Continuous Statistical Model for Telemetry Data

Assume that we have telemetry location data for an individual animal at times . Let be the set of observed locations associated with the animal at all observation time. Often, these measurements do not come at regular temporal intervals, as missing data are common. Johnson et al. [15] provide a means for making inference about the continuous-time movement path of the animal, given such telemetry data. In practice, this approach allows for inference about the location and directional velocity of the animal at an arbitrarily fine temporal resolution at regular temporal intervals. In particular, Johnson et al. [18] provide a means for finding the posterior distribution , where is the quasi-continuous path of the animal at an arbitrarily fine temporal resolution , and the bracket notation ‘’ denotes a probability distribution.

There are advantages to conditioning on , as opposed to directly, to make inference about movement and resource selection. The locations in are typically temporally irregular, since telemetry data are typically collected at irregular intervals. Lagrangian (individual-based) models of movement often use the movement step lengths between locations instead of the actual telemetry locations [23], but the movement steps between successive observations that are close temporally are likely to be quite different than the movement steps between successive observations that are distant temporally. By conditioning on the continuous path , we can obtain values of at temporally regular intervals, providing equal weights to all movement steps. At the same time, the distribution of accounts for the uncertainty in the location and directional velocity at any given time. Thus, times in that are close to times for which we have telemetry observations in will have less uncertainty in the distribution of location and velocity than will times that are temporally distant from any telemetry observations (Figure 1). The ability to sample realizations from at arbitrary temporal resolution also allows us to use a temporal resolution that matches the resolution of available environmental covariates.

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Figure 1. Overview of velocity-based movement modeling framework.

Animal telemetry data (a) are collected at temporally irregular intervals. A continuous time correlated random walk model (b) is used to create a posterior predictive distribution of paths at temporally regular intervals with high temporal resolution. Path realizations from the posterior predictive distribution are transformed to velocity vectors at temporally regular intervals (c). These velocity vectors allow for efficient modeling of environmental and biotic drivers of animal movement.

https://doi.org/10.1371/journal.pone.0022795.g001

Our strategy is to utilize realizations from the continuous posterior path distribution to make inference about environmental and biotic drivers of animal movement and resource selection. In effect, this amounts to a model-based augmentation of the telemetry data . This approach is similar to that of multiple imputation [24], [25], in which missing values are imputed by multiple draws from the posterior predictive distribution of the missing data, and inference is made by averaging over the results obtained by each of these draws. The approach that we take here, which was first described in the movement context by Hooten et al. [21], is to make inference about a set of parameters related to drivers of movement by first proposing a statistical model (i.e., likelihood) for the relationship between the movement path and the environmental drivers of movement:(1)Specifying prior distributions for allows us to find the posterior distribution:(2)and inference about , conditioned only on , can be accomplished by integrating over the distribution of :(3)Within a Bayesian hierarchical modeling (BHM) framework, we can easily accomplish this integration through composition sampling.

Hooten et al. [21] employed a dynamic occupancy model within an agent-based framework for . This framework is flexible and intuitive, but computationally demanding. At this time, the computational costs of this agent-based approach preclude more complicated modeling considerations, such as multiple animals or multiple movement profiles in one path. We thus propose an alternative approach for linking potential drivers of movement to that is similar in spirit, but based on velocity rather than occupancy.

Velocity-Based Movement Model

Let be a continuous representation of an animal's true, but unknown, continuous movement path. A simple transformation, , yields the animal's continuous time-varying velocity vector, which specifies the animal's speed and direction at any time. Given a starting location , this transformation is invertible and preserves all information present in the movement path . In practice we will not have a truly continuous movement path, but instead a quasi-continuous set of location information at temporally regular intervals . Thus, would take the form of a matrix of locations at temporally regular intervals along an animal's movement path, with , and yields the animal's velocity vector at times , which can be collected in a matrix . Thus the -th row of contains the and coordinate of the animal's velocity vector specifying the animal's speed and direction at time . The computationally discrete form for the transformation from location to velocity is essentially a first-differencing, as is common in the analysis of time series data [26] and animal movement [23]. As the temporal resolution of the discrete form approaches zero, the limit is the continuous velocity vector .

We now propose a statistical model for , conditioned on parameters () related to potential drivers of movement and the movement path . Environmental data are commonly available in gridded form (e.g., pixels). As our response variable, , is in velocity vector form, we propose to utilize the gradient of environmental variables as covariates in a statistical model [21], [27]. The gradient of a spatially-referenced covariate is a vector field that points in the direction of the greatest rate of increase in the covariate. If there are environmental or biotic drivers of movement of interest, let be the vectorized gradient of the -th environmental variable at the location , and let . If is a -vector of regression coefficients related to the environmental (and other) covariates whose gradients are in , then we could then specify a multivariate regression model for the relationship between the velocity vector and the environmental and biotic covariates in :(4)Here is normally-distributed with mean (matrix multiplication) and variance .

