title: Goldstein-Kac telegraph processes with random speeds: Path probabilities, likelihoods, and reported Lévy flights.
abstract: The Goldstein-Kac telegraph process describes the one-dimensional motion of particles with constant speed undergoing random changes in direction. Despite its resemblance to numerous real-world phenomena, the singular nature of the resultant spatial distribution of each particle precludes the possibility of any a posteriori empirical validation of this random-walk model from data. Here we show that by simply allowing for random speeds, the ballistic terms are regularized and that the diffusion component can be well-approximated via the unscented transform. The result is a computationally efficient yet robust evaluation of the full particle path probabilities and, hence, the parameter likelihoods of this generalized telegraph process. We demonstrate how a population diffusing under such a model can lead to non-Gaussian asymptotic spatial distributions, thereby mimicking the behavior of an ensemble of Lévy walkers.
authors.affiliation: Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, SW7 2AZ, United Kingdom.
authors.forename: Aaron
authors.initials: A
authors.lastname: Sim
authors.affiliation: Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, SW7 2AZ, United Kingdom.
authors.forename: Juliane
authors.initials: J
authors.lastname: Liepe
authors.affiliation: Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, SW7 2AZ, United Kingdom.
authors.forename: Michael P H
authors.initials: MP
authors.lastname: Stumpf
mesh_terms: D003198:Computer Simulation; D018803:Models, Economic; D008962:Models, Theoretical; D009038:Motion; D011336:Probability