Stochastic process

About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.

A Lévy flight for light

Abstract: Translucent materials such as milk, clouds and biological tissues owe their appearance to the way they interact with light, randomly scattering an incident ray many times before it re-emerges. This process — analogous to the brownian motion of particles in a fluid — is called a random walk, a concept central to statistical physics. It is used, for example, to describe the diffusion of heat, light and sound. An extension of this idea is the Levy flight, where a moving entity can occasionally take unusually large steps, thereby transforming a system's behaviour. Levy flights have been recognized in systems as diverse as earthquakes and animal food searches. Barthelemy et al. have now engineered such behaviour into an optical material (titanium dioxide particles in a glass matrix). In the resulting 'Levy glass', rather than regular diffusion, light waves perform a Levy flight, in which photons spread around extremely efficiently. This will be an ideal model for studying Levy flights, and may also lead to novel optical materials. The cover the photons' path, with the light source top right. Photo by Diederik and Leonardo Wiersma An extension of the concept of a random walk is the Levy flight, in which the moving entity can occasionally take unusually large steps. Pierre Barthelemy and colleagues show how such behaviour can be engineered into an optical material. A random walk is a stochastic process in which particles or waves travel along random trajectories. The first application of a random walk was in the description of particle motion in a fluid (brownian motion); now it is a central concept in statistical physics, describing transport phenomena such as heat, sound and light diffusion1. Levy flights are a particular class of generalized random walk in which the step lengths during the walk are described by a ‘heavy-tailed’ probability distribution. They can describe all stochastic processes that are scale invariant2,3. Levy flights have accordingly turned out to be applicable to a diverse range of fields, describing animal foraging patterns4, the distribution of human travel5 and even some aspects of earthquake behaviour6. Transport based on Levy flights has been extensively studied numerically7,8,9, but experimental work has been limited10,11 and, to date, it has not seemed possible to observe and study Levy transport in actual materials. For example, experimental work on heat, sound, and light diffusion is generally limited to normal, brownian, diffusion. Here we show that it is possible to engineer an optical material in which light waves perform a Levy flight. The key parameters that determine the transport behaviour can be easily tuned, making this an ideal experimental system in which to study Levy flights in a controlled way. The development of a material in which the diffusive transport of light is governed by Levy statistics might even permit the development of new optical functionalities that go beyond normal light diffusion.

Tractable inference for complex stochastic processes

Abstract: The monitoring and control of any dynamic system depends crucially on the ability to reason about its current status and its future trajectory In the case of a stochastic system, these tasks typically involve the use of a belief state--a probability distribution over the state of the process at a given point in time Unfortunately, the state spaces of complex processes are very large, making an explicit representation of a belief state intractable Even in dynamic Bayesian networks (DBNs), where the process itself can be represented compactly, the representation of the belief state is intractable We investigate the idea of maintaining a compact approximation to the true belief state, and analyze the conditions under which the errors due to the approximations taken over the lifetime of the process do not accumulate to make our answers completely irrelevant We show that the error in a belief state contracts exponentially as the process evolves Thus, even with multiple approximations, the error in our process remains bounded indefinitely We show how the additional structure of a DBN can be used to design our approximation scheme, improving its performance significantly We demonstrate the applicability of our ideas in the context of a monitoring task, showing that orders of magnitude faster inference can be achieved with only a small degradation in accuracy

The integer l-shaped method for stochastic integer programs with complete recourse

Abstract: In this paper, a general branch-and-cut procedure for stochastic integer programs with complete recourse and first stage binary variables is presented. It is shown to provide a finite exact algorithm for a number of stochastic integer programs, even in the presence of binary variables or continous random variables in the second stage. (A)