Modeling the relationship between velocity and covariate gradients in this way attempts to capture an animal's response to changes in the covariates (potential drivers of movement in [21]), as opposed to the animal's response to a static level of the covariates (static drivers of movement in [21]). For example, a positive in (4) related to elevation would indicate that the animal moves generally in the direction of increasing elevation, perhaps indicating a migration to higher elevations, a negative would indicate that the animal is moving to lower elevations, and a near zero would indicate that the animal's movement is not correlated with increasing or decreasing elevation.

Environmental data often come in categorical form, which can be incorporated into models as static drivers of movement with the assumption that animal movement and behavior differs between categories. However, the velocity-based framework (4) incorporates potential drivers of movement (gradients) and not static drivers. To illustrate one approach for utilizing a categorical variable in the existing velocity-based framework, consider a categorical variable consisting of land cover types. For each cover type , a new spatial variable could be created consisting of the shortest distance from each spatial location to the -th cover type. The gradients of these new variables would point in the direction of the shortest direct path to each cover type, allowing inference to be made in the velocity-based framework about cover types that animals are drawn to, or away from.

Temporal Heterogeneity in Drivers of Movement

It is often unreasonable to assume that the animal's response to the environment remains homogeneous over the temporal domain of . We propose an approach for modeling heterogeneity in response to potential drivers of movement that is based on partitioning the temporal domain of the movement path into regions in which the animal's response is homogeneous. This allows the effect of an environmental driver of movement to vary over time, and also allows inference on the degree of heterogeneity in the animal's response to the environment. The number of change points, , can be either fixed a priori or random and unknown, as explained below.

Let be a -vector with each entry being the change point between two partitions in which the animal exhibits distinct movement profiles. Then we can write (4) as:(5)In (5), we model changing patterns of movement and resource selection as coming from a bivariate normal distribution with a mean function that is piece-wise linear. Each partition of the movement path represents a contiguous segment of time in which the animal's pattern of movement and selection is homogeneous. Thus , the response to the drivers of movement represented by covariate gradients in is allowed to vary over time, taking different values at each partition of the movement path . We will present an approach that allows inference to be made on the number of partitions in an observed movement path, their temporal locations , and the response to drivers of movement associated with each partition . Allowing the length, location, and number of the partitions to change results in a highly flexible model of heterogeneous response to the environment. In this way, we can make inference about changing behavior over time in individual animal movement and selection.

Univariate Formulation

The velocity-based model of animal movement and selection we have presented (5) is bivariate. The dependent variable is the vector-valued velocity of the animal at time , where and are the components of the vector-valued velocity in two orthogonal directions (e.g., Easting and Northing). The independent variables are also vector-valued:where and are the components of the covariate gradients in the same two orthogonal directions.

In a movement setting, it is reasonable to assume that and are uncorrelated, conditional on . If this were not true, and and were correlated, the animal would prefer moving in northeast or southwest directions (in the case of positive correlation) or southeast or northwest directions (in the case of negative correlation) over other directions. Drift behavior of this form, if present, could be modeled in the first-order effects . The assumption of independence of and allows us to formulate the bivariate model (5) in an equivalent univariate regression form:(6)This model (6) specifies the relationship between the components of an animal's velocity vector over time and the parameters related to drivers of movement in a linear framework that is extensible, allows for easily interpretable results, and is computationally efficient.

Prior Distributions for Model Parameters

We will fit this model using a Bayesian hierarchical modeling (BHM) framework [28] and thus specify prior distributions for all model parameters. For the regression coefficients , we specify a Gaussian prior distribution with mean and variance-covariance matrix .(7)In practice, setting and gives a diffuse prior that allows inference to be driven in large part by the data.

For the variance component in the regression model (5), we specify an inverse-gamma prior distribution with shape parameter and scale parameter :(8)In our analysis, we have set and in such a way that the prior mean of is 10 and the prior variance is , giving a diffuse prior distribution.

For the locations of the change points , we specify a discrete uniform prior distribution, with change points equally likely to occur at any point in the observation window :(9)subject to the constraints that and for each .

If we have fixed a priori, then (6)–(9) is a fully specified BHM. If we allow for an unspecified number of change points, and thus movement partitions, we allow to be random as well, and assume a Poisson prior distribution:(10)

The choice of , the prior mean number of behavioral partitions of the movement path, should be made based on existing knowledge of the animal, the length of the observation period, and the time scale at which inference on changing behavior patterns is desired.

