Moment closure explained

In probability theory, moment closure is an approximation method used to estimate moments of a stochastic process.^[1]

Introduction

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.

History

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.

In 2006, Singh and Hespanha proposed a closure which approximates the population distribution as a log-normal distribution to describe biochemical reactions.^[5]

Applications

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]

Notes and References

1. Gillespie . C. S. . Moment-closure approximations for mass-action models . 10.1049/iet-syb:20070031 . IET Systems Biology . 3 . 1 . 52–58 . 2009 . 19154084.
2. Goodman . L. A. . Leo Goodman. Population Growth of the Sexes . Biometrics . 9 . 2 . 212–225 . 10.2307/3001852 . 3001852. 1953 .
3. Whittle . P. . Peter Whittle (mathematician). On the Use of the Normal Approximation in the Treatment of Stochastic Processes . Journal of the Royal Statistical Society . 19 . 2 . 268–281 . 2983819. 1957 .
4. Matis . T. . Guardiola . I. . 10.3888/tmj.12-2 . Achieving Moment Closure through Cumulant Neglect . The Mathematica Journal . 12 . 2010 . free .
5. Book: Singh . A. . Hespanha . J. P. . 10.1109/CDC.2006.376994 . Lognormal Moment Closures for Biochemical Reactions . Proceedings of the 45th IEEE Conference on Decision and Control . 2063 . 2006 . 978-1-4244-0171-0 . 10.1.1.130.2031 .
6. Matis . J. H. . Kiffe . T. R. . On Approximating the Moments of the Equilibrium Distribution of a Stochastic Logistic Model . Biometrics . 52 . 3 . 980–991 . 10.2307/2533059 . 2533059. 1996 .
7. Marion . G. . Renshaw . E. . Gibson . G. . 10.1093/imammb/15.2.97 . Stochastic effects in a model of nematode infection in ruminants . Mathematical Medicine and Biology . 15 . 2 . 97 . 1998 .
8. Baytaş . Bekir . Bojowald . Martin . Crowe . Sean . Canonical tunneling time in ionization experiments . Physical Review A . American Physical Society (APS) . 98 . 6 . 2018-12-17 . 2469-9926 . 10.1103/physreva.98.063417 . 063417. 1810.12804.