An analytical description of the time-integrated Brownian bridge

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

Differential post-fledging habitat use of Nearctic-Neotropical migratory birds within an urbanized landscape

“…For example, [ 83 ] estimated home range sizes of fledgling wood thrushes ranged between 0.026 and 0.25 km 2 whereas in our study estimated home range sizes ranged between 0.019 and 8.7 km 2 . In light of the development of improved analysis methods for estimating habitat use [ 69 , 72 , 86 ], observed differences are of course, understandable. Although these ranges overlap at the lower end, the discrepancy in differing upper ranges is likely due to differences in the landscapes and habitat composition between the two studies.…”

“…It is determined by the registered locations, the time between those locations and the animal’s mobility, which are not accounted for in some alternative approaches [ 70 , 71 ]. Here, the PDF is constructed for each individual and for each period pre- and post-independence using the analytical solution proposed by [ 72 ] using Mathematica (ver. 11.1.1.0).…”