Moment closure

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. The approximation is particularly useful in models with a very large state space, such as stochastic population models.

The moment closure approximation was first used by Goodman{{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 }} and Whittle{{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 }}{{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 }} who set all third and higher-order cumulants to be zero, approximating the population distribution with a normal distribution.

In 2006, Singh and Hespanha proposed a closure which approximates the population distribution as a log-normal distribution to describe biochemical reactions.{{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 }}

The approximation has been used successfully to model the spread of the Africanized bee in the Americas,{{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 }} nematode infection in ruminants.{{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 }} and quantum tunneling in ionization experiments.{{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}}

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• {{cite book |first=Lesław |last=Socha |chapter=Moment Equations for Nonlinear Stochastic Dynamic Systems |pages=85–102 |title=Linearization Methods for Stochastic Dynamic Systems |location=Berlin |publisher=Springer |year=2008 |isbn=978-3-540-72996-9 |doi=10.1007/978-3-540-72997-6_4 }}

Category:Stochastic processes