Model Implementation

Having specified a statistical model, we now outline the procedure used to fit this model to telemetry data. Markov chain Monte Carlo (MCMC) methods are used extensively to fit BHMs such as we have specified in (6)–(10) by sampling from the posterior distribution of the parameters [29]. In this case, full-conditional distributions can be found for and for , allowing use of the robust Gibbs sampler within an MCMC framework [29]. Updates to are accomplished through Metropolis-Hastings steps in the algorithm.

Allowing to vary results in a parameter space that varies as samples are drawn from the posterior distribution with varying numbers of change points, and thus varying the dimensionality of model parameters . That is, we do not specify beforehand how to partition the movement path into contiguous segments in which the animal's pattern of movement and response to the environment is homogeneous. Rather, we have specified a BHM (6)–(10) so that location data can be used to infer the number and location of these partitions. MCMC methods employ random walks through parameter space to sample correlated realizations from the distribution of interest. In the case of , the MCMC approach will lead to different iterations having different numbers of partitions . As each of the partitions has a distinct pattern of response to environmental and biotic covariates, changing the number of partitions results in adding or deleting the set of parameters describing that partition from the model.

Reversible jump Markov chain Monte Carlo (RJMCMC) methods are one popular approach for sampling from the appropriate stationary distribution in a situation where the parameter space varies [30], but the rate of convergence in the algorithm is highly dependent on the choice of proposal distribution. Birth-death Markov chain Monte Carlo methods (BDMCMC) are continuous-time MCMC methods that are closely related to RJMCMC [22], [31]. One particular advantage of BDMCMC is the ease with which multiple proposal distributions (e.g., birth distributions) can be implemented [22]. In the BDMCMC approach, the parameter space of the change-point model (6)–(10) is viewed as a point process in in which each point is a vector: containing model parameters for one movement partition. New movement partitions are added to the model space in a manner specified by a birth distribution, . Each existing movement partition is allowed to die (be removed from the point process and thus the model) at a Poisson rate specified in such a way that the BDMCMC process is guaranteed to converge to the appropriate stationary distribution [22], [31]. Success of the BDMCMC procedure is dependent on finding a birth distribution that allows for good mixing and thus rapid convergence to the appropriate stationary posterior distribution. In general, BDMCMC is merely an efficient algorithm for implementing the change point model (7). We provide further discussion of the BDMCMC process in the supporting information (Text S1).

The hierarchical model in (6)–(10) is also conditioned on the true quasi-continuous movement path . is unknown, and to make inference on , as opposed to , we need to integrate over the distribution of . This is accomplished through composition sampling; BDMCMC updates of , , and are drawn iteratively with Gibbs updates of , , and to improve mixing. In brief, the algorithm for drawing samples from the posterior distribution in the case where is treated as random and unknown is as follows:

1. Specify a beginning model state with partitions.
2. Sample from .
3. Update and using the BDMCMC procedure.
4. Sample from its full-conditional distribution.
5. Sample from their full-conditional distributions.
6. Update using Metropolis-Hastings steps.
7. Repeat steps 2–6 until convergence is reached and a sufficient number of samples are obtained.

The algorithm for the fixed- case is identical to the variable- case, except that step 3 is omitted.

It should be noted that steps 5–6 in the algorithm are not strictly necessary for convergence to the appropriate posterior distribution as the BDMCMC process updates and in step 3. However, including steps 5–6 greatly improves the mixing and thus computational efficiency of this iterative sampling process.

The supporting information contains example code (Text S1) for implementing this approach and a simulation study (Text S2) illustrating the implementation of this individual-level model of animal movement on simulated data.tpb

Model Comparison and Selection

In regression models, such as (6), parsimony in covariates is often desirable. Multiple methods for comparing models with different sets of regression covariates exist, including the Akaike information criterion (AIC) [32]. For Bayesian models, the Bayesian information criterion (BIC) [33], and the deviance information criterion (DIC) [34] are commonly used. Celeux et al. [35] present eight methods for calculating DIC for models with missing data. These methods are easily applied to the model-based data augmentation approach which we present here. Celeux et al. [35] suggest that the most reliable of the eight methods they present is(11)where , the full set of parameters in (6)–(10), and is the posterior mean of these parameters.

This criterion can be used to compare models with different subsets of covariates, which may aid inference on the relative importance of individual environmental covariates to animal movement. This criterion could also be used for multi-model inference [36].

