In
probability theory
Probability theory 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 expressing it through a set o ...
, 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. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
.
Introduction
Typically,
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
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 ha ...
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
A state space is 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 intelligence and game theory.
For instance, the toy ...
, such as stochastic
population model A population model is a type of mathematical model that is applied to the study of population dynamics.
Rationale
Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can ...
s.
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 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 e^
The parameter \mu ...
.
In 2006, Singh and Hespanha proposed a closure which approximates the population distribution as a
log-normal distribution 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 and known colloquially as the "killer bee", is a hybrid of the western honey bee (''Apis mellifera''), produced originally by crossbreeding of the East African lowland honey bee (''A. ...
in the Americas,
nematode infection
A nematode infection is a type of helminthiasis caused by organisms in the nematode phylum.
An example is enterobiasis. Several antinematodal agents are available. References
External links
Helminthiases
Infection
An infection is th ...
in
ruminants. and
quantum tunneling
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
in
ionization experiments.
References
{{reflist
Stochastic processes