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Background The ability to observe animal movement and possible correlates has increased strongly over the past decades. Methods to analyze trajectories have developed in parallel, but many tools fail to make an immediate connection between a movement model, covariates of the movement, and animal space use. Methods Here I develop a novel method based on the Brownian Bridge Movement Model that facilitates investigating and testing covariates of movement. The model makes it possible to flexibly investigate different covariates including, for example, periodic movement patterns. Results I applied the Brownian Bridge Covariates Model (BBCM) to simulated trajectories demonstrating its ability to reproduce the parameters used for the simulation. I also applied the model to a GPS trajectory of a meerkat, showing its application to empirical data. The value of the model was shown by testing the interaction between maximal daily temperature and the daily movement pattern. Conclusion This model produces accurate parameter estimates for covariates of the movements and location error in simulated trajectories. Application to the meerkat trajectory also produced plausible parameter estimates. This new method opens the possibility to directly test hypotheses about the influence of covariates on animal movement while linking these to space-use estimates.

We propose a method to infer the presence and location of change-points in the distribution of a sequence of independent data taking values in a general metric space, where change-points are viewed as locations at which the distribution of the data sequence changes abruptly in terms of either its Fréchet mean, Fréchet variance or both. The proposed method is based on comparisons of Fréchet variances before and after putative change-point locations and does not require a tuning parameter, except for the specification of cut-off intervals near the endpoints where change-points are assumed not to occur. Our results include theoretical guarantees for consistency of the test under contiguous alternatives when a change-point exists and also for consistency of the estimated location of the change-point, if it exists, where, under the null hypothesis of no change-point, the limit distribution of the proposed scan function is the square of a standardized Brownian bridge. These consistency results are applicable for a broad class of metric spaces under mild entropy conditions. Examples include the space of univariate probability distributions and the space of graph Laplacians for networks. Simulation studies demonstrate the effectiveness of the proposed methods, both for inferring the presence of a change-point and estimating its location. We also develop theory that justifies bootstrap-based inference and illustrate the new approach with sequences of maternal fertility distributions and communication networks.

Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(±T) = 0 conditioned to stay above the semicircle$c_{T}(t)=\\sqrt{T^{2}-t^{2}}$. In the limit of large T, the fluctuation scale of b(t)-cT(t) is T1/3and its time-correlation scale is T2/3. We prove that, in the sense of weak convergence of path measures, the conditioned Brownian bridge, when properly rescaled, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point t = τT, τ ∈ (-1, 1), is only through the second derivative of cT(t) at t = τT. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height Tγ, γ > 1/2. The fluctuation scale is then T(2-γ)/3. More general conditioning shapes are briefly discussed.

Quantifying animals' home ranges is a key problem in ecology and has important conservation and wildlife management applications. Kernel density estimation (KDE) is a workhorse technique for range delineation problems that is both statistically efficient and nonparametric. KDE assumes that the data are independent and identically distributed (IID). However, animal tracking data, which are routinely used as inputs to KDEs, are inherently autocorrelated and violate this key assumption. As we demonstrate, using realistically autocorrelated data in conventional KDEs results in grossly underestimated home ranges. We further show that the performance of conventional KDEs actually degrades as data quality improves, because autocorrelation strength increases as movement paths become more finely resolved. To remedy these flaws with the traditional KDE method, we derive an autocorrelated KDE (AKDE) from first principles to use autocorrelated data, making it perfectly suited for movement data sets. We illustrate the vastly improved performance of AKDE using analytical arguments, relocation data from Mongolian gazelles, and simulations based upon the gazelle's observed movement process. By yielding better minimum area estimates for threatened wildlife populations, we believe that future widespread use of AKDE will have significant impact on ecology and conservation biology.

1. The recently developed Brownian bridge movement model (BBMM) has advantages over traditional methods because it quantifies the utilization distribution of an animal based on its movement path rather than individual points and accounts for temporal autocorrelation and high data volumes. However, the BBMM assumes unrealistic homogeneous movement behaviour across all data. 2. Accurate quantification of the utilization distribution is important for identifying the way animals use the landscape. 3. We improve the BBMM by allowing for changes in behaviour, using likelihood statistics to determine change points along the animal's movement path. 4. This novel extension, outperforms the current BBMM as indicated by simulations and examples of a territorial mammal and a migratory bird. The unique ability of our model to work with tracks that are not sampled regularly is especially important for GPS tags that have frequent failed fixes or dynamic sampling schedules. Moreover, our model extension provides a useful one-dimensional measure of behavioural change along animal tracks. 5. This new method provides a more accurate utilization distribution that better describes the space use of realistic, behaviourally heterogeneous tracks.