Population-Level Inference

Telemetry data for different animals in a population often differ in length and occur on different spatial and temporal domains. The differences in observation windows among animals can make scaling up inference to a population level difficult. One significant advantage of the continuous movement model for animal movement we propose is that a uniform temporal discretization of the movement path can be obtained for all animals. Population-level inference can then be made by aggregating inference from different animals in covariate effect space.

State-switching models of animal movement commonly assume that a population's movement can be described by a fixed number of population-level movement regimes [6], [14], each representing a distinct form of movement. For example, one regime could represent directed, migration-like movement, another regime could represent a movement state of strong response to certain environmental factors, and another regime could represent a state of little movement.

In our individual approach to modeling animal movement, we allowed for temporal heterogeneity in the response to environmental and biotic drivers of movement through a change point model (6)–(10). At a population level, we also adopt a framework that allows for different states of movement and behavior, with the following assumptions:

1. At the population level, patterns of animal movement and response to the environment can be partitioned into a fixed number of regimes, each of which can be represented by a distinct pattern of correlation to gradients of environmental or biotic covariates ( in our individual-level model).
2. Sub-groups of the population may have different propensities for each of these movement regimes, manifested by differences in the proportion of use of the regimes across subgroups.
3. Individual animals may have varying propensities for each of the movement regimes as well, manifested by differences in the proportion of use of the regimes across individuals.
4. Individual animal movement in an observation period can be modeled by breaking the path into an unknown (random) number of partitions, as specified in our individual-level approach (6)–(10). In each of these partitions, animal movement and resource selection follows one of the population-level regimes.

One approach to making population-level inference together with individual-level inference would be to add another level to the hierarchical model (6)–(10). The prior on (8) could be replaced with a hierarchical prior, such as a mixture of normal distributions, with each mixture component representing one population-level movement regime. However, such an approach is currently precluded by computational barriers. Inference in such a unified hierarchical model would require jointly making inference on an individual-level model for each animal in the population, and tuning such a model to ensure efficient mixing of the MCMC algorithm could be infeasible. Instead, we propose an approach for making population-level inference about movement and resource selection based on a post-hoc analysis of the inference made on individual movement paths from all members of the population. We have outlined this approach graphically in Figure 2.

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Figure 2. Overview of population-level inference based on a cluster analysis in the parameter space.

We show here an overview, using simulated data, of the approach for making population-level inference based on the results of individual-level response to covariates over time. Inference is made about the time-varying response of multiple animals to environmental drivers (a), with each animal being analyzed individually. In (b), the mean values of for all members of the population and all times are combined. Inference about population-level drivers of movement can be made using cluster analysis. Examination of the marginal distributions of each multivariate cluster (c) can aid understanding of the movement profile defined by each cluster. Differences between subsets of the population can be inferred from differences in the proportion of time each subset spends in each movement cluster (d).

https://doi.org/10.1371/journal.pone.0022795.g002

Our individual-level model, (6)–(10), results in inference about the time-varying response of the animal to different environmental and ecological drivers, , where indexes time. If we apply the approach to multiple animals in a population, then the results for the -th animal are (Figure 2a), where indexes animals and indexes time. Aggregating the values of for all animals and times in parameter space (Figure 2b) can reveal population-level responses to drivers of movement. Cluster analysis [37] can be used to reveal these profiles, which represent population-level movement states. As clustering is an unsupervised form of learning, interpreting the differences between different movement clusters can be done by examining the marginal distribution of the response to each driver of movement (Figure 2c). For example, a migration to lower elevations would manifest itself as a cluster of with negative values for the element of associated with elevation. Foraging behavior, in which an animal's movement is not driven by environmental factors, could manifest itself as a cluster of near , while a state of high speed movement would manifest itself as a cluster with extreme absolute values of the . After clustering the in parameter space, differences in movement between subsets of the population would manifest as differences in the proportion of time spent in each of the clusters by members of different subsets (Figure 2d). In this way, location data from multiple animals can be combined to make population-level inference about movement and resource selection.

Our simulation study (Text S2) illustrates this population-level approach based on simulated individual-level movement.

Application: Northern Fur Seal Movement

We apply our approach for individual and population-level inference to a study of northern fur seal (NFS) movement and resource selection. Approval for the field research described below was granted by permission of the National Marine Fisheries Service, permit number 782–1455.

Northern fur seals are pelagic foragers found in the North Pacific Ocean and Bering Sea. During the winter months, NFS are migratory, while during the summer months, a large percentage of the NFS population congregate at the Pribilof Islands (St. Paul and St. George Islands, Alaska, USA) to breed [38], [39]. Female NFS, after giving birth to a pup, nurse their dependent offspring for around 4 months by alternating trips to sea to feed with time on shore to nurse. Male NFS, on the other hand, do not participate in pup rearing.

