Milad Rahimi, Mousa Golalizadeh,
Volume 17, Issue 40 (9-2015)
Abstract
Diffusion Processes such as Brownian motions and Ornstein-Uhlenbeck processes are the classes of stochastic processes that have been under considerations of the researchers in various scientific disciplines including biological sciences. It is usually assumed that the outcomes of these processes
are lied on the Euclidean spaces. However, some data are appeared in physical, chemical and biological phenomena that cannot be considered as the observations in Euclidean spaces due to various features
such as the periodicity of the data. Hence, we cannot analysis them using the common mathematical methods available in Euclidean spaces. In addition, studying and analyzing them using common linear statistics are not possible. One of these typical data is the dihedral angles that are utilized to identifying, modeling and predicting the proteins backbones. Because these angles are representatives of points on the surface of torus, it seems that proper statistical modeling of diffusion processes on the torus could be of a great help for the research activities on dynamic molecular simulations in predicting the proteins backbones. In this article, using the Riemannian distance on the torus, the stochastic differential equations to describe the Brownian motions and Ornstein-Uhlenbeck processes on this geometrical objects will be derived. Then, in order to evaluate the proposed models, the statistical simulations will be performed using the equilibrium distributions of aforementioned stochastic processes. Moreover, the link between the gained results with the available concepts in the non-linear statistics will be highlighted.
