更改

[[Pseudo-random number sampling]] algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given [[probability distribution]].

[[Pseudo-random number sampling]] algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given [[probability distribution]].

Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and systems with many coupled degrees of freedom. Areas of application include:

[[Low-discrepancy sequences]] are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called [[quasi-Monte Carlo method]]s.

 

In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's [[RDRAND]] instruction set, as compared to those derived from algorithms, like the [[Mersenne Twister]], in Monte Carlo simulations of radio flares from [[brown dwarfs]].  RDRAND is the closest pseudorandom number generator to a true random number generator. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10⁷ random numbers.<ref>{{cite journal|last1=Route|first1=Matthew|title=Radio-flaring Ultracool Dwarf Population Synthesis|journal=The Astrophysical Journal|date=August 10, 2017|volume=845|issue=1|page=66|doi=10.3847/1538-4357/aa7ede|arxiv=1707.02212|bibcode=2017ApJ...845...66R|s2cid=118895524}}</ref>

[[Low-discrepancy sequences]] are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called [[quasi-Monte Carlo method]]s.

There are ways of using probabilities that are definitely not Monte Carlo simulations – for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a "best guess" estimate.  Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.<ref>{{harvnb|Vose|2000|page=13}}</ref>

In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's [[RDRAND]] instruction set, as compared to those derived from algorithms, like the [[Mersenne Twister]], in Monte Carlo simulations of radio flares from [[brown dwarfs]]. RDRAND is the closest pseudorandom number generator to a true random number generator. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10⁷ random numbers.<ref>{{cite journal|last1=Route|first1=Matthew|title=Radio-flaring Ultracool Dwarf Population Synthesis|journal=The Astrophysical Journal|date=August 10, 2017|volume=845|issue=1|page=66|doi=10.3847/1538-4357/aa7ede|arxiv=1707.02212|bibcode=2017ApJ...845...66R|s2cid=118895524}}</ref>

By contrast, Monte Carlo simulations sample from a [[probability distribution]] for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring.<ref>{{harvnb|Vose|2000|page=16}}</ref> For example, a comparison of a spreadsheet cost construction model run using traditional "what if" scenarios, and then running the comparison again with Monte Carlo simulation and [[triangular distribution|triangular probability distribution]]s shows that the Monte Carlo analysis has a narrower range than the "what if" analysis.{{Examples|date=May 2012}}  This is because the "what if" analysis gives equal weight to all scenarios (see [[Corporate finance#Quantifying uncertainty|quantifying uncertainty in corporate finance]]), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events".

'''Monte Carlo simulation versus "what if" scenarios'''

'''Monte Carlo simulation versus "what if" scenarios'''

</syntaxhighlight>

</syntaxhighlight>

=== Monte Carlo simulation versus "what if" scenarios ===

==Applications==

There are ways of using probabilities that are definitely not Monte Carlo simulations – for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a "best guess" estimate.  Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.<ref>{{harvnb|Vose|2000|page=13}}</ref>

Monte Carlo methods are especially useful for simulating phenomena with significant [[uncertainty]] in inputs and systems with many [[coupling (physics)|coupled]] degrees of freedom. Areas of application include:

By contrast, Monte Carlo simulations sample from a [[probability distribution]] for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring.<ref>{{harvnb|Vose|2000|page=16}}</ref> For example, a comparison of a spreadsheet cost construction model run using traditional "what if" scenarios, and then running the comparison again with Monte Carlo simulation and [[triangular distribution|triangular probability distribution]]s shows that the Monte Carlo analysis has a narrower range than the "what if" analysis.{{Examples|date=May 2012}}  This is because the "what if" analysis gives equal weight to all scenarios (see [[Corporate finance#Quantifying uncertainty|quantifying uncertainty in corporate finance]]), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events".

Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and systems with many coupled degrees of freedom. Areas of application include:

蒙特卡罗方法尤其适用于模拟输入和多自由度耦合系统中具有明显不确定性的现象。申请范围包括:

 

Monte Carlo methods are especially useful for simulating phenomena with significant [[uncertainty]] in inputs and systems with many [[coupling (physics)|coupled]] degrees of freedom. Areas of application include:

===Physical sciences===

===Physical sciences===

===Engineering===

===Engineering===

Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilistic analysis in process design. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,

Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilistic analysis in process design. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,

蒙特卡罗方法被广泛应用于工程设计中的敏感度分析和工艺设计中的定量概率分析。这种需求来源于典型过程模拟的交互性、共线性和非线性行为。比如说,

蒙特卡罗方法被广泛应用于工程设计中的敏感度分析和工艺设计中的定量概率分析。这种需求来源于典型过程模拟的交互性、共线性和非线性行为。比如说,

* In [[microelectronics|microelectronics engineering]], Monte Carlo methods are applied to analyze correlated and uncorrelated variations in [[Analog signal|analog]] and [[Digital data|digital]] [[integrated circuits]].