During the 1999 and 2000 breeding seasons, 45 northern fur seals were captured at two haul-out sites and two rookeries on St. Paul Island. In 1999, thirteen juvenile male NFS were captured, and in 2000, fifteen juvenile male NFS and seventeen lactating female NFS were captured. These animals were fitted with satellite transmitters, and location information on foraging trips was obtained using the Argos system (Argos website (accessed 2011) https://www.argos-system.org/manual/). During the summer months, in which our study was conducted, NFS are central place foragers. Figure 3 shows a compilation of the movement paths from these animals. Sea surface temperatures were colder in 1999 than in 2000, and this may lead to different NFS movement patterns in these two years.

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Figure 3. Movement paths for northern fur seals in the Pacific Ocean.

In 1999 and 2000, 45 northern fur seals on the Pribilof Islands were fitted with satellite transmitters. Movement paths for all 45 northern fur seals are shown.

https://doi.org/10.1371/journal.pone.0022795.g003

We were most interested in at-sea behavior of NFS, and so we separated the full location record for each animal in the study into separate at-sea trips by removing locations at times when the NFS was hauled out one one of the Pribilof islands. For 34 of the NFS, this involved trimming hauled-out telemetry locations from the beginning and end of the observation window, resulting in one at-sea trip for each animal. For the remaining 11 NFS, we split the full location record into two or more at-sea trips beginning (and ending for complete trips) with a single on-land location. All told, the telemetry locations for the 45 NFS were split into 64 at-sea trips. The number of at-sea trips for each animal is shown in the supplemental material (Text S3).

We used three ocean environmental covariates for this study: sea surface temperature (SST), Chlorophyll a level (CHA), and net primary production (NPP). Each of these covariates were measured monthly at a 9 km resolution across the Bering Sea. For each at-sea trip, the mean observation time of the path was found, and the measured covariates from the month before and the month after the mean observation time were averaged, with weights inversely proportional to the time between the mean observation time and the monthly covariate observation times. This resulted in a spatially referenced set of covariates for each at-sea path. Missing values in the covariate data were interpolated using thin plate splines [40] as implemented in the fields package [41] in the R statistical computing environment [42]. For each covariate, the gradient (directional derivative) was calculated and used as a potential driver of movement.

Female NFS are central-place foragers during the summer months, often making trips of 4–10 days at a time to travel to foraging locations. As they provide the only source of nourishment for their pups, the length of a foraging trip is constrained by the amount of time their pups can survive without food [20]. In an attempt to include this potential driver of movement in our study, we include the gradient of the distance to the seal's rookery in the set of model covariates. This is done for both male and female NFS.

We use the velocity-based approach described in the preceding sections to examine telemetry data from each of the 64 at-sea trips made by these NFS. We then make population-level inference based on the individual-level results for for all animals. The aim of our study is to characterize individual seal response to environmental drivers of movement and make inference about potential differences between subsets of the population. In particular, our population-level study focuses on two questions. We first compare movement patterns in male NFS tagged in 1999 to movement patterns of male NFS tagged in 2000. We then examine potential differences between juvenile male NFS and lactating female NFS movement and response to environmental factors.

Individual-Level Methods

A preliminary model fit with a fixed number of change-points () was conducted on each animal to provide reasonable birth distributions. Following [22], we utilize a birth distribution based on the posterior distribution of from these preliminary models. In the full models, in which is allowed to vary, when a new partition is added to the parameter space via the BDMCMC process, it is drawn from the following birth distribution: we first draw , the change-point for the new partition, uniformly from all times at which there is not currently a change-point:We then draw model parameters for the new partition from a Gaussian distribution:(12)where is the posterior mean value at time for from the preliminary fixed- model fit, is the posterior variance at time of in the fixed- model fit at time , and is a tuning parameter which can be modified to improve mixing. We note that a birth distribution in BDMCMC is similar to a proposal distribution in a Metropolis-Hastings update in that the posterior distribution is invariant to the choice of proposal (or birth) distribution, though the rate of convergence can vary widely for different proposal (or birth) distributions. Our use of the data to inform a birth distribution only facilitates mixing in the MCMC process, and does not influence the resulting posterior distribution (see [22] for further discussion). The supporting information (Text S1) contains more details of the BDMCMC algorithm.

