 Review
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When to be discrete: the importance of time formulation in understanding animal movement
Movement Ecology volume 2, Article number: 21 (2014)
Abstract
Animal movement is essential to our understanding of population dynamics, animal behavior, and the impacts of global change. Coupled with highresolution biotelemetry data, exciting new inferences about animal movement have been facilitated by various specifications of contemporary models. These approaches differ, but most share common themes. One key distinction is whether the underlying movement process is conceptualized in discrete or continuous time. This is perhaps the greatest source of confusion among practitioners, both in terms of implementation and biological interpretation. In general, animal movement occurs in continuous time but we observe it at fixed discretetime intervals. Thus, continuous time is conceptually and theoretically appealing, but in practice it is perhaps more intuitive to interpret movement in discrete intervals. With an emphasis on statespace models, we explore the differences and similarities between continuous and discrete versions of mechanistic movement models, establish some common terminology, and indicate under which circumstances one form might be preferred over another. Counter to the overly simplistic view that discrete and continuoustime conceptualizations are merely different means to the same end, we present novel mathematical results revealing hitherto unappreciated consequences of model formulation on inferences about animal movement. Notably, the speed and direction of movement are intrinsically linked in current continuoustime random walk formulations, and this can have important implications when interpreting animal behavior. We illustrate these concepts in the context of statespace models with multiple movement behavior states using northern fur seal (Callorhinus ursinus) biotelemetry data.
Introduction
Animal movement is at the heart of many important ecological processes and considered essential for a better understanding of population dynamics, animal behavior, and the impacts of global change. However, movement is a complex process modulated by many factors acting at different spatial and temporal scales. Our ability to study animal movement has been bolstered by recent advances in animalborne biologging technology that have permitted the collection of detailed location and biotelemetry data [13]. The quality and quantity of information from these devices is rapidly increasing, and there has been a recent flood in the development of sophisticated statistical models that use these data for modelbased inferences about animal movement and associated behaviors [48].
This myriad of new methods for analyzing movement data can make the selection of any particular method (or model) a difficult task, particularly for ecologists and wildlife biologists without formal statistical training. This poses a dilemma because ecologists and biologists constitute the vast majority of scientists collecting the very data for which these methods were developed. The complexities of animal movement and location data require sophisticated analytical techniques, but we believe that the inconsistent mathematical and statistical jargon used to describe these methods may be discouraging their widespread application by nonstatisticians. In our experience, the greatest source of confusion among practitioners, both in terms of implementation and biological interpretation, seems to be the distinction between continuous and discretetime formulations of the movement process.
Here we briefly review several of the modelbased (nonphenomenological) approaches for analyzing animal location data that have been proposed in recent years. We then focus on how time is formulated in these movement process models, establish some common terminology (see Table 1), elucidate the differences and similarities among them, and identify some potential advantages and limitations. We also present novel mathematical results (see Does a continuous or discretetime formulation really matter? ) refuting the overly simplistic view that discrete and continuoustime conceptualizations are merely different means to the same end in terms of inferences about animal movement. We then illustrate these concepts in the context of statespace models with multiple movement behavior states using northern fur seal (Callorhinus ursinus) movement data collected in the Pribilof Islands of Alaska, USA.
Review
Characterization of the movement process
Regardless of the underlying statistical framework, most analyses of animal location data that are based on hierarchical movement models consist of two components: a mechanistic model for the movement process and a statistical model for the observation process. Although earlier methods ignored error in the location of observations [5,9,10], most contemporary approaches simultaneously model both the movement process and observation process using a socalled’ “statespace” framework [6,8,11,12].
Recent technological advances (e.g., GPS) are making location measurement error less of a concern, and this has allowed greater focus on the development of more realistic (and biologically meaningful) models for the movement process. These developments primarily differ in the spatiotemporal conceptualization of the movement process, including discretetime and discretespace [1315], discretetime and continuousspace [5,6], continuoustime and discretespace [16,17], and continuoustime and continuousspace [8,9] movement process models (see Table 2). Although time formulation in continuous space is our primary focus henceforth, discretespace movement models are often employed in the absence of detailed location data (e.g., capturemarkrecapture studies e.g., [14,16]), or resource selection studies in heterogeneous environments e.g., [17]. Latent behaviors associated with different types of movement can also be treated as continuous [18] or discrete [5,6,19,20] states among which individuals transition in response to changes in their internal and external environment. Other approaches go a step further by attempting to combine “macroscopic” resource selection models with “microscopic” discrete or continuoustime movement process models [7,2127].
Before proceeding, we note that hierarchical discretetime, continuousspace movement process models are often referred to as “statespace” models in the literature. This is not a misnomer. However, based on conventional time series jargon, any approach that simultaneously accounts for the system process (i.e., the movement process) and the observation process through time qualifies as a statespace model. In this sense, all of the hierarchical modeling approaches above employ statespace methods. In the contemporary statistical literature, statespace models are now more commonly referred to as hierarchical models; “hierarchical” because the data arise from a probability distribution that depends on a latent process, which, in turn, is modeled stochastically [34,35]. We also note that discretetime movement models where each behavioral state is associated with a distinct random walk [5,6,20,30] can be considered as hidden Markov models, a special class of statespace models with a finite number of latent states [36].
