Statistical analysis of sets of random walks: how to resolve their generating mechanism

• Published in: Bulletin of mathematical biology 2007-11, Vol.69 (8), p.2467-2492
• Main Authors: Coscoy, Sylvie, Huguet, Etienne, Amblard, François
• Format: Article
• Language: English
• Subjects: Animals; Anisotropy; Biometry; Computer Simulation; Diffusion; LLC-PK1 Cells; Mathematics; Microvilli - physiology; Models, Biological; Movement - physiology; Stochastic Processes; Swine
• ISSN: 0092-8240; 1522-9602
• DOI: 10.1007/s11538-007-9227-8

The analysis of experimental random walks aims at identifying the process(es) that generate(s) them. It is in general a difficult task, because statistical dispersion within an experimental set of random walks is a complex combination of the stochastic nature of the generating process, and the possibility to have more than one simple process. In this paper, we study by numerical simulations how the statistical distribution of various geometric descriptors such as the second, third and fourth order moments of two-dimensional random walks depends on the stochastic process that generates that set. From these observations, we derive a method to classify complex sets of random walks, and resolve the generating process(es) by the systematic comparison of experimental moment distributions with those numerically obtained for candidate processes. In particular, various processes such as Brownian diffusion combined with convection, noise, confinement, anisotropy, or intermittency, can be resolved by using high order moment distributions. In addition, finite-size effects are observed that are useful for treating short random walks. As an illustration, we describe how the present method can be used to study the motile behavior of epithelial microvilli. The present work should be of interest in biology for all possible types of single particle tracking experiments.