A stochastic diffusion process for Lochner's generalized Dirichlet distribution
Abstract
The method of potential solutions of FokkerPlanck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner’s generalized Dirichlet distribution as its asymptotic solution. Individual samples of a discrete ensemble, obtained from the system of stochastic differential equations, equivalent to the FokkerPlanck equation developed here, satisfy a unitsum constraint at all times and ensure a bounded sample space, similarly to the process developed in for the Dirichlet distribution. Consequently, the generalized Dirichlet diffusion process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Compared to the Dirichlet distribution and process, the additional parameters of the generalized Dirichlet distribution allow a more general class of physical processes to be modeled with a more general covariance matrix.
 Authors:

 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1233161
 Report Number(s):
 LAUR1321573
Journal ID: ISSN 00222488; JMAPAQ
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Mathematical Physics
 Additional Journal Information:
 Journal Volume: 54; Journal Issue: 10; Journal ID: ISSN 00222488
 Publisher:
 American Institute of Physics (AIP)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; FokkerPlanck equation, Stochastic diffusion, Generalized Dirichlet distribution
Citation Formats
Bakosi, J., and Ristorcelli, J. R. A stochastic diffusion process for Lochner's generalized Dirichlet distribution. United States: N. p., 2013.
Web. doi:10.1063/1.4822416.
Bakosi, J., & Ristorcelli, J. R. A stochastic diffusion process for Lochner's generalized Dirichlet distribution. United States. https://doi.org/10.1063/1.4822416
Bakosi, J., and Ristorcelli, J. R. Tue .
"A stochastic diffusion process for Lochner's generalized Dirichlet distribution". United States. https://doi.org/10.1063/1.4822416. https://www.osti.gov/servlets/purl/1233161.
@article{osti_1233161,
title = {A stochastic diffusion process for Lochner's generalized Dirichlet distribution},
author = {Bakosi, J. and Ristorcelli, J. R.},
abstractNote = {The method of potential solutions of FokkerPlanck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner’s generalized Dirichlet distribution as its asymptotic solution. Individual samples of a discrete ensemble, obtained from the system of stochastic differential equations, equivalent to the FokkerPlanck equation developed here, satisfy a unitsum constraint at all times and ensure a bounded sample space, similarly to the process developed in for the Dirichlet distribution. Consequently, the generalized Dirichlet diffusion process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Compared to the Dirichlet distribution and process, the additional parameters of the generalized Dirichlet distribution allow a more general class of physical processes to be modeled with a more general covariance matrix.},
doi = {10.1063/1.4822416},
journal = {Journal of Mathematical Physics},
number = 10,
volume = 54,
place = {United States},
year = {2013},
month = {10}
}
Web of Science
Works referenced in this record:
Numerical Correlation and Petrographic Variation
journal, July 1962
 Chayes, F.
 The Journal of Geology, Vol. 70, Issue 4
A stochastic diffusion process for the Dirichlet distribution
text, January 2013
 Bakosi, J.; Ristorcelli, J. R.
 arXiv
A Stochastic Diffusion Process for the Dirichlet Distribution
journal, April 2013
 Bakosi, J.; Ristorcelli, J. R.
 International Journal of Stochastic Analysis, Vol. 2013
Concepts of Independence for Proportions with a Generalization of the Dirichlet Distribution
journal, March 1969
 Connor, Robert J.; Mosimann, James E.
 Journal of the American Statistical Association, Vol. 64, Issue 325
Generalized Dirichlet distribution in Bayesian analysis
journal, December 1998
 Wong, TzuTsung
 Applied Mathematics and Computation, Vol. 97, Issue 23
An approximation to the multinomial distribution some properties and applications
journal, January 1960
 Johnson, N. L.
 Biometrika, Vol. 47, Issue 12
On the compound multinomial distribution, the multivariate βdistribution, and correlations among proportions
journal, January 1962
 Mosimann, James E.
 Biometrika, Vol. 49, Issue 12
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 359367, p. 489498
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
Numerical Correlation and Petrographic Variation
journal, July 1962
 Chayes, F.
 The Journal of Geology, Vol. 70, Issue 4
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
Applications of the Dirichlet distribution to forensic match probabilities
journal, June 1995
 Lange, Kenneth
 Genetica, Vol. 96, Issue 12
Multivariate Jacobi process with application to smooth transitions
journal, March 2006
 Gourieroux, Christian; Jasiak, Joann
 Journal of Econometrics, Vol. 131, Issue 12
Assumed βpdf Model for Turbulent Mixing: Validation and Extension to Multiple Scalar Mixing
journal, August 1991
 Girimaji, S. S.
 Combustion Science and Technology, Vol. 78, Issue 46
An explicit transition density expansion for a multiallelic Wright–Fisher diffusion with general diploid selection
journal, February 2013
 Steinrücken, Matthias; Wang, Y. X. Rachel; Song, Yun S.
 Theoretical Population Biology, Vol. 83
Exploring the beta distribution in variabledensity turbulent mixing
journal, January 2010
 Bakosi, J.; Ristorcelli, J. R.
 Journal of Turbulence, Vol. 11
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
Works referencing / citing this record:
Diffusion Processes Satisfying a Conservation Law Constraint
journal, March 2014
 Bakosi, J.; Ristorcelli, J. R.
 International Journal of Stochastic Analysis, Vol. 2014