In general, animal movement occurs in continuous time but we observe it at fixed discretetime intervals. Thus, continuoustime models are conceptually and theoretically appealing, but in practice it is perhaps more intuitive to interpret movement in discrete intervals (e.g., turning angle and step length per unit time). It is easier to conceptualize the movement process as a series of steps and turns sampled from particular distributions than to deal with partial differential equations. This may in part explain why the methodological development and application of discretetime models has thus far exceeded that of continuoustime models.
Whether in discrete or continuous time, most mechanistic movement process models are based on correlated random walks. In discrete time, correlated movement is typically modeled with nonuniform turning angle distributions, usually with mean of zero, which result in shortterm directional persistence between successive time steps. The more highly correlated movement exhibits turning angles tending towards zero [5,6]. In continuous time, correlated movement can be expressed through a special type of diffusion model that accounts for dependence between locations, the OrnsteinUhlenbeck (OU) process [4,10]. The OU process is essentially a continuoustime random walk with a tendency to drift towards a central location. Using an OU process to model movement velocity instead of locations, Johnson et al. [8] developed a correlated random walk model that is a continuoustime analog to the discretetime model of Jonsen et al. [6].
Both discrete and continuoustime random walk models can incorporate directed (or oriented) movement, but this is often referred to as “biased” movement in discretetime models [20,37] and “drift” or “advection” in continuoustime models [4,10]. Directed movements are typically associated with specific locations in space, such as “centers of attraction” or “centers of repulsion,” and can be used to model a general tendency towards the center of a home range [7,10] or patch [4,20,31]. Thus, directional persistence can result from directed movements, but the longterm directional persistence that can result from directed movement is different from the shortterm directional persistence associated with a correlated random walk [38]. Under directed movement, longerterm directional persistence results from an individual being constantly pulled towards (or pushed away from) a particular location or gradient (without explicit consideration of the direction of previous movements).
Without correlated movements, the discretetime models of Morales et al. [5] and Jonsen et al. [6] reduce to simple random walks. Without directed movements, the discretetime model of McClintock et al. [20] reduces to the correlated random walk model of Morales et al. [5]. The OU process models of Dunn and Gipson [10], Blackwell [4,9], Johnson et al. [8], and Harris and Blackwell [31] reduce to Brownian motion (i.e., a continuoustime simple random walk), using a mathematical limit argument. We note that because the directional persistence in a correlated random walk decays exponentially as the time lapse increases, correlated random walks can be approximated at larger scales with a simple diffusion model [16].
To incorporate both correlated and directed movement, the expected direction of movement must reflect a tradeoff between shortterm directional persistence and the strength of bias towards (or away from) a center of attraction (or repulsion). This has been examined in discrete time by modeling the expected direction as a weighted average of the strength of bias in the direction of the center of attraction and the previous movement direction [20,37]. Although a similar approach has yet to be thoroughly investigated in continuous time, this would be akin to modeling the drift parameter of an OU process as a function of both directed and correlated movements.
The metrics of movement
Movement metrics also differ among the aforementioned approaches by specifying the movement process on the positions themselves [7,9,28] or on derived quantities, such as the differences between consecutive locations (i.e., velocities) [6,8,19,32,33], step lengths [18], step lengths and turning angles [5], or step lengths and bearings [20,29] (see Table 2). These movement metrics are important for model specification and interpretation. For example, by modeling velocity, the discretetime model of Jonsen et al. [6] and the continuoustime model of Johnson et al. [8] induce dependence between the speed and direction of movement, so that long steps are possible when turning angles are small, resulting in higherorder autocorrelations than found in standard correlated random walks [5,20]. Although Blackwell [4,9] models position and Johnson et al. [8] model velocity, the speed and direction of movement are intrinsically linked through the drift process of these continuoustime models (see Does a continuous or discretetime formulation really matter? ). By modeling turning angles independent of step lengths in discretetime, Morales et al. [5] could investigate correlated (but not directed) movements independent of speed. By modeling bearings using a similar discretetime movement process model, McClintock et al. [20] could simultaneously investigate both correlated and directed movements independent of speed.
Does a continuous or discretetime formulation really matter?
Outside of fitting them to data and empirically assessing differences, it is not immediately apparent how alternative time formulations of movement models differ analytically. In fact, continuous and discretetime formulations are often over simplistically viewed as merely different means to the same end. But this is not the case, and we derive a partial translation here to compare continuous and discretetime formulations with a common and intuitive language: step length and bearing.
