampled randomly within some finite spatial domain (e.g. standard statistical packages. An iSSA thus holds promise as a within the maximal observed displacement distance; Appendices S2 user-friendly yet versatile approach in the movement ecologist’s and S4). Distance weighted sampling is expected to increase inferential toolbox. In Appendix S4, we provide practical guidelines for the efficiency, resulting in a smaller standard error for a given s value, but is application of iSSA. In the next sections, we explore the utility of not a necessity (Forester, Im & Rathouz 2009). In general, any increase this approach using computer simulations. in T and/or s will result in better approximation of the used and/or availability distributions (respectively), and hence better inference SIMULATIONS (together with larger computational costs). Once sampled, control (available) points, x , are assigned a value of t;i Testing the inferential and predictive capacities of any statistical 0, whereas the observed (used or case) points, x , are assigned a value of space-use model is challenging because we are often ignorant of the true 1. The resulting binomial response variable can now be statistically process giving rise to the observed patterns, as well as of the true distri- modelled using conditional (case–control) logistic regression, as the bution of space-use from which these patterns are sampled (Avgar, likelihood of the observed data is exactly proportional to (Gail, Lubin Deardon & Fryxell 2013a; Van Moorter et al. 2013). To deal with this & Rubinstein 1981; Forester, Im & Rathouz 2009; Duchesne, Fortin & challenge, we employ here a simple process-based movement Rivest 2015): exp½b hðx Þþ ½b þ b yðx Þ  cosða  a Þþ b l þðb þ b D Þ lnðl Þ 3 t 4 5 t1 t1 t 6 t 7 8 t t ; eqn 3 0 0 0 0 exp½b hðx Þþ ½b þ b yðx Þcosða  a Þþ b l þðb þ b D Þ lnðl Þ 3 4 5 t1 t1 6 7 8 t t;i t;i t;i t;i i¼0 t¼3 0 0 simulation framework. We provide full details of the simulation where a is the direction of movement from x to x ,and D is an t1 t t;i t;i procedure and its statistical analysis in Appendix S5. indicator variable having the value 1 when t is daytime and 0 otherwise. Note that the summation in the denominator starts at s = 0(rather Fine-scale space-use dynamics were simulated using stochastic than 1) to indicate that the used step is included in the availability set ‘stepping-stone’ movement across a hexagonal grid of cells. Each (x ¼ x ). Also note that it is the inclusion of turn angles that neces- cell, x, is characterised by habitat quality, h(x) with spatial autocor- t;i¼0 sitates the exclusion of the first two positions (x and x ); if no relation set by an autocorrelation range parameter, q (=0, 1, 5, 10 t =1 t =2 velocity autocorrelation is modelled, only the first position is excluded. and 50). For each q value, 1000 trajectories were simulated and Lastly, note that this formulation implies that the degree of directional then rarefied (by retaining every 100th position). Each of these rar- persistence is affected by the value of y at the onset of the step only; in efied trajectories was then separately analysed using RSA and 10 the next section, we provide an example of modelling habitat effects on different (i)SSA formulations, including one or more of the follow- movement along the step. ing covariates (Table 1): habitat values at the end of each step, h (x ), the average habitat value along each step, h(x ,x ), the step Equation 3 is a discrete-choice approximation of Eqn 1 (specifi- t t1 t length, l (=||x – x ||), its natural-log transformation, ln(l ), and cally tailored according to propositions A–C), and we provide its t t1 t t the statistical interactions between l,ln(l)and h(x ,x ). Models full derivation in Appendix S3. In summary, b is the habitat-selec- t t t1 t that included only h(x)and/or h(x ,x ) correspond to traditionally tion coefficient (corresponding to proposition A and estimating the t t1 t used SSA (relying on empirical movement distributions with no only component of the parameter vector x in Eqn 1), b and b 4 5 movement attribute included as covariates; models a, b and c in are the basal (habitat-independent) and y-dependent directional Table 1), whereas models that additionally included l and ln(l)cor- persistence coefficients (corresponding to proposition B and esti- t t respond to iSSA. The predictive capacity of the models was esti- mating two components of the parameter vector h in Eqn 1), and mated based on the agreement between their predicted utilisation b , b and b are the modifiers of the step-length shape and scale 6 7 8 distributions (UD) and the ‘true’ UD, generated by the true under- coefficients (corresponding to proposition C and estimating the remaining components of the parameter vector h). Once maxi- lying movement process. We refer the reader to Appendix S5 for mum-likelihood estimates are obtained, the shape and scale param- further details. eters of the basal step-length distribution can be calculated A separate simulation study was conducted to evaluate the identifia- (Appendix S3), where the shape is given by: [(b + b ) + b .D ], bility and estimability of the iSSA parameters as function of sample size 1 7 8 t and the scale is given by: [1/(b –b )]. Similarly, b can be shown and habitat-selection strength (Appendix S6). 