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An analytical description of the time-integrated Brownian bridge

Steffie Van Nieuland (UGent) , Jan Baetens (UGent) , Hans De Meyer (UGent) and Bernard De Baets (UGent)
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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.
Keywords
(Time-integrated) Brownian bridge, (Animal) Movement, Brownian bridge movement model (BBMM), ANIMAL MOVEMENTS, MOTION

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Please use this url to cite or link to this publication:

Chicago
Van Nieuland, Steffie, Jan Baetens, Hans De Meyer, and Bernard De Baets. 2017. “An Analytical Description of the Time-integrated Brownian Bridge.” Computational & Applied Mathematics 36 (1): 627–645.
APA
Van Nieuland, S., Baetens, J., De Meyer, H., & De Baets, B. (2017). An analytical description of the time-integrated Brownian bridge. COMPUTATIONAL & APPLIED MATHEMATICS, 36(1), 627–645.
Vancouver
1.
Van Nieuland S, Baetens J, De Meyer H, De Baets B. An analytical description of the time-integrated Brownian bridge. COMPUTATIONAL & APPLIED MATHEMATICS. 2017;36(1):627–45.
MLA
Van Nieuland, Steffie et al. “An Analytical Description of the Time-integrated Brownian Bridge.” COMPUTATIONAL & APPLIED MATHEMATICS 36.1 (2017): 627–645. Print.
@article{8517552,
  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.},
  author       = {Van Nieuland, Steffie and Baetens, Jan and De Meyer, Hans and De Baets, Bernard},
  issn         = {0101-8205},
  journal      = {COMPUTATIONAL \& APPLIED MATHEMATICS},
  language     = {eng},
  number       = {1},
  pages        = {627--645},
  title        = {An analytical description of the time-integrated Brownian bridge},
  url          = {http://dx.doi.org/10.1007/s40314-015-0250-3},
  volume       = {36},
  year         = {2017},
}

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