Kobayashi et al. [39] provides the following necessary result for two independent normallydistributed random variables, A and B. If
then the distance from the origin, \( L=\sqrt{A^2+{B}^2}, \) has a Rice distribution, R(μ, σ ^{2}), where \( \mu =\sqrt{\mu_A^2+{\mu}_B^2}=\left\Vert \boldsymbol{\mu} \right\Vert \) is the distance from the origin to the center of the bivariate normal distribution, σ ^{2} is a variance parameter, and \( \mathcal{N}\left(\right) \) is the Normal probability density function. The Rice distribution is a generalization of the Rayleigh distribution (for μ ≠ 0) whose expected value increases with increasing values of μ. Further, the bearing θ = tan^{− 1}(B/A), has the conditional von Mises distribution
where κ = lμ/σ ^{2}, ω = tan^{− 1}(μ _{ B }/μ _{ A }), and I _{0}() is the modified Bessel function of the first kind and of order 0. The von Mises distribution is symmetric and centered on the angle ω, and dispersion decreases with increasing κ values.
We can now translate a time step of the continuoustime correlated random walk (CTCRW) model of Johnson et al. [8] to a discretetime step length and bearing. First, the transformation of the bivariate velocity process to speed (distance unit per time unit) and direction is given by
and
The resulting distributions are obtained by applying the results in Kobayashi et al. [39] to the CTCRW velocity model equations (see Eqs. 3 and 4 in Continuoustime formulation below). Using the velocity process transformation, the location step length \( {S}_t=\sqrt{{\left({X}_{t+1}{X}_t\right)}^2+{\left({Y}_{t+1}{Y}_t\right)}^2} \) and bearing ϕ _{ t } = tan^{− 1}{(Y _{ t + 1} − Y _{ t })/(X _{ t + 1} − X _{ t })} are distributed as
where Z _{ t } is the latent behavioral state, and q _{ z,t } is the (1,1) (or, (2,2) as they are the same) entry of the covariance matrix for the velocity process (Q _{ z,t }).
There are now notable differences that one can easily distinguish between the continuous and discrete formulations for step length and bearing distributions. First, unlike the discretetime model (see Eqs. 1 and 2 in Discretetime formulation below), the step length and bearing of the continuoustime model are clearly correlated. As step length increases the distribution of the bearing becomes more concentrated around θ _{ t }, the latent velocity bearing. Second, given a constant state process, step lengths are independent in the discretetime formulation. However, in the CTCRW model step lengths are still correlated via the autocorrelated speed process, l _{ t }. Thus, unlike the discretetime model, the CTCRW maintains not just directional persistence, but persistence in speed as well. Note that this result does not depend on latent behavioral state (Z _{ t }) and holds for movement models with a single behavioral state.
We emphasize that these results are not simply attributable to the fact that the CTCRW model is based on an integrated OU velocity. They hold analogously for continuoustime models which use OU process models for position directly [4,9,31], even if X _{ t } and Y _{ t } are modeled independently (i.e., by setting the offdiagonal elements of the covariance matrix for the bivariate OU process to zero). Using the same result from Kobayashi et al. [39], the distributions of the step length and bearing of an OU process directly modeling position, with central location μ = (μ _{ x }, μ _{ y }), are
where D _{ t }(μ) and θ _{ t } are respectively the distance and bearing from the current position to the central location, and \( {\sigma}_t^2 \) is the variance of the OU process at time t. One can see that the OU model directly applied to the positions still maintains correlation between step length and bearing. Moreover, it also possesses the (potentially undesirable) quality that movement rate depends on distance from the point of attraction, thus necessitating rapid movement that slows as the animal approaches the central location.
Potential advantages and disadvantages
Given the various ways by which similar movement properties can be expressed using either discrete or continuoustime process models, some potential advantages and disadvantages are evident. Although animal movement clearly occurs in continuous time, discretetime models are often viewed as more intuitive, and perhaps the biological interpretation of instantaneous movement parameters in continuous time (e.g., those related to OU processes and other diffusion models) can in practice be discouraging to applied ecologists wishing to use or extend continuoustime methods.
Notably, discretetime models that simultaneously incorporate multiple latent movement behavior states, Markov stateswitching, correlated movements, and directed movements have already been developed and fitted to data [5,6,20]. For example, Morales et al. [5] used a discretetime random walk mixture model to examine time allocations and transition probabilities between two latent movement behaviors in elk: a longstep, directionallypersistent “exploratory” state and a shortstep, negativelycorrelated (i.e., with animals tending to move in the opposite direction of the previous move) “encamped” state. Similarly, Jonsen et al. [6] investigated analogous “transit” and “foraging” movement behavior states in seals. Also using seal data, McClintock et al. [20] developed a biased, correlated random walk mixture model with five latent movement behavior states allowing for directed and exploratory movement among foraging and haulout locations.