2 6 4 to be an unbiased estimator of the concentration parameter of the (habitat-independent) von Mises turn angles distribution (Duch- Results esne, Fortin & Rivest 2015). Including movement attributes as covariates in SSA, which we PARAMETERISATION termed here iSSA, thus allows simple likelihood-based estimation of explicit ecological hypotheses within a framework of a mechanistic All models converged in a timely manner and the convergence habitat-mediated movement model. Such hypotheses might include, time for the most complex model (model j in Table 1) was in addition to those mentioned thus far, long- and short-term target approximately 1 CPU sec. Of the 10 (i)SSA formulations speci- prioritisation (Duchesne, Fortin & Rivest 2015), barrier crossing fied in Table 1, AIC ranking indicated support for only four and avoidance behaviour (Beyer et al. 2015), and interactions with (d, f, h and j), all of which include the habitat value at the step’s conspecifics and intraspecifics (Latombe, Fortin & Parrott 2014; endpoint (with coefficient b ) and the step length and its natu- Potts, Mokross & Lewis 2014b; Potts et al. 2014a). In fact, many of 3 the facets of the generic approach developed by Langrock et al. ral logarithm (with coefficients b and b ) as covariates. Hence, 5 6 © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 624 T. Avgar et al. Table 1. The 11 different models fitted here and their relative performance ranking at five different levels of habitat spatial autocorrelation (with 1000 realisations at each level). To enable AIC comparison, RSA’s were run with only those positions included in the SSA (i.e. excluding the first position) Covariates % Scord as best (based on AIC) R (x x ) R (x x ) t, t1 t, t1 Model R (x ) R (x x ) l ln (l ) l ln (l ) q = 0 q = 1 q = 5 q = 10 q = 50 t t, t1 t t t t RSA b 00 0 0 0 0 0 203 457447 RSA SSA a b 00 0 0 0 0 0 0 0 0 b 0 b 0 0 0 0 00 00 0 c b b 0 0 0 0 00 00 0 3 4 iSSA d b 0 b b 00 0 0 0 1111 3 5 6 e 0 b b b 0 0 00 00 0 4 5 6 f b b b b 0010444139240284 3 4 5 6 g 00 b b b b 00 00 0 5 6 7 8 h b 0 b b b b 662 2831075 23 3 5 6 7 8 i 0 b b b b b 00 00 0 4 5 6 7 8 j b b b b b b 234 673551 233235 3 4 5 6 7 8 Bolded numbers mark the best performing model at each level of spatial autocorrelation. iSSAs better explain our simulated data than traditionally used and j), b wasclosertozero(Fig. 3).Interestingly,whenonly SSAs (excluding step length as a covariate), but only as long as the habitat at the end of the step and the habitat along the step an endpoint effect (i.e. selection for/against the habitat value at were included in the model (i.e. model c;acommonly used the end of the step) is included. In fact, models that excluded SSA formulation), and at low q values (=0, 1), b was negative, the habitat value at the step’s endpoint (models b, e, g and i) indicating ‘selection against’ high-quality steps. In fact, this had AIC scores that were typically two orders of magnitude reflects the low probability of observing a ‘used’ step that tra- larger than those including it. In comparison to RSA, iSSA verses high-quality habitat but does not end there. formulations had unequivocal AIC support at low habitat spa- As explained above (and in Appendices S2 and S3), iSSA tial autocorrelation levels, but only partial support at high coefficients affiliated with the step length (b ) and its natural autocorrelation levels (Table 1). logarithm (b ), when combined with the estimated shape and Estimated habitat-selection strengths, as indicated by our scale values of the observed step-length distribution (b and b ; 1 2 RSA and SSA coefficient estimates (b and b respectively), on which sampling was conditioned; Appendices S3 and S5), RSA 3, were appreciably larger than the true habitat-selection strength could be used to infer the shape and scale of the ‘habitat-inde- (x = 1), and more so for RSA estimates than for SSA (Fig. 2). pendent’ step-length distribution [i.e. assuming h(x ,x ) = 0]. t t+1 Note that this in itself does not mean these estimates are ‘bi- Under most imaginable scenarios, we would expect this basal ased’ but rather reflects the inherent difference between the movement kernel to be wider (i.e. with larger mean) than the intensity of the true process and that of the emerging pattern, observed one, as animals tend to linger in preferred habitats at the scale of observation (see further discussion below). These and hence display more restricted movements compared to the estimates showed little sensitivity to the level of habitat spatial basal expectation. Indeed, the mean of these inferred distribu- ðb þb Þ 1 6 