We analyzed each of the 64 at-sea paths from the 45 NFS using the individual velocity-based movement approach. The algorithm for fitting the continuous-time correlated random walk model [15] did not converge for five seal paths (see Text S3 for details). For each of the remaining 59 at-sea paths we used a preliminary model with a fixed number of change-points to generate a birth distribution. The birth distribution was then used in the variable- model. The variable- algorithm was used to generate 11,000 MCMC iterations, with the first 1,000 being discarded as burn-in. Convergence of the Markov chain for each animal was assessed visually by examining the chains for proper mixing. Example trace plots are shown in the supporting information (Text S3). For animals whose Markov chains showed lack of convergence, an additional 100,000 iterations were generated. MCMC algorithms for five NFS paths did not converge after this longer run, and these paths were discarded from the analysis, leaving 54 at-sea paths from 41 animals in the study (see Text S3 for details). Our population-level study is based on the individual results from these 41 animals.

We also conducted an in-depth individual-level study on two NFS, one male (animal 1) and one female (animal 2) to illustrate model comparison procedures. For each of these animals, a set of models with all possible subsets of the four covariates were applied to the data. The MCMC procedure was used twice with unique starting values to produce 11,000 iterations of the composition sampler for each model and animal, and the first 1,000 iterations were discarded as burn-in. Convergence was assessed visually (see Text S3 for additional details). The two chains for each model were combined, resulting in 20,000 iterations for each model and animal. The for each model was computed, using the 20,000 iterations.

Population-Level Methods

Our population-level inference is based on clustering the time-varying coefficients for the 41 animals in the individual-level study. To remove the temporal autocorrelation in the response to the drivers of movement, and to make the cluster analysis more tractable, we thinned the for each path, keeping only 1/100 of the time points. The resulting values of from all animals were combined, and a Gaussian mixture model clustering algorithm was fit using the MCLUST R-package [43]. Models with one to eight mixture components were compared using BIC, and the best model was used for inference on population-level drivers of NFS movement.

Differences in movement by subsets of the NFS population were assessed by examining differences in the time spent by animals in the different subsets (e.g., male and female NFS) in each of the resulting movement states. As the location data vary in length across animals, we examined differences in the proportion of time each animal spent in each movement cluster. If an animal's telemetry location data were split into multiple at-sea trips, all trips were aggregated by animal and the proportion of time spent in each movement cluster for the animal was the proportion of all at-sea time. In this way we can examine differences in the proportional use of the different movement clusters between individuals and subgroups of the population.

To test for differences between male and female NFS movement patterns, we applied a classification tree [44] to the proportion data with sex as a response variable. A classification tree was also used to test for differences between movement patterns of males tagged in 1999 and males tagged in 2000.

Northern Fur Seal Discussion

For both animal 1 and animal 2, the model with the best DIC includes only the distance to rookery covariate, indicating that these animals were not strongly influenced by the environmental factors represented by the other covariates (Table 1).

The results concerning for animal 1, shown in Figure 4a and Figure 5a, confirm that the distance to rookery covariate does have a significant effect which changes over time. At the beginning of the seal's trip, the positive effect of this covariate indicates a desire to move away from the rookery, possibly towards memorized foraging locations. During the middle of the trip, the lack of any significant covariates indicates that movement is not being driven by the covariates in our study, but by some other drivers. Potential drivers of movement not examined in this study include subsurface temperatures and behavioral drivers (e.g., prey and predator locations). At around hour 475, the effect of the distance to rookery covariate becomes negative, driving the male to return to the rookery.

The results for animal 2 (Figure 4b and Figure 5b) show that this female NFS is first pulled away from the rookery (hours 3–10), then is not driven by any of the covariates in this study for a period of time (hours 19–38). After approximately 38 hours, the animal begins to return to the rookery, though the return is quite different than that of animal 1. The second half of the female's trip (after approximately 50 hours) is dominated by little response to any of the covariates in the study. The female appears to be foraging on the return trip, and forages close to the rookery between hours 66 and 83. In contrast, the path of animal 1 shows a relatively long period in which the animal is drawn out away from the rookery, followed by a fairly abrupt change in response to the distance to rookery driver (around hour 475), and a short time in which the animal returns to the rookery with little distraction. This juvenile male NFS appears to have a fairly distinct turnaround time, in which it decides to cease foraging and return quickly to the rookery.

A comparison of the scale of in Figure 4 shows that the female's absolute response to all covariate gradients is larger in magnitude than the male's response. This indicates that the female traveled at a higher swim velocity than the male in these example trips. The female's trip is also much shorter than that of animal 1. This is typical of the NFS studied here (see Text S3).

One possible explanation for these differences may be tied to the different roles male and female northern fur seals play in rearing pups. The female northern fur seals are the primary providers for unweaned pups, and may be more constrained in their foraging, as they must return to feed their pups. Male northern fur seals, like animal 1, may be able to forage in a less-constrained manner, and thus are less driven to return within a certain time interval, leading to a slower swim speed and longer duration of foraging trips. Additional individual-level results are in the supplemental material (Text S3).