Similar applications of multistate mixture models have yet to appear in continuoustime (but see Example: northern fur seal ). Blackwell [9] assumed movement behavior states were known, and Johnson et al. [8] assumed states were defined by known covariates, hence neither of these approaches included an estimation framework for both latent movement states and switching behavior. Hanks et al. [19] extended the framework of Johnson et al. [8] and Hooten et al. [17] to accommodate inhomogeneous movement characteristics along the movement path using a changepoint model. However, because this approach does not explicitly incorporate distinct movement behavior states or stateswitching mechanisms with direct biological interpretation, post hoc cluster analyses were used to identify potential movement behavior states. Harris and Blackwell [31] recently described a continuoustime multistate mixture modeling framework, but fitting these models is challenging, and they have yet to be demonstrated using real data. Part of the difficulty of multistate mixture models in continuous time is due to the underlying relationships these models typically impose on the movement characteristics (e.g., speed or directional persistence) commonly used to distinguish movement behavior states (see Does a continuous or discretetime formulation really matter? and Example: northern fur seal ). Because multistate models are of great practical importance for investigating time allocations to different behaviors (i.e., “activity budgets”), this currently remains an advantage of discretetime models.
Two important disadvantages of discretetime models are related to the necessary discretization of the movement path into a finite number of temporallyregular time steps [40]. The time step length must be specified a priori, but inferences about animal movement from a discretetime analysis are not time scaleinvariant. For example, inferences about bumblebee movement characteristics from discretetime analyses using 30second versus 30minute time steps would likely be dramatically different. The 30second analysis would reveal finegrain movement properties but could potentially mask coarsergrain properties. The 30minute analysis could reveal coarsegrain properties, but would completely miss finegrain properties. The specification of time step length in a discretetime analysis is therefore critical and requires very careful consideration [4143], and it is particularly important that the time step is chosen to match the scale at which behavioral decisions are made [40]. A major advantage of continuoustime models is that they avoid dependence on a particular timescale. Within reasonable limits, a continuoustime analysis will yield the same results regardless of the temporal resolution of observations; if so desired, movement properties from a continuoustime analysis may be summarized a posteriori for time steps of any length. However, we note that for any continuous or discretetime approach to be useful, the temporal resolution of the observed data must be relevant to the specific movement behaviors of interest.
Discretetime movement models can also be more computationally demanding than continuoustime models. Unless observations exactly match the regular time steps required of a discretetime model, the movement path must be predicted at temporallyregular intervals. Perfectly observed, temporallyregular observations are very rare in animal telemetry data (especially for marine species). For longer time series, this can result in thousands of additional location parameters that must be estimated. As movement process models incorporate more details and realism, model fitting becomes more complex. This is particularly true for multistate mixture models. Therefore, once multistate model development and fitting in continuous time has caught up with that in discrete time, the computational advantages of continuoustime formulations are likely to be significant.
Example: northern fur seal
To illustrate the concepts elaborated above in the context of statespace models with latent movement behavior states, we apply comparable multistate movement models in discrete and continuous time to a northern fur seal track in the Pribilof Islands of Alaska, USA. The animal was a nursing female equipped with a Mk10AF satellite tag from Wildlife Computers (see [44] for full study deployment details). The Mk10AF tag has both Fastloc GPS and timedepth recording capabilities. Using both location and diving activity data, we wish to identify and characterize three latent movement behavior states: “resting,” “foraging,” and “transit”. We define foraging (state F) as movement that is characteristic of area restricted searches and includes foraging dives, where a foraging dive must have a max depth >5 m and at least 5 changes in vertical direction (i.e., sinuosities or “wiggles”). The sinuosities are a characteristic of the animal chasing prey during the dive. We define transit (state T) as predominantly travelling with little to no foraging dives, noting that seals may opportunistically feed while travelling. Resting (state R) is defined by types of movement that do not fall under foraging or transit states, including resting at haulouts and resting at sea. In terms of trajectory, we would expect speeds to be low during resting and low to moderate during foraging, with little directional persistence. During transit, we would expect higher speeds and greater directional persistence.
The diving activity data were summarized as the number of foraging dives for each of N = 242 1hour time steps between 7–17 October 2007. Although diving data were logged continuously, location data were obtained opportunistically at 15minute intervals. There are therefore frequent missing location data due to an inability to obtain locations while the seal was underwater. Because the tag possessed GPS capabilities, rather than ARGOS technology, we expect location measurement error to be minimal. The raw location data consist of 241 observations during a single foraging trip (Figure 1), with 40% of the 1hour time steps containing no observed locations.
Discretetime formulation
With the location data being temporally irregular, a discretetime analysis requires that the movement path be estimated at regular time steps. We chose 1hour time steps to exactly match the temporal resolution of the foraging dive data. Using the same statespace formulation as McClintock et al. [20], for time step t = 1, …, N, and observation i = 1, …, k _{ t }, we relate the irregularly observed locations (x _{ t,i }, y _{ t,i }) to the temporally regular model locations (X _{ t }, Y _{ t }) using
and
where j _{ t,i } ∈ [0, 1) is the proportion of the time interval between locations (X _{ t − 1}, Y _{ t − 1}) and (X _{ t }, Y _{ t }) at which the i ^{th} observation between times t 1and t was obtained, \( \left[{\epsilon}_{x_{t,i}}\right]=\mathcal{N}\left(0,{\sigma}_x^2\right), \) \( \left[{\epsilon}_{y_{t,i}}\right]=\mathcal{N}\left(0,{\sigma}_y^2\right),\kern0.5em \left[\dots \right] \) indicates the probability density function for the random variable in brackets, and \( \mathcal{N}\left(\right) \) is the Normal (Gaussian) density. Time steps with no observations (i.e., k _{ t } = 0) do not contribute to the observation model.