autocorrelation, although a substantial increase in variance is tions (the product of their shape and scale: ) corre- ðb b Þ 2 5 observed in the RSA case (Fig. 2a). As found before by Fores- sponds exactly to the observed mean, as long as no other ter, Im & Rathouz (2009), the strength of SSA-inferred habitat covariates are included in the analysis (model x in Fig. 4). selection is larger when step lengths are included as a covariate Once other covariates are included in the model (and hence in the analysis (iSSA), but this effect is fairly weak and dimin- habitat selection is at least partially accounted for), inferred ishesasthe habitat’sspatial autocorrelation increases mean step-length values were significantly higher from the (Fig. 2b). observed values, showed little sensitivity to model structure, Overall, SSA-inferred habitat selections were substantially but increased with q (as do the observed mean step lengths). less variable than RSA-based estimates and showed little sensi- One exception is model g, which strongly underestimated the tivity to the inclusion or exclusion of other covariates in the mean step length at moderate-high q valuesasitdoesnot model fit (Fig. 2). This is not the case, however, for the effect include any main habitat effects. of the mean habitat value along the step (b ), which varied sub- Even at high q values, inferred mean step length slightly but stantially with both the level of habitat spatial autocorrelation consistently underestimates the true habitat-independent step- length distribution (calculated by simulating the process based and the inclusion of an endpoint effect (b ). Where b was not 3 3 included in the model fit (models b, e and i in Table 1), b on Eqn S51with x = 0; Fig. 4). This bias is a result of an increased with q,whereas where b was included (models c, f iSSA’s limited capacity to account for the full movement © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 Integrated step selection analysis 625 (a) (b) Fig. 2. Statistically inferred habitat-selection coefficient estimates for RSA (a) and SSA (b; letters along the x-axis refer to the SSA formulations listed in Table 1), for five levels of habitat spatial autocorrelation, q. Each box-and-whiskers is based on 1000 independent estimates, where the cen- tral mark is the median, the edges of the box are the 25th and 75th percentiles, the whiskers extend to the most extreme data points not considered outliers (i.e. within approximately 99% coverage if the data are normally distributed), and outliers are plotted individually. Horizontal dashed lines represent the true habitat-selection intensity, x = 1. See Appendix S5 for further details. Fig. 3. Statistically inferred effects of the mean habitat along the step. The dashed line represents no effect. Other details are as in Fig 2. process as it unfolds in between observations. The animal does of the underlying movement process, the animal’s true move- not actually travel along the straight lines that we term ‘steps’ ment capacity is never fully manifested in the observed reloca- and, even if it would, the mean habitat value along the step tion pattern and is thus always underestimated. Note, does not exactly correspond to its probability to travel farther. however, that this bias is negligibly small where the spatial As long as the scale of the observation is coarser than the scale autocorrelation of habitats is high (q > 1; Fig. 4). © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 626 T. Avgar et al. Fig. 4. Mean of gamma step-length distributions (displacement in spatial units per Dt; Appendix S5) inferred based on the different iSSA formula- tions (see Table 1). Model x is a null model, including only the step length and its natural logarithm (with no habitat effects), added here to demon- strate that the conditional logistic regression produces unbiased MLEs. The dotted lines correspond to the observed mean step length across all 1000 realisations at each of the five levels of habitat spatial autocorrelation. The dashed line corresponds to the ‘true’ habitat-free mean step length, calcu- lated by simulating the process using Eqn S5.1 but with x = 0. Other details are as in Fig 2. Finally, despite apparent support for iSSA formulations predictions were robust to these scale mismatches, with G KLD including interaction between the step length and habitat qual- scores of ~098 and ~096, for the q = 50 landscape (with ity along the step (Table 1; models h and j), the estimated val- parameter estimates based on q = 0 data) and the q = 0 ues of the interaction coefficients, b and b , mostly overlapped landscape (with parameter estimates based on q = 50 data), 7 8 zero (Appendix S7). Generally speaking, the mean habitat respectively. Similarly, all step selection models including an value along the step has a weak negative effect on both the endpoint effect performed well, with G scores ranging KLD shape (through b ) and the scale (through its