* In [[microelectronics|microelectronics engineering]], Monte Carlo methods are applied to analyze correlated and uncorrelated variations in [[Analog signal|analog]] and [[Digital data|digital]] [[integrated circuits]].

{{anchor|Monte Carlo tree search}}

{{anchor|Monte Carlo tree search}}

===Artificial intelligence for games===

===Artificial intelligence for games===

#Play a simulated game starting with that node.

#Play a simulated game starting with that node.

#Play a simulated game starting with that node.  以该节点开始玩一个模拟游戏。

#Play a simulated game starting with that node.  以该节点开始玩一个模拟游戏。

#Use the results of that simulated game to update the node and its ancestors.

#Use the results of that simulated game to update the node and its ancestors.

#Use the results of that simulated game to update the node and its ancestors.  使用模拟游戏的结果来更新节点及其祖先。

#Use the results of that simulated game to update the node and its ancestors.  使用模拟游戏的结果来更新节点及其祖先。

The net effect, over the course of many simulated games, is that the value of a node representing a move will go up or down, hopefully corresponding to whether or not that node represents a good move.

The net effect, over the course of many simulated games, is that the value of a node representing a move will go up or down, hopefully corresponding to whether or not that node represents a good move.

{{See also|Computer Go}}

{{See also|Computer Go}}

===Design and visuals===

===Design and visuals===

Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in [[global illumination]] computations that produce photo-realistic images of virtual 3D models, with applications in [[video game]]s, [[architecture]], [[design]], computer generated [[film]]s, and cinematic special effects.<ref>{{harvnb|Szirmay–Kalos|2008}}</ref>

Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in [[global illumination]] computations that produce photo-realistic images of virtual 3D models, with applications in [[video game]]s, [[architecture]], [[design]], computer generated [[film]]s, and cinematic special effects.<ref>{{harvnb|Szirmay–Kalos|2008}}</ref>

A Monte Carlo approach was used for evaluating the potential value of a proposed program to help female petitioners in Wisconsin be successful in their applications for [[Harassment Restraining Order|harassment]] and [[Domestic Abuse Restraining Order|domestic abuse restraining orders]].  It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk of [[rape]] and [[physical assault]].  However, there were many variables in play that could not be estimated perfectly, including the effectiveness of restraining orders, the success rate of petitioners both with and without advocacy, and many others.  The study ran trials that varied these variables to come up with an overall estimate of the success level of the proposed program as a whole.<ref name="montecarloanalysis">{{cite web|url=http://legalaidresearch.org/wp-content/uploads/Research-Increasing-Access-to-REstraining-Order-for-Low-Income-Victims-of-DV-A-Cost-Benefit-Analysis-of-the-Proposed-Domestic-Abuse-Grant-Program.pdf| title=Increasing Access to Restraining Orders for Low Income Victims of Domestic Violence: A Cost-Benefit Analysis of the Proposed Domestic Abuse Grant Program |publisher=[[State Bar of Wisconsin]] |date=December 2006 |accessdate=2016-12-12|last1=Elwart|first1=Liz|last2=Emerson|first2=Nina|last3=Enders|first3=Christina|last4=Fumia|first4=Dani|last5=Murphy|first5=Kevin|url-status=dead|archive-url=https://web.archive.org/web/20181106220526/https://legalaidresearch.org/wp-content/uploads/Research-Increasing-Access-to-REstraining-Order-for-Low-Income-Victims-of-DV-A-Cost-Benefit-Analysis-of-the-Proposed-Domestic-Abuse-Grant-Program.pdf|archive-date=6 November 2018}}</ref>

A Monte Carlo approach was used for evaluating the potential value of a proposed program to help female petitioners in Wisconsin be successful in their applications for [[Harassment Restraining Order|harassment]] and [[Domestic Abuse Restraining Order|domestic abuse restraining orders]].  It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk of [[rape]] and [[physical assault]].  However, there were many variables in play that could not be estimated perfectly, including the effectiveness of restraining orders, the success rate of petitioners both with and without advocacy, and many others.  The study ran trials that varied these variables to come up with an overall estimate of the success level of the proposed program as a whole.<ref name="montecarloanalysis">{{cite web|url=http://legalaidresearch.org/wp-content/uploads/Research-Increasing-Access-to-REstraining-Order-for-Low-Income-Victims-of-DV-A-Cost-Benefit-Analysis-of-the-Proposed-Domestic-Abuse-Grant-Program.pdf| title=Increasing Access to Restraining Orders for Low Income Victims of Domestic Violence: A Cost-Benefit Analysis of the Proposed Domestic Abuse Grant Program |publisher=[[State Bar of Wisconsin]] |date=December 2006 |accessdate=2016-12-12|last1=Elwart|first1=Liz|last2=Emerson|first2=Nina|last3=Enders|first3=Christina|last4=Fumia|first4=Dani|last5=Murphy|first5=Kevin|url-status=dead|archive-url=https://web.archive.org/web/20181106220526/https://legalaidresearch.org/wp-content/uploads/Research-Increasing-Access-to-REstraining-Order-for-Low-Income-Victims-of-DV-A-Cost-Benefit-Analysis-of-the-Proposed-Domestic-Abuse-Grant-Program.pdf|archive-date=6 November 2018}}</ref>