We then model movement between the temporally regular locations using a multistate correlated random walk model [5,20]. Specifically, we assume that, conditional on the behavioral state, Z _{ t }, the step length at time t, S _{ t }, is distributed as
where S _{ t } ≥ 0, z ∈ {R, F, T} is the unknown latent behavioral state, and a _{ z } and b _{ z } are statedependent scale and shape parameters, respectively. The Weibull distribution is popular for modeling step length because of its flexibility; it has fat tails when b _{ z } < 1, reduces to an exponential distribution when b _{ z } = 1, has exponential tails when b _{ z } > 1, and can resemble a normal distribution when b _{ z } ≈ 3.4. The bearing of movement, ϕ _{ t }, is modeled with the wrapped Cauchy distribution
where 0 ≤ ϕ _{ t } < 2π, ϕ _{ t − 1} is the previous bearing, and − 1 < ρ _{ z } < 1 is the statedependent dispersion parameter. Unfamiliar to most nonstatisticians, the wrapped Cauchy distribution converges to a uniform distribution over the circle as ρ _{ z } goes to zero. As ρ _{ z } goes to 1 (or − 1), the distribution tends to a point mass concentrated towards (or away from) the previous bearing. Standard correlated movement is typically modeled with the wrapped Cauchy distribution by constraining 0 ≤ ρ _{ z } < 1 [5,45].
It can be difficult to distinguish resting, foraging, and transit states for seals based on trajectory alone [45], particularly because northern fur seals can forage opportunistically while travelling and will often rest at sea or in the vicinity of breeding rookeries. We therefore incorporate the number of foraging dives during each time step, δ _{ t }, to help inform the foraging state. Specifically, we assume
with the constraints λ _{ R } = 0 and λ _{ F } > λ _{ T }. This model therefore assumes a priori that time steps with foraging dives are never assigned to resting, and steps with relatively many foraging dives are more likely to be assigned to foraging than transit. Note that by constraining λ _{ F } > λ _{ T }, we still allow some possibility for steps with foraging dives to be assigned to transit.
Finally, we model switches between behavior states as a firstorder Markov process. We assign the conditional distribution to the latent state variable Z _{ t }
where for \( z,\ {z}^{\prime}\in \left\{R,F,T\right\},{\psi}_{z,{z}^{\prime }} \) is the probability of switching from state z at time t – 1 to state z′ at time t.
Using Bayesian analysis methods, the joint posterior distribution for our statespace model in discrete time is
where (X _{0}, Y _{0}) is the initial (latent) location. Note that, conditional on Z _{ t }, this discretetime model assumes step length, bearing, and the number of foraging dives are independent. Weakly informative priors were used for all parameters, including the conjugate priors \( \left[{\sigma}_x^2\right]={\Gamma}^{1}\left(0.01,0.01\right), \) \( \left[{\sigma}_y^2\right]={\Gamma}^{1}\left(0.01,0.01\right), \) [λ _{ z }] = Γ(0.01, 0.01) for z ∈ {F, T}, and [ψ _{ z }] = Dirichlet(1, 1, 1) for z ∈ {R, F, T}, where Γ() and Γ^{− 1}() are the gamma and inverse gamma probability density functions, respectively. For [X _{0}, Y _{0}], we specified a joint uniform prior over the region defined by the Bering Sea. We specified a maximum sustainable speed of 3 m/s, such that S _{ t } ≤ 10800m, with [a _{ z }] = Unif(0, 10800), [b _{ z }] = Unif(0, 30), and [ρ _{ z }] = Unif(0, 1) for z ∈ {R, F, T}. Similar to McClintock et al. [20,45], we used a MetropoliswithinGibbs Markov chain Monte Carlo algorithm written in the C programming language [46] to obtain samples from the posterior distribution, performing pre and postprocessing in R via the .C interface [47]. The only notable difference from the MCMC algorithm for the individuallevel model of McClintock et al. [45] results from our model for δ _{ t }, for which the conjugate prior on λ _{ z } yields the full conditional distributions \( \left[{\lambda}_F\left\cdot \right.\right]={\Gamma}_{\left({\lambda}_T,\infty \right)}\left(0.01+{\displaystyle {\sum}_{t=1}^N{\delta}_t{I}_{\left\{{Z}_t=F\right\}},}0.01+{\displaystyle {\sum}_{t=1}^N{I}_{\left\{{Z}_t=F\right\}}}\right) \) and \( \left[{\lambda}_T\left\cdot \right.\right]={\Gamma}_{\left(0,{\lambda}_F\right)}\left(0.01+{\displaystyle {\sum}_{t=1}^N{\delta}_t{I}_{\left\{{Z}_t=T\right\}},}0.01+{\displaystyle {\sum}_{t=1}^N{I}_{{}_{\left\{{Z}_t=T\right\}}}}\right), \) where Γ_{(l,u)} is the renormalized gamma density truncated at l and u, 0 ≤ l < u, and I() is the indicator function. When full conditional distributions were analytically intractable, random walk MetropolisHastings parameter updates were used. After initial pilot tuning and burnin, a single chain of 5 million iterations was attained for posterior summaries. The algorithm required approximately 3 hours to run on a machine running 64bit Windows 7 (3.4GHz Intel Core i7 processor, 16Gb RAM).