inverse relation- from ~094 (model f)to ~098 (model h)for the q = 0land- ship with b ) of the step-length distribution – long steps are less scape (with parameter estimates based on q = 50 data), and likely through high-quality habitats. G scores ranging from ~083 (model j)to ~097 (model a) KLD for the q = 50 landscape (with parameter estimates based on q = 0 data). Overall, iSSA-based predictive capacity PREDICTIVE CAPACITIES remained mostly unaltered by mismatches between the data’s At approximate steady state, RSA-based UD predictions are landscape structure and the structure of the landscape on slightly more accurate and precise than SSA-based predictions which projections are made. As can be expected, step selection models predict transient (Fig. 5 and Appendix Table S8). The RSA’s predictive capac- ity increases with q (while its precision dramatically decreases), UDs better than the inherently stationary RSA (except when whereas the opposite is true for SSA predictions, where the q = 50; Appendix Table S8). In comparison with steady-state minimum KLD value (Kullback-Leibler Divergence; see predictions, complex iSSA-based predictions perform better Appendix S5) is reached when q = 0 (Appendix Table S8). than simpler ones (Appendix Table S8). As for the steady-state KLD values coarsely mirror the AIC ranking of the different predictions, all iSSA formulations including an endpoint effect SSA formulations in distinguishing those that include an end- performed well in predicting transient UDs, with G scores KLD point effect (b ), but the best performing formulations based ranging from ~89% (model h when q = 0) to ~098 (model d on KLD are simpler than the ones selected based on AIC when q = 1). G scores for RSA-based predictions showed KLD (Tables 1 and S8). That said, all iSSA formulations including substantial sensitivity to the level of spatial autocorrelation, an endpoint effect performed well overall, with G scores (a ranging from ~069 (q = 0) to ~097 (q = 50). KLD measure of goodness of fit; Appendix S5) ranging from ~084 (model f when q = 50) to ~098 (model d when q = 0). For ref- ISSA IDENTIFIABILITY AND ESTIMABILITY erence, the G scores for RSA-based predictions ranged KLD from ~098 (q = 0) to ~099 (q = 50). Results are presented in detail in Appendix S6. In short, our To test the sensitivity of the models’ predictions to the analysis revealed that, under the test scenario, all iSSA sampling scale (see Appendix S9 for relating q to the sam- parameters are fully identifiable, that estimates are unbiased pling scale), we generated predicted UDs, both SSA-based in relation to the true values of the kernel generating func- and RSA-based, across a highly autocorrelated landscape tions, andthatan increasein samplesizebeyond ~400 (q = 50) using parameter estimates obtained from samples of observed positions does not seem to substantially enhance a random landscape (q = 0), and vice versa. RSA-based precision (and hence estimability). That said, our results also © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 Integrated step selection analysis 627 Fig. 5. Log–log plots of the true UDs vs the predicted UDs. Each dot represents the utilisa- tion probability of a single spatial cell. Black dots correspond to the median parameter esti- mates, whereas grey dots correspond to the 25 and 975 percentiles of the estimated parame- ters distribution. Black diagonal lines repre- sent a perfect 1:1 mapping – dots appearing above these lines are spatial cells where the true UD value exceeded the predicted UD value (under-predictions), whereas dots appearing below these lines represent over- predictions. Note that iSSA results are pre- sented for the simplest iSSA including an end- point effect, formulation d in Table 1. indicate that inferential accuracy of movement related param- to the true UD when the time-scale is long (and hence eters may be highly variable, leading to compromised preci- approaches the steady-state limit). sion (with up to 1000% departure from the true value; Two caveats are in place here. First, in our analysis the defi- Appendix S6) even at a fairly largesamplesize. This maybe nition of the availability set for the RSA was exact (i.e. the particularly true given the inherent trade-off between sam- entire domain), a situation that seldom occurs in empirical pling extent and frequency (Fieberg 2007). Estimability of cer- studies where availability is unknown. This is not the case for tain parameters, under certain scenarios, may thus be weak iSSA where the availability set always can be adequately andmustbeevaluatedonacase-by-case basis. defined (but is conditional on the temporal resolution of the positional data). Secondly, the high variability characterising the RSA coefficient estimates, and its resulting predictions Discussion (Figs 2 and 5) indicate substantial risk of erroneous inference. This may be particularly true when sample size is smaller than The ideas, simulations and results