Abstract
In animal movement research, the probability density function (PDF) of the time-integrated Brownian bridge (TIBB) is used to delineate important regions on the basis of tracking data. Here, it is assumed that an animal performs a Brownian bridge between the data points. As such, the location at any moment in time of an individual performing a Brownian bridge is described by a normal distribution. The (time-independent) marginal probability density at a given point, i.e., the value of the PDF of the TIBB at that point, is obtained by averaging these normal distributions over time. To the best of our knowledge, the PDF of the TIBB is thus far always computed through the use of numerical integration methods. Here, we demonstrate that it is nevertheless possible to derive its analytical expression. Although the two-dimensional setting is the most interesting one for animal movement studies, also the one- and, in general, the n-dimensional setting are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz M, Stegun IA (1972) Error Function and Fresnel Integrals. Ch. 7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York
Benhamou S (2011) Dynamic approach to space and habitat use based on biased random bridges. PLoS One 6:e14592
Borodin A (1989) Brownian local time. Russian Math Surv 44:1–51
Buchin K, Sijben S, Arseneau T, Willems E (2012) Detecting movement patterns using Brownian bridges. Proceedings of the 20th International Conference on Advances in Geographic Information Systems. Redondo Beach, CA, USA, pp 119–128
Bullard F (1991) Estimating the home range of an animal: a Brownian bridge approach. Master’s thesis, Johns Hopkins University
Byrne M, Clint M, Hinton J, Chamberlain M, Collier B (2014) Using dynamic Brownian bridge movement modelling to measure temporal patterns of habitat selection. J Anim Ecol 83:1234–1243
Chi Z, Pozdnyakov V, Yan J (2015) On expected occupation time of brownian bridge. Stat Prob Lett 97:83–87
Gradshteyn I, Ryzhik I (2007) Table of integrals, series, and products, 7th edn. Academic Press, New York
Horne J, Garton E, Krone S, Lewis J (2007) Analyzing animal movements using Brownian bridges. Ecology 88:2354–2363
Janssen R (2011) Vondst kraamkolonie Bechsteins vleermuis. Nieuweling op rode lijst? Zoogdier 22:13–16
Kranstauber B, Smolla M (2014) Visualizing and analyzing animal track data. http://computational-ecology.com/main-move.htmlcran.r-project.org/web/packages/move/index.html, r package version 1.2.475
Kranstauber B, Kays R, LaPoint S, Wilkelski M, Safi K (2012) A dynamic Brownian bridge movement model to estimate utilization distributions for heterogeneous animal movement. Anim Ecol 81:738–746
Kranstauber B, Safi K, Bartumeus F (2014) Bivariate Gaussian bridges: directional factorization of diffusion in Brownian bridge models. Mov Ecol 2:5
Miller H (2010) The data avalanche is here. Shouldn’t we be digging? J Reg Sci 50:181–201
Nielson MR, Sawyer H, McDonald TL (2012) BBMM: Brownian bridge movement model. http://CRAN.R-project.org/package=BBMM, r package version 2.3
Noog E, Geller M (1969) A table of integrals of the error functions. J Res Nat Bur Stand-B Math Sci 73:1–20
Pitman J (1998) The distribution of local times of a Brownian bridge. Tech. Rep. 539, Department of Statistics, University of California
Pozdnyakov v, Meyer T, Wang Y, Yan J (2014) On modeling animal movements using brownian motion with measurement error. Ecology 95:247–253
Pàges JF, Bartumeus F, Hereu B, López-Sanz À, Romero J, Alcoverro T (2013) Evaluating a key herbivorous fish as a mobile link: a Brownian bridge approach. Marine Ecol Prog Ser 492:199–210
Ross S (1996) Stochastic processes, 2nd edn. Wiley, New York
Sawyer H, Kauffman M, Nielson R, Horne J (2009) Identifying and prioritizing ungulate migration routes for landscape-level conservation. Ecol Appl 19:2016–2025
Walck C (1996) Handbook on statistical distributions for experimentalists. Particle Physics Group Fysikum—University of Stockholm
Wikelski M, Kays R (2014) Movebank: archive, analysis and sharing of animal movement data. World Wide Web electronic publication. http://www.movebank.org, Accessed on 25 Sep 2014
Yan J, Chen Y, Lawrence-Apfel K, Ortega I, Pozdnyakov V, Williams S, Meyer T (2014) A moving-resting process with an embedded Brownian motion for animal movements. Popul Ecol 56:401–415
Acknowledgments
The computational resources (Stevin Supercomputer Infrastructure) and services used in this work were provided by Ghent University, the Hercules Foundation, and the Flemish Government—department EWI (Economie, Wetenschap en Innovatie). A special thanks goes to René Janssens for providing data on Bechstein’s bat.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Josselin Garnier.
Rights and permissions
About this article
Cite this article
Van Nieuland, S., Baetens, J.M., De Meyer, H. et al. An analytical description of the time-integrated Brownian bridge. Comp. Appl. Math. 36, 627–645 (2017). https://doi.org/10.1007/s40314-015-0250-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-015-0250-3