In our population-level study, the movement of all NFS studied was clustered into seven mixture components, representing seven distinct movement profiles for this population (Figure 6). Cluster 1 is the most prevalent cluster of NFS response to the four drivers of movement, and contains values of that are fairly representative for all covariates. Cluster 2 differs from cluster 1 primarily in its responses to CHA, SST, and NPP, which are all more closely distributed around zero than in cluster 1, indicating a state of movement where influence of environmental factors is minimal. Cluster 4 is characterized by extreme response (both positive and negative) to all covariates, indicating a state of fast absolute speed, as extreme responses to any covariate correspond to high velocity in the animal's movement. Cluster 5 is characterized by small (near zero) response to all covariates, indicating a state of slow absolute speed. extreme response to NPP.

The clustering algorithm used for this population-level study of NFS movement patterns is based solely on the aggregated responses to drivers of movement , and does not take into account the animal or the time in the trip. However, the arise from our individual-based change point model of animal movement, which assumes a partition of the movement path into contiguous regions in which the response to the environment is similar. Thus it is not surprising to see that the animal movement paths in Figure 7b and the supporting information (Text S3) show both animals tend to remain in one movement cluster for a time and then switch to another cluster.

Cluster 5, which is a movement state of low absolute response to the environment, is a predominantly male response to the environment (Figure 8). Cluster 4, which is a predominantly female movement state (Figure 8), is characterized by larger absolute response to covariates. This cluster is a state of faster absolute movement and stronger response to the environment than is found in the predominantly male cluster 5. This may indicate that female NFS are more intense foragers than males, as the length of their foraging trips are constrained by the amount of time their pups can survive between feedings. Males, on the other hand, have no such constraints, and may be able to forage in a more relaxed fashion.

The results from the male-only comparison of movement between seals tagged in 1999 and 2000 show no significant differences in response to the drivers of movement used in this study. The velocity-based approach we have used for modeling animal movement is based on gradients of covariates, and responses to these gradients were similar for males tagged in both years. Sea surface temperatures were colder in 1999 than in 2000, but male NFS movement patterns were not greatly affected by this change in temperature.

Modeling Approach

We have presented an approach for modeling animal movement based on random walks (CTCRW) that allows for inference on changing movement profiles exhibited by the animal throughout the time for which we have location information. This approach for linking animal telemetry data to environmental and biotic drivers of movement is flexible, computationally efficient, and yields easily interpretable results. A general overview of our approach to individual and population-level inference could be summarized as follows.

1. Obtain time-referenced location data for multiple animals in a population.
2. For each animal, utilize the CTCRW model [15], or some comparable method, to obtain realizations from the posterior predictive path distribution , with locations at regular time intervals.
3. Obtain spatially-referenced covariates that are hypothesized drivers of movement, and calculate their gradients.
4. Make inference about drivers of movement for each individual animal using the velocity-based movement approach (6)–(10). This approach accounts for the uncertainty in the movement path distribution and allows for temporal heterogeneity in the drivers of movement.
5. Aggregate values of for all animals and at all times and cluster these aggregated effects in parameter space. Insight into the interpretation of each cluster's movement profile can be obtained by comparison of the marginal distributions of the effects in each cluster.
6. Make inference about differences in how population subsets respond to drivers of movement by using the proportion of time spent by individual animals in each movement cluster as predictor variables. This could be done using parametric (e.g., logistic regression) or nonparametric (e.g., classification trees) statistical methods.

Care should be taken when making conclusions based on the posterior number of partitions , as this parameter can be sensitive to model departures [22]. The change point model we have used assumes that an animal's response to a covariate is constant on each of partitions of the observation period. This model can approximate much more complicated time-varying behavior, as shown in our simulation study (Text S2), but if the true response to a covariate is not piece-wise constant, then the location data will likely drive to be a large number to best approximate the changing behavior over time. For example, the distance to rookery covariate seems to vary smoothly over time for animal 1 in our application, while our model assumes a finite number of break points in which the animal changes behavior. If the covariate does vary smoothly, the model would tend to favor a large number of partitions in an effort to best capture the animal's true response to the driver of movement. Stephens [22] also notes that inference on can be influenced by choice of prior distribution and birth distribution. We have found this to be true, though inference on , the focus of our study, is quite resilient to these variations.

A similar, but alternative approach to our change-point model would be a temporally varying coefficient model [45]–[47], which could be used to obtain similar inference about environmental drivers of movement within the velocity-based framework we have described here.