presented above are aimed the relatively large sample used here, resulting in data that are at providing a comprehensive assessment of using integrated not adequate unbiased samples of the steady-state UD, which step selection analysis, iSSA, with emphasis on its predictive is likely the case in most empirical studies. The more mechanis- capacity. The iSSA allows simultaneous inference of habitat- tic nature of the iSSA makes it less sensitive to stochastic differ- dependent movement and habitat selection and is hence a ences between specific realisations of the space-use process powerful tool for both evaluating ecological hypotheses and (e.g. due to differences in landscape configuration) and thus predicting ecological patterns. We have shown that iSSA- leads to more precise inference. Hence, even if the sole objec- based habitat-selection inference is relatively insensitive to tive of a given study is to predict the long-term (steady-state) model structure and landscape configuration, and that iSSA- utilisation distribution, the more complicated iSSA-based pre- based UD predictions perform well across different temporal dictions mightbemorereliablethanthose based onRSA. and spatial scales (we discuss the connection between the Moreover, in many real-world ecological scenarios, a steady temporal resolution of the data and the habitat spatial auto- state is never reached, and consequently, the static RSA-based correlation in Appendix S9). On the other hand, our results approach is less appropriate than the dynamical iSSA. We thus indicate that movement and habitat selection may not be com- conclude that iSSA should be the method of choice whenever: pletely separable once observations are collected at a coarser (i) RSA availability cannot be properly defined, (ii) predicting temporal resolution than the underlying behavioural process. across a landscape different from the landscape used for Consequently, stationary RSA-based predictions, whereas parametrisation, (iii) the data used for parametrisation are not much simpler to obtain, provide slight but consistent better fit © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 628 T. Avgar et al. an adequate sample of the true steady-state UD or (iv) predict- et al. 2005; Getz & Saltz 2008; Avgar, Deardon & Fryxell ing transient space-use dynamics. 2013a; Potts et al. 2014a; Beyer et al. 2015). Unlike the Many movement and selection processes could be consid- iSSA, however, fitting these kernel-based models to empiri- ered plausible, and the particular details of the mechanistic cal data relies on complex, and often specifically tailored model used to simulate space-use data might substantially alter likelihood maximisation algorithms (namely discrete-space our conclusions. Our aim here was to use the simplest, and approximations of the integral in Eqn 1). The statistical hence most general, mechanistic process imaginable, leading machineryused iniSSA, basedonobtaining asmall setof us to choose a stepping-stone movement process as our pat- random samples from an inclusive availability domain, is tern-generating process. Interestingly, this simple process, gov- accessible to most ecologists because it relies on software erned by only two parameters (Eqn S5.1), gave rise to complex that is already used (Thurfjell, Ciuti & Boyce 2014). patterns once rarefied. In particular, the emerging step-length Through the addition of appropriate covariates and interac- distributions fit remarkably well with a gamma distribution, tion terms, iSSA can moreover address many of the ques- with shape and scale that reflect the underlying landscape tions that were the focus of other kernel-based approaches, such as home-range behaviour (Rhodes et al. 2005), mem- structure. Note that this is a purely phenomenological descrip- tion of the movement kernel, as the true underlying process ory-use (Avgar, Deardon & Fryxell 2013a; Merkle, Fortin had a fixed, habitat-independent movement parameter & Morales 2014; Avgar et al. 2015; Schl€agel & Lewis 2015), (Appendix S5). Ideally, a truly mechanistic approach will habitat-dependent habitat selection (Potts et al. 2014a) and involve maximising the likelihood over all possible paths the barrier effects (Beyer et al. 2015). Hence, iSSA allows ecolo- animal might have taken between two observed locations, and gists to tackle complicated questions using simple tools. hence allowing inference of the true underlying process (Mat- To conclude, our work complements several recent contri- thiopoulos 2003). In most cases, however, this approach is for- butions advocating the use of movement covariates within step biddingly computationally expensive. We showed that the selection analysis (Forester, Im & Rathouz 2009; Johnson, approximation based on samples of straight-line movements Hooten & Kuhn 2013; Warton & Aarts 2013; Duchesne, For- between observed positions, which is the underlying assump- tin & Rivest 2015). We believe a convincing