Moment closure: Difference between revisions
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In [[probability theory]], '''moment closure''' is an approximation method used to estimate [[moment (mathematics)|moments]] of a [[stochastic process]].<ref name="c-gillespie">{{Cite journal | last1 = Gillespie | first1 = C. S. | title = Moment-closure approximations for mass-action models | doi = 10.1049/iet-syb:20070031 | journal = IET Systems Biology | volume = 3 | issue = 1 | pages = 52–58 | year = 2009 | pmid = 19154084 |
In [[probability theory]], '''moment closure''' is an approximation method used to estimate [[moment (mathematics)|moments]] of a [[stochastic process]].<ref name="c-gillespie">{{Cite journal | last1 = Gillespie | first1 = C. S. | title = Moment-closure approximations for mass-action models | doi = 10.1049/iet-syb:20070031 | journal = IET Systems Biology | volume = 3 | issue = 1 | pages = 52–58 | year = 2009 | pmid = 19154084}}</ref> |
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==Introduction== |
==Introduction== |
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==History== |
==History== |
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The moment closure approximation was first used by Goodman<ref>{{Cite journal | last1 = Goodman | first1 = L. A. | |
The moment closure approximation was first used by Goodman<ref>{{Cite journal | last1 = Goodman | first1 = L. A. | author-link = Leo Goodman| title = Population Growth of the Sexes | journal = Biometrics | volume = 9 | issue = 2 | pages = 212–225 | doi = 10.2307/3001852 | jstor = 3001852| year = 1953 }}</ref> and Whittle<ref>{{Cite journal | last1 = Whittle | first1 = P. | author-link = Peter Whittle (mathematician)| title = On the Use of the Normal Approximation in the Treatment of Stochastic Processes | journal = Journal of the Royal Statistical Society | volume = 19 | issue = 2 | pages = 268–281 | jstor = 2983819| year = 1957 }}</ref><ref>{{Cite journal | last1 = Matis | first1 = T. | last2 = Guardiola | first2 = I. | doi = 10.3888/tmj.12-2 | title = Achieving Moment Closure through Cumulant Neglect | journal = The Mathematica Journal | volume = 12 | year = 2010 | doi-access = free }}</ref> who set all third and higher-order cumulants to be zero, approximating the population distribution with a [[normal distribution]].<ref name="c-gillespie" /> |
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In 2006, Singh and Hespanha proposed a closure which approximates the population distribution as a [[log-normal distribution]] to describe biochemical reactions.<ref>{{Cite book | last1 = Singh | first1 = A. | last2 = Hespanha | first2 = J. P. | doi = 10.1109/CDC.2006.376994 | chapter = Lognormal Moment Closures for Biochemical Reactions | title = Proceedings of the 45th IEEE Conference on Decision and Control | pages = 2063 | year = 2006 | isbn = 1-4244-0171- |
In 2006, Singh and Hespanha proposed a closure which approximates the population distribution as a [[log-normal distribution]] to describe biochemical reactions.<ref>{{Cite book | last1 = Singh | first1 = A. | last2 = Hespanha | first2 = J. P. | doi = 10.1109/CDC.2006.376994 | chapter = Lognormal Moment Closures for Biochemical Reactions | title = Proceedings of the 45th IEEE Conference on Decision and Control | pages = 2063 | year = 2006 | isbn = 978-1-4244-0171-0 | citeseerx = 10.1.1.130.2031 }}</ref> |
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==Applications== |
==Applications== |
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The approximation has been used successfully to model the spread of the [[Africanized bee]] in the Americas<ref>{{Cite journal | last1 = Matis | first1 = J. H. | last2 = Kiffe | first2 = T. R. | title = On Approximating the Moments of the Equilibrium Distribution of a Stochastic Logistic Model | journal = Biometrics | volume = 52 | issue = 3 | pages = 980–991 | doi = 10.2307/2533059 | jstor = 2533059| year = 1996 | |