Estimated activity budgets to the three movement behavior states were 0.28 (95% HPDI: 0.220.37) to resting, 0.36 (0.260.39) to foraging, and 0.36 (0.290.45) to transit (Figure 2a). Estimated state transition probabilities were \( {\widehat{\psi}}_{R,R} \) = 0.81 (0.710.92), \( {\widehat{\psi}}_{F,F} \). = 0.78 (0.670.88), and \( {\widehat{\psi}}_{T,T} \) = 0.78 (0.650.89), with stateswitches more likely to occur between foraging \( {\widehat{\psi}}_{F,T} \) = 0.15 (0.050.27) and transit \( {\widehat{\psi}}_{T,F} \) = 0.14 (0.060.22). The bivariate posterior densities for step length and turning angle (Figure 3a) indicate some opportunistic foraging during travelling, with foraging movements often exhibiting high speed and directional persistence typically associated with transit. As expected, time steps with >1 foraging dives were rarely assigned to the transit state (Figure 4a). Also as expected, we found lower speeds and less directional persistence during resting movements and higher speed and more directional persistence during transitory movements.
The estimated error (in meters) for the observation process model was similar between longitude (σ _{ x } = 472; 360 − 596) and latitude (σ _{ y } = 489; 381 − 617) coordinates. Although relatively small, these errors are larger than would typically be expected of GPS location measurement error. We therefore suspect the additional error is attributable to deviations from the simple linear model used to relate the temporally irregular observed locations to temporallyregular predicted locations.
Continuoustime formulation
We analyzed the same fur seal data set using a continuoustime model to assess what inferential differences might result by extending the correlated random walk (CRW) models of Jonsen et al. [6] (discretetime, latent states) and Johnson et al. [8] (continuoustime, state model with known covariates) to a continuoustime CRW model with latent states. The continuoustime correlated random walk (CTCRW) is described by modeling the velocity (instantaneous rate of change) of movement with a bivariate OrnsteinUhlenbeck (OU) process. The OU process is the continuoustime version of the bivariate autoregressive model Jonsen et al. [6] use to model position difference. The CTCRW locations are then modeled by integrating the velocity process (i.e., the positions are the solution to the stochastic differential equation used to model velocity).
To make the inference comparable between each analysis, we maintained the same hourly structure for the transitions of behavior states. Thus, the models [Z _{ t }ψ Z _{ t − 1} = z] and [δ _{ t }λ Z _{ t } = z] are the same as in the previous discretetime analysis with the minor technical change that the state Z _{ t } is assumed to be held constant within the interval [t, t + 1). Also, we use the notation t _{ i } to represent the time of the ith observed location in the interval [t, t + 1).
The CTCRW model is defined by a stochastic differential equation model of velocity \( {\boldsymbol{\upnu}}_{t_i}=\left({V}_{x,{t}_i},\ {V}_{y,{t}_i}\right) \) at time t _{ i }, such that
for each coordinate axis c ∈ {x, y}, where t ≤ t _{ i } < t _{ i + 1} ≤ t + 1, Δ_{i} = t _{ i + 1} − t _{ i }, γ _{ c } is the mean velocity (or drift), β _{ t } is an autocorrelation parameter, \( \zeta \left({\Delta}_{\mathrm{i}}\right)=\mathcal{N}\left(0,{\sigma}_t^2\left[1 \exp \left(2{\beta}_t{\Delta}_{\mathrm{i}}\right)\right]\ /2{\beta}_t\right), \) and σ _{ t } is a parameter controlling the overall variability in velocity. The solution to this autoregressive differential equation is the location \( {\boldsymbol{\upmu}}_{t_i}=\left({X}_{t_i},\ {Y}_{t_i}\right) \). Johnson et al. [8] provide details to illustrate that the CTCRW model can be formulated as a linear, Gaussian statespace model that allows efficient calculation of the CTCRW likelihood. For t ≤ t _{ i } < t _{ i + 1} ≤ t + 1, observation \( {\mathbf{y}}_{t_i}=\left({x}_{t_i},\ {y}_{t_i}\right), \) and the vector of the true location and velocity process \( {\boldsymbol{\upalpha}}_{t_i}=\left({\boldsymbol{\upmu}}_{t_i},\ {\boldsymbol{\upnu}}_{t_i}\right), \) the statespace model is given by
where \( \left[{\boldsymbol{\upvarepsilon}}_{t_i}\right]=\mathcal{N}\left(\mathbf{0},{\tau}^2\mathbf{I}\right) \) and \( \left[{\boldsymbol{\eta}}_{z,{t}_i}\right]=\mathcal{N}\left(\mathbf{0},{\mathbf{Q}}_{z,{t}_i}\right) \). The entries of \( {\mathbf{T}}_{z,{t}_i} \) and \( {\mathbf{Q}}_{z,{t}_i} \) are functions of Δ_{i} and the movement parameters β _{ t } and σ _{ t } (see [8] for details), and as in the discretetime analysis, the movement parameters depend on the latent state Z _{ t } = z via β _{ t } = β _{ z } and σ _{ t } = σ _{ z }.