body of theoretical tion of any SSA, performs well over a range of conditions. An evidence now indicates the suitability of integrated step selec- iSSA thus provides a reasonable compromise between compu- tion analysis as a general, flexible and user-friendly approach tationally intensive mechanistic models and the purely phe- for both evaluating ecological hypotheses and predicting nomenological RSA. future ecological patterns. Our work highlights the importance According to Barnett & Moorcroft (2008), the steady-state of including endpoint effects in the analysis together with some UD should scale linearly with the underlying habitat-selection caveats regarding the interpretation of SSA results, specifically function Ψ (Eqn 1) when informed movement capacity when dealing with the effects of the habitat along the step. We exceeds the scale of spatial variation in Ψ, but should scale with also recommend careful consideration of parameter estimabil- the square ofΨ if informed movement capacity is much shorter ity, particularly with regard to the movement components of than the scale of habitat variation. In the particular case of the the model, which may be prone to strong cross-correlations (as exponential habitat-selection function used here (Eqn S5.1), discussed in Appendix S6). Based on our current experience in we would thus expect the following loglinear relationship: ln applying iSSA to empirical data (T. Avgar, work in progress) [UD(x)] = a + b∙x∙h(x), where a is a scaling parameter [the we have provided practical guidelines in Appendix S4. Addi- utilisation probability where h(x) = 0], and b (1 ≤ b ≤ 2) is tional theoretical work is needed to investigate the effects of some increasing function of the habitat spatial autocorrelation, the underlying movement process on iSSA performance, as q. Our results, emerging from a very different model than the well as to come up with computationally efficient iSSA-based continuous-space continuous-time analytical approximation simulations to enable rapid generation of predicted utilisation of Barnett & Moorcroft (2008), corroborate this expectation. distribution (as discussed in Appendix S1). Most importantly, The slope of the loglinear regression model described above the utility of iSSA must now be evaluated by applying it to real increases from b  14to b  2as q increases from 0 to 50 data sets, and using it to solve real ecological problems. (Appendix S10). RSA-based coefficient estimates, b , clo- RSA sely mirror this pattern, increasing from ~16to ~2as q Acknowledgements increases (Fig. 2a). Hence, as can be expected from a phe- TA gratefully acknowledges supported by the Killam and Banting Postdoctoral nomenological model, RSA-based inference reflects the Fellowships. MAL gratefully acknowledges NSERC Discovery and Accelerator steady-state UD rather than the underlying habitat-selection Grants and a Canada Research Chair. MSB thanks NSERC and the Alberta Conservation Association for funding. The authors thank L. Broitman for process. designing Fig. 1 and C. Prokopenko for helpful editorial comments, and Dr. Recent years have seen a proliferation of sophisticated Geert Aarts, Dr. John Fieberg and an anonymous reviewer, for their instructive modelling approaches aimed at mechanistically capturing comments and suggestions. animal space-use behaviours. Many of these models share the theoretical underpinning of iSSA (as formulated in Data accessibility Eqn 1), relying on a depiction of animal space-use as All data used in this paper were simulated as described in the online Supporting emerging from the product of a resource-selection process Information. and a selection-independent movement kernel (e.g. Rhodes © 2015 The Authors. 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Jonsen, I.D., Myers, R.A. & Flemming, J.M. (2003) Meta-analysis of animal Received 18 August 2015; accepted 4 December 2015 movement using state-space models. Ecology, 84, 3055–3063. Handling Editor: Luca Borg € er Kareiva, P.M. & Shigesada, N. (1983) Analyzing insect movement as a correlated random-walk. Oecologia, 56, 234–238. © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630 630 T. Avgar et al. Supporting Information Appendix S6. Evaluating the iSSA parameter identifiability and estima- Additional Supporting Information may be found in the online version bility. of this article. Appendix S7. b7and b8. Appendix S1. From step selection to utilisation distribution. Appendix S8. Predictive performance. Appendix S2. Inferring step-length distributions. Appendix S9. Interpreting q. Appendix S3. Deriving an iSSA likelihood function. Appendix S10. Habitat selection and utilisation distribution. Appendix S4. iSSA practical user guide. Appendix S11. Appendices reference list. Appendix S5. Simulation experiments. © 2015 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society, Methods in Ecology and Evolution, 7, 619–630

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