The approximation has been used successfully to model the spread of the [[Africanized bee]] in the Americas,<ref>{{Cite journal | last1 = Matis | first1 = J. H. | last2 = Kiffe | first2 = T. R. | title = On Approximating the Moments of the Equilibrium Distribution of a Stochastic Logistic Model | journal = Biometrics | volume = 52 | issue = 3 | pages = 980–991 | doi = 10.2307/2533059 | jstor = 2533059| year = 1996 }}</ref> [[nematode infection]] in [[ruminant]]s.<ref>{{Cite journal | last1 = Marion | first1 = G. | last2 = Renshaw | first2 = E. | last3 = Gibson | first3 = G. | doi = 10.1093/imammb/15.2.97 | title = Stochastic effects in a model of nematode infection in ruminants | journal = Mathematical Medicine and Biology | volume = 15 | issue = 2 | pages = 97 | year = 1998 }}</ref> and [[quantum tunneling ]] in [[ionization]] experiments.<ref>{{cite journal | last=Baytaş | first=Bekir | last2=Bojowald | first2=Martin | last3=Crowe | first3=Sean | title=Canonical tunneling time in ionization experiments | journal=Physical Review A | publisher=American Physical Society (APS) | volume=98 | issue=6 | date=2018-12-17 | issn=2469-9926 | doi=10.1103/physreva.98.063417 | page=063417|arxiv=1810.12804}}</ref> |
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and [[nematode infection]] in [[ruminant]]s.<ref>{{Cite journal | last1 = Marion | first1 = G. | last2 = Renshaw | first2 = E. | last3 = Gibson | first3 = G. | doi = 10.1093/imammb/15.2.97 | title = Stochastic effects in a model of nematode infection in ruminants | journal = Mathematical Medicine and Biology | volume = 15 | issue = 2 | pages = 97 | year = 1998 | pmid = | pmc = }}</ref> |
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==References== |
==References== |
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Latest revision as of 01:03, 5 June 2023
 | This article focuses only on one specialized aspect of the subject. (June 2023) |
In probability theory, moment closure is an approximation method used to estimate moments of a stochastic process.[1]
Introduction
[edit]Typically, differential equations describing the i-th moment will depend on the (i + 1)-st moment. To use moment closure, a level is chosen past which all cumulants are set to zero. This leaves a resulting closed system of equations which can be solved for the moments.[1] The approximation is particularly useful in models with a very large state space, such as stochastic population models.[1]
History
[edit]The moment closure approximation was first used by Goodman[2] and Whittle[3][4] who set all third and higher-order cumulants to be zero, approximating the population distribution with a normal distribution.[1]
In 2006, Singh and Hespanha proposed a closure which approximates the population distribution as a log-normal distribution to describe biochemical reactions.[5]
Applications
[edit]The approximation has been used successfully to model the spread of the Africanized bee in the Americas,[6] nematode infection in ruminants.[7] and quantum tunneling in ionization experiments.[8]
References
[edit]- ^ a b c d Gillespie, C. S. (2009). "Moment-closure approximations for mass-action models". IET Systems Biology. 3 (1): 52–58. doi:10.1049/iet-syb:20070031. PMID 19154084.
- ^ Goodman, L. A. (1953). "Population Growth of the Sexes". Biometrics. 9 (2): 212–225. doi:10.2307/3001852. JSTOR 3001852.
- ^ Whittle, P. (1957). "On the Use of the Normal Approximation in the Treatment of Stochastic Processes". Journal of the Royal Statistical Society. 19 (2): 268–281. JSTOR 2983819.
- ^ Matis, T.; Guardiola, I. (2010). "Achieving Moment Closure through Cumulant Neglect". The Mathematica Journal. 12. doi:10.3888/tmj.12-2.
- ^ Singh, A.; Hespanha, J. P. (2006). "Lognormal Moment Closures for Biochemical Reactions". Proceedings of the 45th IEEE Conference on Decision and Control. p. 2063. CiteSeerX 10.1.1.130.2031. doi:10.1109/CDC.2006.376994. ISBN 978-1-4244-0171-0.
- ^ Matis, J. H.; Kiffe, T. R. (1996). "On Approximating the Moments of the Equilibrium Distribution of a Stochastic Logistic Model". Biometrics. 52 (3): 980–991. doi:10.2307/2533059. JSTOR 2533059.
- ^ Marion, G.; Renshaw, E.; Gibson, G. (1998). "Stochastic effects in a model of nematode infection in ruminants". Mathematical Medicine and Biology. 15 (2): 97. doi:10.1093/imammb/15.2.97.
- ^ Baytaş, Bekir; Bojowald, Martin; Crowe, Sean (2018-12-17). "Canonical tunneling time in ionization experiments". Physical Review A. 98 (6). American Physical Society (APS): 063417. arXiv:1810.12804. doi:10.1103/physreva.98.063417. ISSN 2469-9926.