para-meters of the simulated correlated random walks. The movement scales (i.e., step sizes) are indicated on the x axis while correlation of the turning angles is on the y axis. On the upper plots colour represents the variance values, while on the lower plots colour indicates Id and the performance index. The lines represent isoclines as a visual guide for investigating the differences. Note that both axes are log transformed.
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Fisher (Indiv #1) (Martes pennanti)
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Fisher (Indiv #2) (Martes pennanti)
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Straw-coloured fruit bat (Eidolon helvum)
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Figure 4.4: Directionality index Id over time for different tracks. The red and
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Figure 4.5: Utilisation density (UD) contours (for 0.5, 0.95 and 0.99 space use probabilities) for one African fruit bat (Eidolon helvum). The contours are narrower around the track for the dBGB, this is especially visible in the zoomed in section of the track. The grey lines show the track of the bat, the panels from left to right show the different UD contours.
the UD, and highlights the constraints of initial Brownian bridge models (Horne et al., 2007) that assume isotropic and homogeneous diffusion across time. The method presented here relaxes both assumptions by: i) locally characterizing dif-fusion, thus becoming variable in time, andii) analysing the diffusion properties across two major and orthogonal axes of motion, thus becoming variable in space.
The directional bias in trajectories can be a product of various processes (e.g. cor-related random walks, biased random walks, landscape features). Although we only formally investigate correlated random walks we think that dBGB performs equally well if the directional bias has a different cause because the variance estim-ates are adjusted to the observed trajectory.
Biased Random Bridges (Benhamou, 2011) assume an advective and a diffus-ive component, which can incorporate the effect on the directional bias attribut-able to diffusion anisotropy. More mechanistic insight can be extracted from a bridging model by decoupling advection from diffusion, and thus, Biased Ran-dom Bridges represent a clear conceptual improvement with respect to standard BBMM (Horneet al., 2007). Nevertheless, Biased Random Bridges do not modify nor improve the estimated UD. Instead, BGB can approximate the idea of advec-tion/diffusion by considering the diffusive non-isotropic process, and thereby im-prove the accuracy of space use and utilisation density distributions. We did not conduct a direct comparison between the dBGB and the Biased Random Bridges because for the latter no dynamic version is defined. It would be hard to assess where to attribute the differences to.
Worth noting are the jagged contours that appear when the directionality index (Id) is high, such as the high probabilities right in front and behind the observed locations. Within strongly directed movement periods these probabilities overlap with the previous and next segment. It is likely that these probabilities are visible in contour lines when an animal starts or stops a directed movement period, where the transition to a stop causes extension of the contours behind the observed stop location. These jagged contours can for example be observed in figure 4.5 in the up-per right corner where shuttling between the localities occurs. Other bridge mod-els suffer from similar artefacts but they might be less conspicuous because the resulting contours are smoother. In any case, it is important to note, that despite these artefacts, UD estimates are overall more accurate than with former models, and that the previous models also contain biases even though perhaps less con-spicuous.
performance over dBGB may be due to the fact that the estimation of more vari-ables (σm,pandσm,ovs. σm) increases the noise. TheIdprovides a measure of dir-ectedness independent of the step sizes of the movement. It could also be used as an indication of where the largest differences between dBBMM and the dBGB are to be expected: essentially where the largest deviations ofIdfrom 0 are observed.
When the time interval between observations increase, the directional persistence of the correlated random walk decreases (Benhamou, 2004). We would therefore expect thatId →0 if the time interval between observations increase, this would mean the difference between the dBGB and dBBMM decreases. When the loca-tion error is of the same order of magnitude as the movement variances, the divi-sion between movement variance and location error becomes more difficult, res-ulting in estimates where one of the variances becomes 0. Also the directionality index (Id) becomes scale-dependent when location errors are high.
Although we do not investigate it here, it is likely that the estimatedId,σm,pand σm,o contain relevant intra and inter individual variation. For example, migrating versus sedentary herds of caribou have a far narrower turning angle distribution (Bergmanet al., 2000) which would result in a higherId. Hence, studyingId,σm,p, andσm,o spatio-temporal patterns across individuals, ecological contexts, or spe-cies, could provide more mechanistic insights into animal home range and space use behaviour.
