On numerical calculation of probabilities according to Dirichlet distribution
The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation of the probability values according to the underlying multivariate probability distribution. In addition, when we are using a nonlinear programming algorithm for the solution of...
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| Veröffentlicht in: | Annals of operations research 2010-06, Vol.177 (1), p.185-200 |
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| creator | Gouda, Ashraf A. Szántai, Tamás |
| description | The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation of the probability values according to the underlying multivariate probability distribution. In addition, when we are using a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives may also be necessary.
In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of these formulae is that they involve only lower dimensional cumulative distribution function calculations. Numerical test results will also be presented. |
| doi_str_mv | 10.1007/s10479-009-0601-9 |
| format | Article |
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In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of these formulae is that they involve only lower dimensional cumulative distribution function calculations. Numerical test results will also be presented.</description><identifier>ISSN: 0254-5330</identifier><identifier>EISSN: 1572-9338</identifier><identifier>DOI: 10.1007/s10479-009-0601-9</identifier><language>eng</language><publisher>Boston: Springer US</publisher><subject>Algorithms ; Business and Management ; Combinatorics ; Consumers ; Life cycles ; Monte Carlo method ; Monte Carlo simulation ; Nonlinear programming ; Operations research ; Operations Research/Decision Theory ; Probability ; Probability distribution ; Random variables ; Studies ; Theory of Computation</subject><ispartof>Annals of operations research, 2010-06, Vol.177 (1), p.185-200</ispartof><rights>Springer Science+Business Media, LLC 2009</rights><rights>COPYRIGHT 2010 Springer</rights><rights>Springer Science+Business Media, LLC 2010</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c450t-feaa7109138046ee348d17afa3e7d71ea3fe4766bdbdf82cdc263f140201594a3</citedby><cites>FETCH-LOGICAL-c450t-feaa7109138046ee348d17afa3e7d71ea3fe4766bdbdf82cdc263f140201594a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10479-009-0601-9$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10479-009-0601-9$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Gouda, Ashraf A.</creatorcontrib><creatorcontrib>Szántai, Tamás</creatorcontrib><title>On numerical calculation of probabilities according to Dirichlet distribution</title><title>Annals of operations research</title><addtitle>Ann Oper Res</addtitle><description>The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation of the probability values according to the underlying multivariate probability distribution. In addition, when we are using a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives may also be necessary.
In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of these formulae is that they involve only lower dimensional cumulative distribution function calculations. 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In addition, when we are using a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives may also be necessary.
In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of these formulae is that they involve only lower dimensional cumulative distribution function calculations. Numerical test results will also be presented.</abstract><cop>Boston</cop><pub>Springer US</pub><doi>10.1007/s10479-009-0601-9</doi><tpages>16</tpages></addata></record> |
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| subjects | Algorithms Business and Management Combinatorics Consumers Life cycles Monte Carlo method Monte Carlo simulation Nonlinear programming Operations research Operations Research/Decision Theory Probability Probability distribution Random variables Studies Theory of Computation |
| title | On numerical calculation of probabilities according to Dirichlet distribution |
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