Utilizing the velocity-based model with an unknown number of change points requires tuning the birth distribution by adjusting in (14). An alternative that would require less human input would be to fit multiple fixed change point models with different numbers of change points () and rank models using standard model selection criteria [36]. For example, Steele and Raftery [48] have found that BIC can outperform other Bayesian selection criteria (including DIC) in choosing the number of mixture components (K) in a model. In the data augmentation case, BIC can be formulated as(13)where is the total number of free parameters in the model with movement states and is the number of observations. In the fixed- case, this formulation of BIC can be used to compare models with differing numbers of change points.

Our individual-level model allows for heterogeneous behavior over time, as implemented with a change point model. In its current form, we estimate for each partition (6) independent of all other partitions. This allows animal movement behavior in each partition, as characterized by the animal's response to the potential drivers of movement, to be different from behavior in all other partitions. It may be more realistic to assume that there are a fixed (or random and unknown) number of movement states that an animal can exhibit, and the observed path can be partitioned in such a way that movement behavior in each temporal partition follows one of the fixed (or random) movement states. For example, we might expect animals to act similarly at specific times of the day each day (e.g., dormant behavior during the day for nocturnal animals), resulting in partitions each day with similar movement behavior. Our simulation study (Text S2) shows that our individual-level approach can make inference on recurring behavior of this form, but explicitly incorporating this into our model, perhaps through a hierarchical prior on the that is shared by the population, could make our approach more robust. The main difficulties here are computational, as tuning the nested birth-death processes for individual and population-level heterogeneity may be infeasible.

In our approach to modeling individual and population-level movement, we have assumed that each animal moves independently of all other animals. While this is not likely to be true in all situations, jointly modeling the relationship between movement patterns of conspecifics is challenging. One major difficulty is that only a sample of a population is typically monitored, leading to little overlap between the animal movement paths in space and time. One way the relationship between conspecific movement patterns could be incorporated into the current framework (5)–(6) would be to jointly model all movement steps from all animals and specify a covariance structure based on the spatial and temporal separation between the movement steps of different animals. For example, the correlation between and , two movement steps from different animals at possibly different times, could be modeled with an exponentially-decaying structure: , where and are range parameters for correlation between movement in time and space, respectively. In this way, animals that are close together in time and space could exhibit correlated movement and response to the environment.

In the estimation of resource selection functions, a distribution of available habitat is typically specified [15], [49]. The definition of available habitat can be set by the study area [9], [50] or by an underlying movement model [12], [21], [51]. Inference on use versus availability of resources can vary as the scale of what is defined as available changes [9], [50], [52]. Our approach to modeling animal movement and resource selection could be seen as defining availability of resources at two scales. Our velocity-based model with covariate gradients assumes that animals are motivated only by what is in their immediate neighborhood, thus our availability distribution amounts to the adjacent habitat at the scale of the environmental covariates (e.g., the adjacent pixels of land cover types). Jointly with this fine-scale definition of availability, we employ a stochastic CTCRW model of animal movement to stochastically integrate over our uncertainty in the animal's path between location fixes. Thus, in our approach, availability of resources is defined by local environmental gradients of the posterior distribution of movement paths in the CTCRW model.

In this study, we have modeled environmental drivers of movement as local covariate gradients fixed in time, but the integration of environmental factors in various formats could facilitate inference on effects not easily addressed in the current framework. Three possible extensions to the existing framework are the inclusion of temporally-varying environmental covariates, known barriers to animal movement, and larger scale gradients. We consider each of these in turn.

The approach we have presented allows the effect of environmental factors (e.g., ) to vary over time, but the environmental factors themselves (e.g., SST) are held constant in time. This may not always be the case, especially in a marine environment. Currents are constantly shifting, and the ocean is always in flux. The use of potential environmental covariates in the existing framework for making inference about drivers of animal movement will allow for more realistic inference.

Our approach does not currently take into account potential barriers to animal movement, such as land masses for marine animals or cliffs for land animals. Such barriers could be included through rejection sampling of in the data augmentation step (3), accepting a continuous path realization only if it does not cross the known movement barrier.

Our approach to expressing environmental drivers as gradients currently considers only local gradients, though animal movement may be motivated by memory of desirable destinations at a greater distance. A possible extension to our current approach could be to use larger scale (coarser resolution) gradients simultaneously with local environmental gradients. This could provide insight into when an animal's movement is being motivated by local environmental factors, as opposed to distant attractions. This could allow for models that predict animal movement through less desirable landscape features to destinations that are highly desirable. Our velocity-based approach for animal movement modeling is flexible and extensible, and its computational efficiency enables modeling significant complexity in an animal's response to the environment.