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Title: A Stochastic Diffusion Process for the Dirichlet Distribution

Abstract

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.

Authors:
 [1];  [1]
  1. Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1198484
Alternate Identifier(s):
OSTI ID: 1233159
Report Number(s):
LA-UR-12-26980
Journal ID: ISSN 2090-3332; PII: 842981; 842981
Grant/Contract Number:  
AC52-06NA25396
Resource Type:
Published Article
Journal Name:
International Journal of Stochastic Analysis
Additional Journal Information:
Journal Name: International Journal of Stochastic Analysis Journal Volume: 2013; Journal ID: ISSN 2090-3332
Publisher:
Hindawi Publishing Corporation
Country of Publication:
Egypt
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Fokker-Planck equation; Stochastic diffusion; Dirichlet distribution; Monte Carlo simulation

Citation Formats

Bakosi, J., and Ristorcelli, J. R. A Stochastic Diffusion Process for the Dirichlet Distribution. Egypt: N. p., 2013. Web. doi:10.1155/2013/842981.
Bakosi, J., & Ristorcelli, J. R. A Stochastic Diffusion Process for the Dirichlet Distribution. Egypt. doi:10.1155/2013/842981.
Bakosi, J., and Ristorcelli, J. R. Wed . "A Stochastic Diffusion Process for the Dirichlet Distribution". Egypt. doi:10.1155/2013/842981.
@article{osti_1198484,
title = {A Stochastic Diffusion Process for the Dirichlet Distribution},
author = {Bakosi, J. and Ristorcelli, J. R.},
abstractNote = {The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.},
doi = {10.1155/2013/842981},
journal = {International Journal of Stochastic Analysis},
number = ,
volume = 2013,
place = {Egypt},
year = {2013},
month = {4}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1155/2013/842981

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Works referenced in this record:

PDF methods for turbulent reactive flows
journal, January 1985


An approximation to the multinomial distribution some properties and applications
journal, January 1960


Numerical Correlation and Petrographic Variation
journal, July 1962

  • Chayes, F.
  • The Journal of Geology, Vol. 70, Issue 4
  • DOI: 10.1086/626835

Applications of the Dirichlet distribution to forensic match probabilities
journal, June 1995


An explicit transition density expansion for a multi-allelic Wright–Fisher diffusion with general diploid selection
journal, February 2013

  • Steinrücken, Matthias; Wang, Y. X. Rachel; Song, Yun S.
  • Theoretical Population Biology, Vol. 83
  • DOI: 10.1016/j.tpb.2012.10.006

Multivariate Jacobi process with application to smooth transitions
journal, March 2006


Bayesian methods for categorical data under informative general censoring
journal, January 1995

  • Paulino, Carlos Daniel Mimoso; De BraganÇA Pereira, Carlos Alberto
  • Biometrika, Vol. 82, Issue 2
  • DOI: 10.1093/biomet/82.2.439

The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes
journal, September 2008


Geochronology of pluvial Lake Cochise, southern Arizona; [Part] 3, Pollen statistics and Pleistocene metastability
journal, April 1965

  • Martin, P. S.; Mosimann, J. E.
  • American Journal of Science, Vol. 263, Issue 4
  • DOI: 10.2475/ajs.263.4.313