Moment closure

Last updated

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

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 of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

In mathematics, a moment is a specific quantitative measure of the shape of a function. It is used in both mechanics and statistics. If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the zeroth moment is the total probability, the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.

Stochastic process mathematical object usually defined as a collection of random variables

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.

Contents

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]

In probability theory and statistics, the cumulantsκn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have identical cumulants as well, and similarly the cumulants determine the moments.

State space term in the theory of discrete dynamical systems; set of values which a process can take; set of states a system; (in a discrete system) countable and often finite

In the theory of discrete dynamical systems, a state space is the set of all possible configurations of a system. For example, a system in queueing theory defining the number of customers in a line would have state space {0, 1, 2, 3, ...}. State spaces can be either infinite or finite. An example of a finite state space is that of the toy problem Vacuum World, in which there are a limited set of configurations that the vacuum and dirt can be in.

A population model is a type of mathematical model that is applied to the study of population dynamics.

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]

Normal distribution probability distribution

In probability theory, the normaldistribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

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

Log-normal distribution probability distribution

In probability theory, a log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Likewise, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.

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]

Africanized bee common name

The Africanized bee, also known as the Africanised honey bee, and known colloquially as the "killer bee", is a hybrid of the western honey bee species, produced originally by cross-breeding of the East African lowland honey bee (A. m. scutellata) with various European honey bees such as the Italian honey bee A. m. ligustica and the Iberian honey bee A. m. iberiensis.

Nematode infection Human disease

A nematode infection is a type of helminthiasis caused by organisms in the nematode phylum.

Ruminant kind of mammals, by digestive process

Ruminants are mammals that are able to acquire nutrients from plant-based food by fermenting it in a specialized stomach prior to digestion, principally through microbial actions. The process, which takes place in the front part of the digestive system and therefore is called foregut fermentation, typically requires the fermented ingesta to be regurgitated and chewed again. The process of rechewing the cud to further break down plant matter and stimulate digestion is called rumination. The word "ruminant" comes from the Latin ruminare, which means "to chew over again".

Related Research Articles

In probability theory and statistics, kurtosis is a measure of the "tailedness" of the probability distribution of a real-valued random variable. In a similar way to the concept of skewness, kurtosis is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Depending on the particular measure of kurtosis that is used, there are various interpretations of kurtosis, and of how particular measures should be interpreted.

In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location.

Skewness measure of the asymmetry of random variables

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or undefined.

Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.

In probability theory, the Gillespie algorithm generates a statistically correct trajectory of a stochastic equation. It was created by Joseph L. Doob and others, presented by Dan Gillespie in 1976, and popularized in 1977 in a paper where he uses it to simulate chemical or biochemical systems of reactions efficiently and accurately using limited computational power. As computers have become faster, the algorithm has been used to simulate increasingly complex systems. The algorithm is particularly useful for simulating reactions within cells, where the number of reagents is low and keeping track of the position and behaviour of individual molecules is computationally feasible. Mathematically, it is a variant of a dynamic Monte Carlo method and similar to the kinetic Monte Carlo methods. It is used heavily in computational systems biology.

A stochastic simulation is a simulation that traces the evolution of variables that can change stochastically (randomly) with certain probabilities.

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal and gamma distributions, the purely discrete scaled Poisson distribution, and the class of mixed compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. For any random variable Y that obeys a Tweedie distribution, the variance var(Y) relates to the mean E(Y) by the power law,

This page lists articles related to probability theory. In particular, it lists many articles corresponding to specific probability distributions. Such articles are marked here by a code of the form (X:Y), which refers to number of random variables involved and the type of the distribution. For example (2:DC) indicates a distribution with two random variables, discrete or continuous. Other codes are just abbreviations for topics. The list of codes can be found in the table of contents.

In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution. They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively. Standardised L-moments are called L-moment ratios and are analogous to standardized moments. Just as for conventional moments, a theoretical distribution has a set of population L-moments. Sample L-moments can be defined for a sample from the population, and can be used as estimators of the population L-moments.

In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. The test is named after Carlos Jarque and Anil K. Bera. The test statistic is always nonnegative. If it is far from zero, it signals the data do not have a normal distribution.

John Gordon Skellam (1914-1979) was a statistician and ecologist, who discovered the Skellam distribution.

In queueing theory, a discipline within the mathematical theory of probability, an M/G/1 queue is a queue model where arrivals are Markovian, service times have a General distribution and there is a single server. The model name is written in Kendall's notation, and is an extension of the M/M/1 queue, where service times must be exponentially distributed. The classic application of the M/G/1 queue is to model performance of a fixed head hard disk.

In queueing theory, a discipline within the mathematical theory of probability, an M/G/k queue is a queue model where arrivals are Markovian, service times have a General distribution and there are k servers. The model name is written in Kendall's notation, and is an extension of the M/M/c queue, where service times must be exponentially distributed and of the M/G/1 queue with a single server. Most performance metrics for this queueing system are not known and remain an open problem.

In probability theory, a piecewise-deterministic Markov process (PDMP) is a process whose behaviour is governed by random jumps at points in time, but whose evolution is deterministically governed by an ordinary differential equation between those times. The class of models is "wide enough to include as special cases virtually all the non-diffusion models of applied probability." The process is defined by three quantities: the flow, the jump rate, and the transition measure.

In queueing theory, a discipline within the mathematical theory of probability, the G/G/1 queue represents the queue length in a system with a single server where interarrival times have a general distribution and service times have a (different) general distribution. The evolution of the queue can be described by the Lindley equation.

References

  1. 1 2 3 4 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. CiteSeerX   10.1.1.130.2031 . doi:10.1109/CDC.2006.376994. ISBN   978-1-4244-0171-0.
  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.
  8. ↑ Baytas, B.; Bojowald, M.; Crowe, S. (2018). "Canonical tunneling in ionization experiments". Phys. Rev. A. 98 (6): 063417.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.