==Use in mathematics==

==Use in mathematics==

=== Integration ===

=== Integration ===

{{Main|Monte Carlo integration}}[[File:Monte-carlo2.gif|thumb|Monte-Carlo integration works by comparing random points with the value of the function|链接=Special:FilePath/Monte-carlo2.gif]]

{{Main|Monte Carlo integration}}[[File:Monte-carlo2.gif|thumb|Monte-Carlo integration works by comparing random points with the value of the functionMonte-Carlo integration works by comparing random points with the value of the function

[[File:Monte-Carlo method (errors).png|thumb|Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math>|链接=Special:FilePath/Monte-Carlo_method_(errors).png]]

 

[[File:Monte-Carlo method (errors).png|thumb|Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math>Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math>

 

错误减少一个因素 < math > scriptstyle 1/sqrt { n } <nowiki></math ></nowiki>|链接=Special:FilePath/Monte-Carlo_method_(errors).png]]

Deterministic [[numerical integration]] algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then [[googol|10¹⁰⁰]] points are needed for 100 dimensions—far too many to be computed. This is called the [[curse of dimensionality]]. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an [[iterated integral]].<ref name="Press">{{harvnb|Press|Teukolsky|Vetterling|Flannery|1996}}</ref> 100 [[dimension]]s is by no means unusual, since in many physical problems, a "dimension" is equivalent to a [[degrees of freedom (physics and chemistry)|degree of freedom]].

Deterministic [[numerical integration]] algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then [[googol|10¹⁰⁰]] points are needed for 100 dimensions—far too many to be computed. This is called the [[curse of dimensionality]]. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an [[iterated integral]].<ref name="Press">{{harvnb|Press|Teukolsky|Vetterling|Flannery|1996}}</ref> 100 [[dimension]]s is by no means unusual, since in many physical problems, a "dimension" is equivalent to a [[degrees of freedom (physics and chemistry)|degree of freedom]].

A similar approach, the [[quasi-Monte Carlo method]], uses [[low-discrepancy sequence]]s. These sequences "fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on the integral more quickly.

A similar approach, the [[quasi-Monte Carlo method]], uses [[low-discrepancy sequence]]s. These sequences "fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on the integral more quickly.

Another class of methods for sampling points in a volume is to simulate random walks over it ([[Markov chain Monte Carlo]]). Such methods include the [[Metropolis–Hastings algorithm]], [[Gibbs sampling]], [[Wang and Landau algorithm]], and interacting type MCMC methodologies such as the [[Particle filter|sequential Monte Carlo]] samplers.<ref>{{Cite journal|title = Sequential Monte Carlo samplers|journal = Journal of the Royal Statistical Society, Series B|doi=10.1111/j.1467-9868.2006.00553.x|volume=68|issue = 3|pages=411–436|year = 2006|last1 = Del Moral|first1 = Pierre|last2 = Doucet|first2 = Arnaud|last3 = Jasra|first3 = Ajay|arxiv = cond-mat/0212648|s2cid = 12074789}}</ref>

Another class of methods for sampling points in a volume is to simulate random walks over it ([[Markov chain Monte Carlo]]). Such methods include the [[Metropolis–Hastings algorithm]], [[Gibbs sampling]], [[Wang and Landau algorithm]], and interacting type MCMC methodologies such as the [[Particle filter|sequential Monte Carlo]] samplers.<ref>{{Cite journal|title = Sequential Monte Carlo samplers|journal = Journal of the Royal Statistical Society, Series B|doi=10.1111/j.1467-9868.2006.00553.x|volume=68|issue = 3|pages=411–436|year = 2006|last1 = Del Moral|first1 = Pierre|last2 = Doucet|first2 = Arnaud|last3 = Jasra|first3 = Ajay|arxiv = cond-mat/0212648|s2cid = 12074789}}</ref>

=== Simulation and optimization ===

=== Simulation and optimization ===