We used an MCMC sampler for Bayesian inference of movement parameters and states. Similar to Johnson et al. [8], we assumed no drift (i.e., γ _{ c } = 0) and similar movement processes in both coordinates (i.e., β _{ c,t } = β _{ t } and σ _{ c,t } = σ _{ t } for c ∈ {x, y}). The same priors were used for all common variables between the two analyses (e.g., diving rates, behavior states). For the CTCRW movement parameters, we used vague priors on the log scale with the following constraints: β _{ R } > β _{ F } > β _{ T } and σ _{ R } < σ _{ F } < σ _{ T }. These constraints imply that movement is typically faster and more correlated as one moves from R to T. The flat prior [log τ] > 10 m was used for the measurement error parameter. The sampler was custom coded in R [47] making use of the FORTRAN coded CTCRW likelihood and posterior track simulation in the R package crawl [48]. The CTCRW likelihood computed via the Kalman filter allowed us to sample from the marginal posterior distribution of the states and movement parameters without having to sample the unobserved α _{ t } values. The sampled posterior distribution is given by
where the right handside of the product is the CTCRW likelihood. Note that the true locations \( \left({X}_{t_i},\ {Y}_{t_i}\right) \) and velocities \( \left({V}_{x,{t}_i},\ {V}_{y,{t}_i}\right) \). have been integrated from the posterior. The benefit of this is that the MCMC sampler for the states and parameters converges more quickly to the approximate posterior distribution. The full algorithm took 66 hours to run (due to coding in R rather than C), however, only 20,000 iterations were necessary to obtain an effective sample of ≥ 4,000 posterior draws. To compare step lengths and turning angles of the CTCRW model to the discrete time model, we needed a sample of hourly locations. To obtain a posterior sample of α _{ t }, t = 1, …, N, on the hour, the sampling method of Johnson et al. [49] was used at each MCMC iteration as if α _{ t } was a derived parameter. From the sampled α _{ t } values, step length and turning angle were calculated for comparison to the equivalent discretetime quantities.
Estimated activity budgets to the three movement behavior states were 0.10 (95% HPDI: 0.030.15) to resting, 0.29 (0.230.34) to foraging, and 0.61(0.530.67) to transit (Figure 2b). Estimated state transition probabilities were \( {\widehat{\psi}}_{R,R} \) = 0.52 (0.100.86), \( {\widehat{\psi}}_{F,F} \) = 0.75 (0.620.86), and \( {\widehat{\psi}}_{T,T} \) = 0.82 (0.750.89). Stateswitches to transit were most likely, with \( {\widehat{\psi}}_{R,T} \) = 0.40 (0.090.81) and \( {\widehat{\psi}}_{F,T} \) = 0.23 (0.120.35). These are noticeably different from the discretetime analysis, with much less time spent “resting.” The bivariate posterior densities for step length and turning angle (Figure 3b) also reflect this reduction in state R, with more small steps associated with the travel state. However, there were also more large steps associated with the resting state. This calls into question the designation of these states as actually “resting” when using the continuoustime multistate movement model. As in the discretetime analysis, time steps with >1 foraging dives were rarely assigned to the transit state (Figure 4b). The estimated error (in meters) for the observation process model was \( \widehat{\tau}=64 \) m (55 m75 m). Because the observed data linear interpolation does not need to be accounted for, the measurement error variance is noticeably smaller here than in the discretetime analysis.
Although inferences about time spent foraging were similar between the two approaches, we found considerable differences between the discretetime and continuoustime formulations with respect to resting and travelling activity. This is counter to the simplistic view that time formulations are merely different means to the same end. The reasons for these differences lie in the underlying relationships of the metrics of movement (speed and directional persistence) that are used to define resting and travelling. Because these metrics are dependent and speed is autocorrelated in the continuoustime model (see Does a continuous or discretetime formulation really matter? ), the lack of auxiliary information (such as metabolic rate) to help distinguish these movement behavior states induces a tendency for the “resting” state to be associated with sudden switches (or changepoints) in movement properties during periods with no foraging dives. In other words, instead of identifying periods of slow movement with no foraging dives as intended, the “resting” state serves to break the momentum of the continuoustime movement process.
