An analytical description of the time-integrated Brownian bridge
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An analytical description of the time-integrated Brownian bridge Steffie Van Nieuland1 · Jan M. Baetens1 · Hans De Meyer2 · Bernard De Baets1
Received: 3 February 2015 / Revised: 2 May 2015 / Accepted: 4 June 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract In animal movement research, the probability density function (PDF) of the timeintegrated 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) Mathematics Subject Classification
60Gxx · 33B20 · 33C10 · 92B99
1 Introduction A Brownian motion, i.e., a random motion of particles or individuals, conditioned by a starting and ending location is called a Brownian bridge (BB) (Ross 1996). The particle’s position at any moment in time is described by a normal distribution. The (time-independent)
Communicated by Josselin Garnier.
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Steffie Van Nieuland [email protected]
1
KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, 9000 Ghent, Belgium
2
Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281 S9, 9000 Ghent, Belgium
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S. Van Nieuland et al.
marginal probability density at each point is obtained by averaging these normal distributions over time. The probability density function (PDF) of this so-called time-integrated Brownian bridge (TIBB) is used in animal movement research to delineate important regions on the basis of tracking data describing the locations of individuals at certain moments in time (Bullard 1991; Horne et al. 2007). Such a delineation of important regions is realized by the Brownian Bridge Movement Model (BBMM) through the use of TIBBs (Horne et al. 2007; Nielson et al. 2012). More specifically, the BBMM constructs the PDF of a weighted average of TIBBs between every consecutive pair of observations (Bullard 1991; Horne et al. 2007). The resulting PDF describes the probability density that an animal is located at a certain point at an arbitrary moment in time within the considered time interval. It is determined by the registered locations, the time between tho
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