It was shown by Le Jan that the occupation field of a Poisson ensemble of Markov loops (\"loop soup\") of parameter $\\frac{1}{2}$ associated to a transient symmetric Markov jump process on a network is half the square of the Gaussian free field. We construct a coupling between these loops and the free field such that an additional constraint holds: the sign of the free field is constant on each cluster of loops. As a consequence of our coupling we deduce that the loop clusters of parameter $\\frac{1}{2}$ do not percolate on periodic lattices. We also construct a coupling between the random interlacement on ℤd, d ≥ 3, introduced by Sznitman, and the Gaussian free field, such that the set of vertices visited by the interlacement is contained in a one-sided level set of the free field. We deduce an inequality between the critical level for the percolation by level sets of the free field and the critical parameter for the percolation of the vacant set of the random interlacement.

We outfitted six male Hawaiian geese, or nene (Branta sandvicensis), with 45-g solar-powered satellite transmitters and collected four location coordinates d−1 from 2010 to 2012. We used 6193 coordinates to characterize migration corridors, habitat preferences and temporal patterns of displacement for 16 migration events with Brownian bridge utilization distributions (BBUD). We used 1552 coordinates to characterize stopovers from 37 shorter-distance movement events with 25% BBUDs. Two subpopulations used a well-defined common migration corridor spanning a broad gradient of elevation. Use of native-dominated subalpine shrubland was 2.81 times more likely than the availability of this land-cover type. The nene differed from other tropical and temperate-zone migrant birds in that: (1) migration distance and the number of stopovers were unrelated (Mann–Whitney test W = 241, P < 0.006), and; (2) individual movements were not unidirectional suggesting that social interactions may be more important than refuelling en route; but like other species, nene made more direct migrations with fewer stopovers in return to breeding areas (0.58 ± 0.50) than in migration away from breeding areas (1.64 ± 0.48). Our findings, combined with the direction and timing of migration, which is opposite that of most other intratropical migrants, suggest fundamentally different drivers of altitudinal migration.

We propose a method for estimating first passage time densities of one-dimensional diffusions via Monte Carlo simulation. Our approach involves a representation of the first passage time density as the expectation of a functional of the three-dimensional Brownian bridge. As the latter process can be simulated exactly, our method leads to almost unbiased estimators. Furthermore, since the density is estimated directly, a convergence of order 1 / √N, where N is the sample size, is achieved, which is in sharp contrast to the slower nonparametric rates achieved by kernel smoothing of cumulative distribution functions.

By studying animal movements, researchers can gain insight into many of the ecological characteristics and processes important for understanding population-level dynamics. We developed a Brownian bridge movement model (BBMM) for estimating the expected movement path of an animal, using discrete location data obtained at relatively short time intervals. The BBMM is based on the properties of a conditional random walk between successive pairs of locations, dependent on the time between locations, the distance between locations, and the Brownian motion variance that is related to the animal's mobility. We describe two critical developments that enable widespread use of the BBMM, including a derivation of the model when location data are measured with error and a maximum likelihood approach for estimating the Brownian motion variance. After the BBMM is fitted to location data, an estimate of the animal's probability of occurrence can be generated for an area during the time of observation. To illustrate potential applications, we provide three examples: estimating animal home ranges, estimating animal migration routes, and evaluating the influence of fine-scale resource selection on animal movement patterns.

We consider a one-dimensional superprocess with a supercritical local branching mechanism $\\psi$ , where particles move as a Brownian motion with drift $-\\rho$ and are killed when they reach the origin. It is known that the process survives with positive probability if and only if $\\rho<\\sqrt{2\\alpha}$ , where $\\alpha=-\\psi'(0)$ . When $\\rho<\\sqrt{2 \\alpha}$ , Kyprianou et al. [18] proved that $\\lim_{t\\to \\infty}R_t/t =\\sqrt{2\\alpha}-\\rho$ almost surely on the survival set, where $R_t$ is the rightmost position of the support at time t. Motivated by this work, we investigate its large deviation, in other words, the convergence rate of $\\mathbb{P}_{\\delta_x} (R_t >\\gamma t+\\theta)$ as $t \\to \\infty$ , where $\\gamma >\\sqrt{2 \\alpha} -\\rho$ , $\\theta \\ge 0$ . As a by-product, a related Yaglom-type conditional limit theorem is obtained. Analogous results for branching Brownian motion can be found in Harris et al. [13].