# Moment closure

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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.[1] The approximation is particularly useful in models with a very large state space, such as stochastic population models.[1]

## 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.[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

The approximation has been used successfully to model the spread of the Africanized bee in the Americas[6] and nematode infection in ruminants.[7]

## References

1. ^ 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.
2. ^ Goodman, L. A. (1953). "Population Growth of the Sexes". Biometrics. 9 (2): 212–225. doi:10.2307/3001852. JSTOR 3001852.
3. ^ 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.
4. ^ Matis, T.; Guardiola, I. (2010). "Achieving Moment Closure through Cumulant Neglect". The Mathematica Journal. 12. doi:10.3888/tmj.12-2.
5. ^ Singh, A.; Hespanha, J. P. (2006). "Lognormal Moment Closures for Biochemical Reactions". Proceedings of the 45th IEEE Conference on Decision and Control. p. 2063. doi:10.1109/CDC.2006.376994. ISBN 1-4244-0171-2.
6. ^ 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.
7. ^ 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.