Specification of biased correlated random walks

BCRW is considered as a highly flexible and powerful discrete-step model of single-scale single mode animal movements, as it combines long and short term attractiveness with various degrees of stochasticity [23]. The long-term attractiveness can, for instance, be associated to a patch that is rich in nutrients while directional persistence accounts for short-term effect (i.e., directional persistence implies that the expected direction at a given point in time is that of the previous move bearing) [14]. When implementing a BCRW, the path of the animal is discretized and the movement between times t and t+1 is a 2D step vector, for t = 0,…,T. It is convenient to use a polar representation of this vector, (y_t, d_t) where the step angle y_t ∈ [−π, π) gives the direction of the movement with respect to a reference direction, e.g. north, and d_t is the step length.

In a BCRW, let ψ_t be an angle giving the direction toward the target (i.e., the long-term attractor) at time t; this is the angle, with respect to the reference direction, of a line joining the animal’s position at time t and the target’s position. The step angle y_t is the result of a compromise between directional persistence, i.e., the innate tendency to move in the same direction as the previous step [7], and movement in response to the target in direction ψ_t. The compromise nature of the random step angle y_t can be modeled by setting its mean direction μ_t to the vector direction of 1 Thus, μ_t = atan{sin(y_t−1) + β sin(ψ_t), cos(y_t−1) + β cos(ψ_t)}, where atan(a,b) = atan (a/b) for b>0 and atan (a/b) + π for b<0 [24,25]. Many ecological papers, e.g. [12,14,15,22], sets the mean direction μ_t equal to a convex combination, μ_t = λy_t−1 + (1−λ)ψ_t for some λ in (0,1). As McClintock et al. (2012) pointed out, this is not satisfactory, especially if |y_t−1−ψ_t| > π, because adding, or substracting, 2 π to either y_t-1 or ψ_t leads to different numerical values for μ_t. Using Eq 1 gives a more stable definition, invariant to the addition or the subtraction of 2π to either angle. It can therefore be easily extended to more than one directional bias as illustrated below.

Parameter β weights the attractive strength of the target relative to directional persistence in the compromise that the animal makes when it chooses the movement angle for its next step. When β = 0, μ_t = y_t-1 and the model reduces to a correlated random walk, whereas when β goes to infinity μ_t = ψ_t−1 and we get a biased random walk (Benhamou, 2014). Overall we demonstrate that the expected movement direction in BCRW should be defined using Eq 1, because it gives a more stable definition than one that uses a convex combination of y_t-1 and ψ_t.

Estimation of the parameter β of biased correlated random walk

Given some data {(y_t, ψ_t): t = 0,…, T}, we consider two circular regression models to estimate the parameter β of the BCWR, viz., the angular model and the consensus model.

The angular model is obtained by assuming that the step angles y_t have a von Mises distribution, with a mean direction μ_t and a concentration parameter κ > 0. The corresponding density is 2 where I₀ denotes a modified Bessel function [26]. The angular estimation method consists of estimating the parameters (β, κ) by maximizing the likelihood that is constructed with Eq 2 [24,25].

The consensus model assumes that the error concentration parameter depends upon the degree of agreement between the two possible goals of the animal, i.e., moving forward versus going towards a target. The agreement is measured by the length of vector v_t in Eq 1, . When β is positive, maximum length, , occurs when directional persistence leads directly to the target, i.e., ψ_t = y_t−1, whereas minimum length, , is obtained when directional persistence pushes the animal in a direction opposite to that of the target, i.e., ψ_t = y_t−1 +π. As an alternative to Eq 2, the consensus model assumes that the step angles y_t have a von Mises distribution with a mean direction μ_t and a concentration parameter for some κ > 0. This simultaneous modeling of the mean direction and the concentration parameter has been advocated by Presnell et al. [27], in particular due to the ease with which the likelihood can be numerically maximized. Given that and , the von Mises density for the consensus model simplifies to: 3 where κ₁ = κ and κ₂ = κβ. This density belongs to the exponential family and the estimation of (κ₁, κ₂), and of β = κ₂/κ₁ is easily carried out by maximum likelihood. This defines the consensus estimation method. In summary, the parameter β in a BCRW can be directly estimated by maximum likelihood by maximizing the likelihood of one of the two models (Eq 2 or Eq 3), respectively the angular and consensus models.

