Distributions associated with general runs and patterns in hidden Markov models

This paper gives a method for computing distributions associated with patterns in the state sequence of a hidden Markov model, conditional on observing all or part of the observation sequence. Probabilities are computed for very general classes of patterns (competing patterns and generalized later patterns), and thus, the theory includes as special cases results for a large class of problems that have wide application. The unobserved state sequence is assumed to be Markovian with a general order of dependence. An auxiliary Markov chain is associated with the state sequence and is used to simplify the computations. Two examples are given to illustrate the use of the methodology. Whereas the first application is more to illustrate the basic steps in applying the theory, the second is a more detailed application to DNA sequences, and shows that the methods can be adapted to include restrictions related to biological knowledge.Comment: Published in at http://dx.doi.org/10.1214/07-AOAS125 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

How to read probability distributions as statements about process

Probability distributions can be read as simple expressions of information. Each continuous probability distribution describes how information changes with magnitude. Once one learns to read a probability distribution as a measurement scale of information, opportunities arise to understand the processes that generate the commonly observed patterns. Probability expressions may be parsed into four components: the dissipation of all information, except the preservation of average values, taken over the measurement scale that relates changes in observed values to changes in information, and the transformation from the underlying scale on which information dissipates to alternative scales on which probability pattern may be expressed. Information invariances set the commonly observed measurement scales and the relations between them. In particular, a measurement scale for information is defined by its invariance to specific transformations of underlying values into measurable outputs. Essentially all common distributions can be understood within this simple framework of information invariance and measurement scale.Comment: v2: added table of contents, adjusted section numbers v3: minor editing, updated referenc

Generalized Master Equations for Non-Poisson Dynamics on Networks

The traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network. This implicitly assumes that the edges are governed by Poisson processes, which is not typically the case in empirical temporal networks. Consequently, we examine the effects of non-Poisson inter-event statistics on the dynamics of edges, and we apply the concept of a generalized master equation to the study of continuous-time random walks on networks. We show that the equation reduces to the standard rate equations when the underlying process is Poisson and that the stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate. We discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature

Five Years of Continuous-time Random Walks in Econophysics

This paper is a short review on the application of continuos-time random walks to Econophysics in the last five years.Comment: 14 pages. Paper presented at WEHIA 2004, Kyoto, Japa

Truncated Levy statistics for transport in disordered semiconductors

Probabilistic interpretation of transition from the dispersive transport regime to the quasi-Gaussian one in disordered semiconductors is given in terms of truncated Levy distributions. Corresponding transport equations with fractional order derivatives are derived. We discuss physical causes leading to truncated waiting time distributions in the process and describe influence of truncation on carrier packet form, transient current curves and frequency dependence of conductivity. Theoretical results are in a good agreement with experimental facts.Comment: 6 pages, 4 figures, presented in "Nonlinear Science and Complexity - 2010" (Turkey, Ankara

L\'evy walks

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The L\'{e}vy walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and behavioral science demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction to L\'{e}vy walks, surveys their existing applications, including latest advances, and outlines further perspectives.Comment: 50 page