113,173 research outputs found
Nonlinear Markov Processes in Big Networks
Big networks express various large-scale networks in many practical areas
such as computer networks, internet of things, cloud computation, manufacturing
systems, transportation networks, and healthcare systems. This paper analyzes
such big networks, and applies the mean-field theory and the nonlinear Markov
processes to set up a broad class of nonlinear continuous-time block-structured
Markov processes, which can be applied to deal with many practical stochastic
systems. Firstly, a nonlinear Markov process is derived from a large number of
interacting big networks with symmetric interactions, each of which is
described as a continuous-time block-structured Markov process. Secondly, some
effective algorithms are given for computing the fixed points of the nonlinear
Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff
center, the Lyapunov functions and the relative entropy are used to analyze
stability or metastability of the big network, and several interesting open
problems are proposed with detailed interpretation. We believe that the results
given in this paper can be useful and effective in the study of big networks.Comment: 28 pages in Special Matrices; 201
Fast MCMC sampling for Markov jump processes and extensions
Markov jump processes (or continuous-time Markov chains) are a simple and
important class of continuous-time dynamical systems. In this paper, we tackle
the problem of simulating from the posterior distribution over paths in these
models, given partial and noisy observations. Our approach is an auxiliary
variable Gibbs sampler, and is based on the idea of uniformization. This sets
up a Markov chain over paths by alternately sampling a finite set of virtual
jump times given the current path and then sampling a new path given the set of
extant and virtual jump times using a standard hidden Markov model forward
filtering-backward sampling algorithm. Our method is exact and does not involve
approximations like time-discretization. We demonstrate how our sampler extends
naturally to MJP-based models like Markov-modulated Poisson processes and
continuous-time Bayesian networks and show significant computational benefits
over state-of-the-art MCMC samplers for these models.Comment: Accepted at the Journal of Machine Learning Research (JMLR
Using imprecise continuous time Markov chains for assessing the reliability of power networks with common cause failure and non-immediate repair.
We explore how imprecise continuous time Markov
chains can improve traditional reliability models based
on precise continuous time Markov chains. Specifically,
we analyse the reliability of power networks under very
weak statistical assumptions, explicitly accounting for
non-stationary failure and repair rates and the limited
accuracy by which common cause failure rates can be
estimated. Bounds on typical quantities of interest
are derived, namely the expected time spent in system
failure state, as well as the expected number of
transitions to that state. A worked numerical example
demonstrates the theoretical techniques described.
Interestingly, the number of iterations required for
convergence is observed to be much lower than current
theoretical bounds
Elimination of Intermediate Species in Multiscale Stochastic Reaction Networks
We study networks of biochemical reactions modelled by continuous-time Markov
processes. Such networks typically contain many molecular species and reactions
and are hard to study analytically as well as by simulation. Particularly, we
are interested in reaction networks with intermediate species such as the
substrate-enzyme complex in the Michaelis-Menten mechanism. These species are
virtually in all real-world networks, they are typically short-lived, degraded
at a fast rate and hard to observe experimentally.
We provide conditions under which the Markov process of a multiscale reaction
network with intermediate species is approximated in finite dimensional
distribution by the Markov process of a simpler reduced reaction network
without intermediate species. We do so by embedding the Markov processes into a
one-parameter family of processes, where reaction rates and species abundances
are scaled in the parameter. Further, we show that there are close links
between these stochastic models and deterministic ODE models of the same
networks
Fast MCMC sampling for Markov jump processes and continuous time Bayesian networks
Markov jump processes and continuous time Bayesian networks are important
classes of continuous time dynamical systems. In this paper, we tackle the
problem of inferring unobserved paths in these models by introducing a fast
auxiliary variable Gibbs sampler. Our approach is based on the idea of
uniformization, and sets up a Markov chain over paths by sampling a finite set
of virtual jump times and then running a standard hidden Markov model forward
filtering-backward sampling algorithm over states at the set of extant and
virtual jump times. We demonstrate significant computational benefits over a
state-of-the-art Gibbs sampler on a number of continuous time Bayesian
networks
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