A Stochastic Diffusion Process for the Dirichlet Distribution

Bakosi, J.; Ristorcelli, J. R.

doi:10.1155/2013/842981

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 ofNcoupled 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 ofNvariables 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:

Bakosi, J. ^[1]; Ristorcelli, J. R. ^[1]

Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Publication Date:: Wed Apr 10 00:00:00 EDT 2013

Research Org.:: Los Alamos National Laboratory (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.

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                    Bakosi, J., & Ristorcelli, J. R. A Stochastic Diffusion Process for the Dirichlet Distribution.  Egypt.  https://doi.org/10.1155/2013/842981

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                    Bakosi, J., and Ristorcelli, J. R. Wed .  
"A Stochastic Diffusion Process for the Dirichlet Distribution".  Egypt.  https://doi.org/10.1155/2013/842981.

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@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 ofNcoupled 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 ofNvariables 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         = {Wed Apr 10 00:00:00 EDT 2013},

  month        = {Wed Apr 10 00:00:00 EDT 2013}

}

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Works referenced in this record:

PDF methods for turbulent reactive flows
journal, January 1985

Pope, S. B.
Progress in Energy and Combustion Science, Vol. 11, Issue 2
DOI: 10.1016/0360-1285(85)90002-4

Mathematical contributions to the theory of evolution.—On a form of spurious correlation which may arise when indices are used in the measurement of organs
journal, January 1897

Pearson, Karl
Proceedings of the Royal Society of London, Vol. 60, Issue 359-367, p. 489-498
DOI: 10.1098/rspl.1896.0076

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

Johnson, N. L.
Biometrika, Vol. 47, Issue 1-2
DOI: 10.1093/biomet/47.1-2.93

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

Lange, Kenneth
Genetica, Vol. 96, Issue 1-2
DOI: 10.1007/BF01441156

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

Gourieroux, Christian; Jasiak, Joann
Journal of Econometrics, Vol. 131, Issue 1-2
DOI: 10.1016/j.jeconom.2005.01.014

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

Forman, Julie Lyng; SØRensen, Michael
Scandinavian Journal of Statistics, Vol. 35, Issue 3
DOI: 10.1111/j.1467-9469.2007.00592.x

On the compound multinomial distribution, the multivariate β-distribution, and correlations among proportions
journal, January 1962

Mosimann, James E.
Biometrika, Vol. 49, Issue 1-2
DOI: 10.1093/biomet/49.1-2.65

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

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Title: A Stochastic Diffusion Process for the Dirichlet Distribution

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