In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, moment closure is an approximation method used to estimate
moments of a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
.
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
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s 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
In computer science, a state space is a discrete space representing the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial ...
, such as stochastic
population models.
History
The moment closure approximation was first used by Goodman and Whittle who set all third and higher-order cumulants to be zero, approximating the population distribution with a
normal distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
f(x) = \frac ...
.
In 2006, Singh and Hespanha proposed a closure which approximates the population distribution as a
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
to describe biochemical reactions.
Applications
The approximation has been used successfully to model the spread of the
Africanized bee
The Africanized bee, also known as the Africanized honey bee (AHB) and colloquially as the "killer bee", is a Hybrid (biology), hybrid of the western honey bee (''Apis mellifera''), produced originally by crossbreeding of the African bee, East A ...
in the Americas,
nematode infection in
ruminant
Ruminants are herbivorous grazing or browsing artiodactyls belonging to the suborder Ruminantia that are able to acquire nutrients from plant-based food by fermenting it in a specialized stomach prior to digestion, principally through microb ...
s. and
quantum tunneling in
ionization
Ionization or ionisation is the process by which an atom or a molecule acquires a negative or positive Electric charge, charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged at ...
experiments.
References
Further reading
*{{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
Stochastic processes