Step selection function for estimating the parameter β of biased correlated random walk

We now provide a third method that, based on step selection function (SSF) [10], can be used to estimate the parameter β of Eq 1. SSF uses, for each time point t, J control step angles , j = 1,…,J that are randomly selected among the T step angles {y_t} in the data set. The SSF for the BCRW of Eq 1 has two explanatory variables, the cosines of the turning angle y_t—y_t-1 and of the angle to the target y_t−ψ_t. Its likelihood, which is derived in section 8.4 of [28], is proportional to

4

The SSF estimation method consists of maximizing L_a(κ_1, κ₂) to estimate κ_1, κ₂, and thus, β = κ₂/κ₁. We now show that this estimation method is related to the consensus estimation method.

Suppose that the control step angles are uniformly distributed over (0, 2 π). The denominator of L_a(κ₁, κ₂) for time t is equal to

When J is large, the term involving y_t is negligible and the denominator at time step t is

This equation derives from the Law of Large Numbers [29]: as J goes to ∞, the sample average of converges to the expectation .

The SSF likelihood that is given in Eq 4 is approximately equal to:

This is the likelihood function for the consensus model of Eq 3. Thus, the empirical SSF estimates for κ₁, κ₂, and β = κ₂/κ₁ should be close to those that are obtained by the consensus estimation method, as long as the control angles have a uniform distribution; this property is investigated in the Monte Carlo study that is presented in the next section.

Monte Carlo study comparing β estimators

Three methods are available to estimate the parameter β, which quantifies the importance of the target on animal movement relative to directional persistence. They are: (i) the angular estimation method, which uses the likelihood that is constructed using Eq 2; (ii) the consensus estimation method, which uses the likelihood that is constructed using Eq 3; and (iii) the SSF estimation method, with cos(y_t—y_t-1) and cos(y_t−ψ_t) as explanatory variables, see Eq 4. We used Monte Carlo simulations to compare the β values that were provided by the three estimation methods.

To avoid multicollinearity problems that arise when simulating data for a single animal, all simulations involved two animals moving independently on a 2D simulated landscape of 1024 x 1024 pixels. The simulated landscape was developed following [9]. Directional bias was towards the center pixel of the landscape. The starting locations of the two animals were randomly sampled, one in the northwest corner of the landscape map and the other in the southeast corner. Each animal was observed for 61 steps, yielding 120 observations contributing to the likelihood of each estimation method. This approach ensured that the series of data were long enough to make the model parameters estimable, yet not long to the point where the animals would spend the latter part of the series meandering about their long term target.

Three sets of simulations were carried out, each with 500 repetitions. The first simulation model used a map that was comprised of three habitat types occupying respectively 80%, 10%, and 10% of the simulated landscape (see S1 Appendix). The map was constructed by discretizing the realization of a Gaussian random spatial process (details available in S1 Appendix). An animal’s displacement at time t was simulated using a discrete choice model, which enables us to assess the robustness of the proposed methods, i.e., to verify whether they are able to identify the directional biases in situations where the noise is not exactly as modeled in Eq 2 and 3. First, among the pixels that were more than 60 pixels away from the animal’s current location, the closest pixels of habitats 2 and 3 are identified as candidates for the next step. One pixel is selected randomly among the candidate pixels; the probability of selecting pixel k among these candidate pixels is proportional to , where is the angle of the segment joining the animal’s position at time t to candidate pixel k, whereas ψ_t is the angle of the segment joining the animal’s position at time t to the long term target. Thus, pixels in the direction of the previous displacement or in the direction of the target are assigned higher probabilities. Three sets of values for (κ₁, κ₂) were used in the simulations, viz., (2, 2), (2, 0.5) and (0.5, 0.5). The pair (κ₁, κ₂) = (0.5, 2) was found to be uninteresting since the animals reached the target in a few steps and spent most of their time meandering around.

The animals’ displacement rules in the second and third simulation models did not depend upon the type of habitat and all steps had a length of d_t = 60 units. In the second set of simulations, the movement direction, y_t, was simulated using the consensus model of Eq 3, with κ = κ₁ and β = κ₂/κ₁. For the third simulation model, y_t was simulated according to the angular regression model of Eq 2, with β = κ₂/κ₁ and von Mises errors with respective concentration parameters 3, 2, and 0.75 for the three (κ₁, κ₂) pairs, i.e., (2, 2), (2, 0.5) and (0.5, 0.5).