Although continuoustime formulations necessarily induce dependence between step length and bearing, the differences between our discrete and continuoustime analyses are not entirely attributable to time formulation per se. In order to account for shortterm directional persistence in continuous time, Johnson et al. [8] used correlation in the velocity process (Jonsen et al. [6] use the same correlation model in discrete time). Whether in continuous or discrete time, the modelling of velocity clearly induces additional dependence between speed and bearing. Correlated random walk models with two latent movement behavior states can be relatively easy to fit in continuous time (D. Johnson, unpublished data) or when modeling velocity in discrete time [6,50]. However, the modelling of velocity can make it more difficult to characterize and identify >2 distinct movement behavior states with straightforward biological interpretation. While this can be easily avoided in discrete time by modelling step length and bearing independently (as was done here), most continuoustime CRW models are formulated on the velocity process [8,31] (but see [22]).
Conclusions
Modern tracking and biologging devices allow us to record detailed information on animal location and physiology, thus opening the possibility to better understand the role of movement in population dynamics, animal behavior, and the environment [51,52]. To make the most of these hardearned data and learn about important aspects of animal movement such as activity budgets, space use, and behavioral responses to landscape features, sophisticated data analysis tools have been proposed. Statespace models, where one explicitly accounts for the fact that the observed data arise from a mechanistic or “biological” model that is in turn sampled by an observation model, are currently regarded as the most correct and elegant methods to fit movement models to data [12,52]. We have shown that there exist underappreciated differences among the current available formulations, and although our northern fur seal example focused on statespace models with multiple movement behavior states, our findings have important implications for singlestate mechanistic movement process models, including (discretetime) stepselection or (continuoustime) partial differential equation resource selection models (e.g., see recent reviews by [26,27]).
Although movement is a continuoustime process, it is perhaps more intuitive to think about (and formulate models for) movement in discrete time. In our experience, practitioners find a discretetime model (Eqs. 1 and 2) and its parameters easier to interpret than its continuoustime counterpart (Eqs. 3 and 4). As we have demonstrated, current discretetime formulations also provide both flexibility and feasibility for identifying latent behavioral states and incorporating auxiliary biotelemetry or environmental data to inform these states. However, these advantages of discretetime models do indeed come at a cost. Because inferences from discretetime models are not time scaleinvariant, it is absolutely critical that the chosen time scale between movement steps appropriately matches the animal’s behavioral scales and the frequency of observations.
In addition to loss of resolution, when observations are irregular and/or the frequency of observations greatly exceeds that of the chosen time scale, discretetime models can suffer from additional lack of fit due to the need to discretize the movement path into temporallyregular locations. This was apparent in the magnitudes of the measurement error terms in our northern fur seal example, where the discretetime model had larger errors than would normally be expected for GPS data. The need for temporallyregular positions for the entire movement path can also make it more difficult to deal with missing data in a discretetime framework. While this is less of a problem for terrestrial animals, missing data is a major issue for marine animals due to our inability to obtain locations while underwater.
Continuous time is clearly a more natural representation of movement than discrete time. These models are not dependent on any particular time scale and do not require temporallyregular observations. It is therefore far easier to deal with missing data or changing observational frequencies in continuous time. However, as demonstrated by our northern fur seal example and Does a continuous or discretetime formulation really matter? , current continuoustime formulations may not be well suited for identifying >2 latent movement behavior states. This is unfortunate because the identification of different behaviors, activity budgets, and how these potentially relate to habitat use and demographic parameters is among the most interesting aspects of movement ecology [51].
Although discretetime approaches thus far have seen greater development and application, we believe further development of continuoustime models is needed to facilitate more widespread application of these models to real data. For example, the continuous formulations of Blackwell [9], Johnson et al. [8], and Harris and Blackwell [31] could potentially be extended to accommodate “stops” where animals can reorient and change movement state, thereby curbing the momentum inherent to these continuoustime movement process models. By overcoming the hurdles identified here and making latent stateswitching models more feasible in continuous time, the best of both worlds may soon be within grasp.
Abbreviations
 CRW:

Correlated random walk
 CTCRW:

Continuoustime correlated random walk
 MCMC:

Markov chain Monte Carlo
 OU:

OrnsteinUhlenbeck
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Acknowledgements
The findings and conclusions in the paper are those of the author(s) and do not necessarily represent the views of the National Marine Fisheries Service, NOAA. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. JMM was funded by CONICET and PICT 20110790.
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All authors conceived, drafted, read, and approved the manuscript. BTM conducted the discretetime analysis of the northern fur seal data. DSJ conducted the continuoustime analysis of the northern fur seal data. All authors read and approved the final manuscript.
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McClintock, B.T., Johnson, D.S., Hooten, M.B. et al. When to be discrete: the importance of time formulation in understanding animal movement. Mov Ecol 2, 21 (2014). https://doi.org/10.1186/s4046201400216
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Keywords
 Animal location data
 Diffusion
 Movement model
 Random walk
 Statespace model
 Switching behavior
 Telemetry