It is clear that many observed trajectories do not adhere to the assumptions of isotropic, homogeneous Brownian motion. Our model had the highest perform-ance gain for correlated random walks with high directional correlation. Further work defining analytical descriptions of bridge functions for frequently used ran-dom walk models (e.g., correlated ranran-dom walks, Lévy walks, or continuous-time random walks) is needed since the dBGB does not formally describe the probab-ility density of any of these random walks.
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. 4.4 Methods
For notational convenience we followed the notation and variable definitions of Horne et al. Horneet al.(2007).
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. 4.4.1 Probability density function
Zi denotes the observed locations of the animal, at timesti with a normal dis-tributed observation error with standard deviation ofδi. Tidenotes the time gap between two observations and is calculated asTi =ti+1−ti. The expected center of the distribution of possible positions of the animal at timetin the time interval
Ti
μ(t) = Zi+α(t)(Zi+1−Zi).
The standard deviations are assumed to be independent in the orthogonal and par-allel direction and increase between locations
σ2p(t)=α(t)(1−α(t))Tiσ2m,p, σ2o(t)=α(t)(1−α(t))Tiσ2m,o.
In a first step, we transform the coordinates into a parallel and an orthogonal dis-tance using Eq. 4.1 by projecting the vectorμ−eon toμ−d. This equation gives the parallel and orthogonal distances fromμtoewhen heading towarddfromμ.
D = μ−d
∥μ−d∥
fpo(e,μ,d) = ( D⋅(μ−e)
∥μ−e−(D⋅(μ−e))D∥).
(4.1)
The following equation defines the probability density function of the multivariate Gaussian distribution, wherekis the number of dimensions,μthe center of the distribution andxakdimensional vector for which the density is calculated.
fx(x1,. . .,xk)= 1 wherezis any location in the space.
Furthermore, if we assume the orthogonal and perpendicular distances to be un-correlated, thenρ = 0. The probability density function of the bivariate normal distribution is:
and can be simplified to:
In order to include the locations errors we need to redefine ourσfunctions as, σ2p(t)=α(t)(1−α(t))Tiσ2m,p+(1−α(t))2δ2i +α2(t)δ2i+1
In order to estimateσm,pandσm,oby omitting every second location we haveiin 1,3,5,. . .,n−1. We get the following set of equations:
Making the same assumption that there is no correlation between parallel and or-thogonal variation (ρ=0) and filling out the log-likelihood equation we get
ln(L)= −ln(2π)−12ln(∣(σ2p(ti) 0
1
2(fpo(Zi,μ(ti),Zi+1)⊺(σ2p1(t) 0 0 σ21
o(t))fpo(Zi,μ(ti),Zi+1)).
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. 4.5 Competing interests
The authors declare that they have no competing interests.
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. 4.6 Author contributions
All authors contributed to the design of the study and production of the manu-script.
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. 4.7 Acknowledgement
The International Max Planck Research School for Organismal Biology for sup-port of BK; F.B. acknowledges Grants RyC-2009-04133, and BFU2010-22337 from the Ministry of Science and Innovation (Spain), and the Human Frontier Science Program (Ref. RGY0084/2011); Rechenzentrum Garching of the Max Planck Society for support and maintenance of the computational cluster; Jordi Pages (CEAB-CSIC) for initial discussions about variance decomposition and Biased random walks with FB, and Riek van Noordwijk and Scott LaPoint for comments on the manuscript; Track owners for providing sample data: Fisher data, Scott LaPoint, supported by NSF (Grant 0756920 to Roland Kays) & Na-tional Geographic Society Waitt Grant Program (Grant W157-11 to SL), Eagle owl data, commissioned by Paul Voskamp (province of Limburg, the Netherlands) collected by René Janssen (Bionet natuuronderzoek), Stork Data, Martin Wikel-ski, HUJ-MPIO (Hebrew University Jerusalem, PI Ran Nathan) and MPIO joint Deutsch-Israelisches Project Grant and the Storchenhof Loburg, Turkey Vulture data, Keith Bildstein and the Hawk Mountain Sanctuary, Waved Albatross data, Sebastian Cruz and the Galapagos National Park Service, Straw-coloured fruit bat, Dina Dechmann, Jakob Fahr and Martin Wikelski. We also thank three anonym-ous reviewers for their valuable comments.
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. 4.A The performance of the dBGB on correlated random walks with increased location errors
Figure 4.A.1: Repetition of the analyses with a higher location error (0.05) showing the effect on the directionality index.
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Figure 4.A.2: Repetition of the analyses with a higher location error (0.1) showing the effect on the directionality index.