The SSF estimation method involves a comparison between observed step angles, y_t, with control step angles as in Eq 4. The likelihood L_a(κ₁, κ₂) in (4) was calculated using, for each t, J = 10 random step angles, which were randomly sampled among the 120 observed step angles for that simulation. We used Kuiper’s test of uniformity [26] to verify whether or not the distribution of the observed y_t’s from which we sampled the control step angles was close to the uniform distribution for each of the 9 simulation scenarios. Except in the case of the angular models with (κ₁, κ₂) = (2, 0.5) and (κ₁, κ₂) = (0.5, 0.5) that yielded P < 0.10, all tests had P > 0.15, indicating no significant differences between the empirical and uniform distributions. The conditions for the asymptotic equivalence of the SSF estimators and those that were obtained from the consensus model were approximately met.

Simulation results

Fig 1 compares the three estimators of β = κ₂/κ₁ for the 9 simulation scenarios. All three methods estimate β well (Fig 1). The consensus and SSF estimators are very similar. The random control locations entering into the SSF likelihood, however, make the SSF estimator slightly more variable than the consensus estimator. For the discrete choice simulation model, the angular estimation method exhibits a small bias when κ₁ = 2, κ₂ = 0.5. It is superior to some degree, however, to the other two when κ₁ = 0.5, κ₂ = 0.5, i.e., when the effect of directional persistence and the target’s attraction are small. For the consensus simulation model, the three methods are virtually equivalent, with a small advantage to the consensus estimator as it is the maximum likelihood estimator under the simulation model. For the angular simulation model, the consensus estimator is slightly better than the SSF estimator. The angular estimator, which is the maximum likelihood estimator under the simulation model, is somewhat better than the other two.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Comparison of the three estimators of β = κ₂/κ₁ for 9 simulation scenarios.

Each boxplot summarizes values of the estimates of parameters β = κ₂/κ₁ that were obtained with the angular (Ang.), consensus (Cons.) and empirical SSF (SSF) estimation methods from 500 simulations under the discrete choice model (row 1), the consensus simulation model (row 2), and the angular simulation model (row 3). The red horizontal dashed line indicates the true parameter value.

https://doi.org/10.1371/journal.pone.0122947.g001

Ethics Statement

The research was conducted in Prince Albert National Park, a Canadian National Park, in accordance with the research permit PA-2010–4552 provided by Parks Canada. Given that our study only relies on bison trails, data were collected without any manipulation of animals.

Field data

We now estimate movement taxis based on each of the three methods, which were applied to bison trail data collected in Prince Albert National Park, Saskatchewan, Canada. The park is dominated by forest, with a bison trail network linking the discrete meadows that are found on the forest mosaic. Because we used the field data of Dancose et al. [30], with the addition of 22 bison trails, we will only summarize their method. Bison trails were systematically located by walking around the edges of meadows. Trails were then followed using a handheld Global Positional System unit until they reached another meadow. The information was then transferred to a Geographic Information System with a 10-m resolution. The trails were transformed into a string of successive pixels (10 × 10 m).

Analysis

In the SSF analysis reported in their Table 5, Dancose et al. [30] investigate the relative importance of directional persistence, the direction of the next meadow, and the direction of the next canopy gap in the determination of a bison trail’s trajectory. The “time” variable is the position along the trail at time t, as if the observer were traveling on the trail towards the target meadow [30]. Given the position at time t, the position at time t+1 could in principle be one of the 8 pixels that neighbor the location occupied at time t. But the pixel that is occupied at time t-1 is excluded so that there are 7 pixels that can be used at time t+1; one of the 7 gives the observed movement and the other 6 are used as controls in the SSF analysis [30]. The 218 trails generated a data set containing 5696 pixels and their controls.

The approach used to determine the link between animal movement and habitat characteristics often entails the development of candidate movement models that account for various combinations of habitat features that could influence movement decisions. The candidate models are then compared based on their relative empirical support. Given our purpose of contrasting the three methods, our analysis simply considered two movement taxes, together with directional persistence. For the consensus model, the generalization of Eq 3 to include two movement taxes is where ψ_1t and ψ_2t are the angles of the directions to the target meadow (TM) and to the closest canopy gap (CG) at time t. A generalization of the angular model (Eq 2) to two directional biases is constructed in a similar way. Estimates of the parameters κ₁, κ₂, and κ₃ and β₁ = κ₂/κ₁ and β₂ = κ₃/κ₁ that were obtained with three estimation methods, whereas the robust standard errors that were calculated using sandwich estimates of the variance covariance matrix, which is valid regardless of the true error structure [31], are provided for the models that are defined by Eq 2 and 3. For the SSF estimates, the standard errors were obtained from the conditional logistic regression output. Finally, the standard errors of the β parameters for the SSF and the consensus estimates were calculated using the multivariate delta method (see S2 Appendix and S1 and S2 R codes).