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J Klafter - One of the best experts on this subject based on the ideXlab platform.

  • from diffusion to anomalous diffusion a century after einstein s brownian motion
    Chaos, 2005
    Co-Authors: I M Sokolov, J Klafter
    Abstract:

    Einstein’s explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian Property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker–Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

  • from diffusion to anomalous diffusion a century after einstein s brownian motion
    arXiv: Statistical Mechanics, 2004
    Co-Authors: I M Sokolov, J Klafter
    Abstract:

    Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian Property and exhibit to anomalous diffusion. We consider here the case of subdiffusive processes, which are semi-Markovian and correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to know fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power-law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

I M Sokolov - One of the best experts on this subject based on the ideXlab platform.

  • from diffusion to anomalous diffusion a century after einstein s brownian motion
    Chaos, 2005
    Co-Authors: I M Sokolov, J Klafter
    Abstract:

    Einstein’s explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian Property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker–Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

  • from diffusion to anomalous diffusion a century after einstein s brownian motion
    arXiv: Statistical Mechanics, 2004
    Co-Authors: I M Sokolov, J Klafter
    Abstract:

    Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian Property and exhibit to anomalous diffusion. We consider here the case of subdiffusive processes, which are semi-Markovian and correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to know fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power-law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

James R. Morrison - One of the best experts on this subject based on the ideXlab platform.

  • Markovian Property for the delays in multiclass deterministic flow lines with random arrivals
    Conference on Automation Science and Engineering, 2017
    Co-Authors: Sangyoon Bae, James R. Morrison
    Abstract:

    We consider multi-class deterministic service flow lines with random arrivals. A Markovian Property for the customer delays is provided. The Property generalizes previous results and subsumes them. It may lead to methods for determining the equilibrium probabilities in such systems.

  • Markovian modeling of multiclass deterministic flow lines with random arrivals: The case of a single-channel
    2015 IEEE International Conference on Automation Science and Engineering (CASE), 2015
    Co-Authors: James R. Morrison
    Abstract:

    Although there has been some success in the exact analysis of tandem queueing networks with finite intermediate buffers, equilibrium probabilities for the waiting time of customers remain elusive. Recently, for deterministic flow lines with random arrivals and a single customer class, exact channel decomposition has enabled Markovian modeling of the waiting time probabilities. Although exact channel decomposition results have been obtained for certain types of multi-class deterministic flow lines, stochastic analysis of customer delays remains unresolved. Here we demonstrate that certain types of single channel multi-class flow lines also possess a Markovian Property for their customer delays. The explicit recursive relationship between the delays from one customer to the next is developed. Due to the complexity of the recursive relationship, we provide some guidance for constructing the state space and transition probabilities of a Markov chain modeling the delays. A computational example is provided. As flow lines can serve as good models for certain types of semiconductor manufacturing equipment, the results may ultimately lead to useful analytic models for such systems.

Friel Nial - One of the best experts on this subject based on the ideXlab platform.

  • Choosing the number of groups in a latent stochastic block model for dynamic networks
    'Cambridge University Press (CUP)', 2019
    Co-Authors: Rastelli Riccardo, Latouche Pierre, Friel Nial
    Abstract:

    Latent stochastic block models are flexible statistical models that are widely used in social network analysis. In recent years, efforts have been made to extend these models to temporal dynamic networks, whereby the connections between nodes are observed at a number of different times. In this paper we extend the original stochastic block model by using a Markovian Property to describe the evolution of nodes cluster memberships over time. We recast the problem of clustering the nodes of the network into a model-based context, and show that the integrated completed likelihood can be evaluated analytically for a number of likelihood models. Then, we propose a scalable greedy algorithm to maximise this quantity, thereby estimating both the optimal partition and the ideal number of groups in a single inferential framework. Finally we propose applications of our methodology to both real and artificial datasets.Science Foundation IrelandInsight Research Centre2 year embargo to cover until published, checkdate -A

  • Choosing the number of groups in a latent stochastic block model for dynamic networks
    Cambridge Journals, 2018
    Co-Authors: Rastelli Riccardo, Latouche Pierre, Friel Nial
    Abstract:

    International audienceLatent stochastic block models are flexible statistical models that are widely used in social network analysis. In recent years, efforts have been made to extend these models to temporal dynamic networks, whereby the connections between nodes are observed at a number of different times. In this paper we extend the original stochas-tic block model by using a Markovian Property to describe the evolution of nodes' cluster memberships over time. We recast the problem of clustering the nodes of the network into a model-based context, and show that the integrated completed likelihood can be evaluated analytically for a number of likelihood models. Then, we propose a scalable greedy algorithm to maximise this quantity, thereby estimating both the optimal partition and the ideal number of groups in a single inferential framework. Finally we propose applications of our methodology to both real and artificial datasets

Lars S Jermiin - One of the best experts on this subject based on the ideXlab platform.

  • a likelihood ratio test for lumpability of phylogenetic data is the Markovian Property of an evolutionary process retained in recoded dna
    Systematic Biology, 2021
    Co-Authors: Victor A Veraruiz, John Robinson, Lars S Jermiin
    Abstract:

    In molecular phylogenetics, it is typically assumed that the evolutionary process for DNA can be approximated by independent and identically distributed Markovian processes at the variable sites and that these processes diverge over the edges of a rooted bifurcating tree. Sometimes the nucleotides are transformed from a 4-state alphabet to a 3- or 2-state alphabet by a procedure that is called recoding, lumping, or grouping of states. Here, we introduce a likelihood-ratio test for lumpability for DNA that has diverged under different Markovian conditions, which assesses the assumption that the Markovian Property of the evolutionary process over each edge is retained after recoding of the nucleotides. The test is derived and validated numerically on simulated data. To demonstrate the insights that can be gained by using the test, we assessed two published data sets, one of mitochondrial DNA from a phylogenetic study of the ratites (Syst. Biol. 59:90-107 [2010]) and the other of nuclear DNA from a phylogenetic study of yeast (Mol. Biol. Evol. 21:1455-1458 [2004]). Our analysis of these data sets revealed that recoding of the DNA eliminated some of the compositional heterogeneity detected over the sequences. However, the Markovian Property of the original evolutionary process was not retained by the recoding, leading to some significant distortions of edge lengths in reconstructed trees.