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===Climate change and radiative forcing ===

===Climate change and radiative forcing ===

The [[IPCC|Intergovernmental Panel on Climate Change]] relies on Monte Carlo methods in [[probability density function]] analysis of [[radiative forcing]].

The [[IPCC|Intergovernmental Panel on Climate Change]] relies on Monte Carlo methods in [[probability density function]] analysis of [[radiative forcing]].

{{Quote|text=Probability density function (PDF) of ERF due to total GHG, aerosol forcing and total anthropogenic forcing. The GHG consists of WMGHG, ozone and stratospheric water vapour. The PDFs are generated based on uncertainties provided in Table 8.6. The combination of the individual RF agents to derive total forcing over the Industrial Era are done by Monte Carlo simulations and based on the method in Boucher and Haywood (2001). PDF of the ERF from surface albedo changes and combined contrails and contrail-induced cirrus are included in the total anthropogenic forcing, but not shown as a separate PDF. We currently do not have ERF estimates for some forcing mechanisms: ozone, land use, solar, etc.<ref>{{cite book|title=Climate Change 2013 The Physical Science Basis|date=2013|publisher=Cambridge University Press|isbn=978-1-107-66182-0|page=697|url=http://www.climatechange2013.org/images/report/WG1AR5_ALL_FINAL.pdf|accessdate=2 March 2016}}</ref>}}

{{Quote|text=Probability density function (PDF) of ERF due to total GHG, aerosol forcing and total anthropogenic forcing. The GHG consists of WMGHG, ozone and stratospheric water vapour. The PDFs are generated based on uncertainties provided in Table 8.6. The combination of the individual RF agents to derive total forcing over the Industrial Era are done by Monte Carlo simulations and based on the method in Boucher and Haywood (2001). PDF of the ERF from surface albedo changes and combined contrails and contrail-induced cirrus are included in the total anthropogenic forcing, but not shown as a separate PDF. We currently do not have ERF estimates for some forcing mechanisms: ozone, land use, solar, etc.<ref>{{cite book|title=Climate Change 2013 The Physical Science Basis|date=2013|publisher=Cambridge University Press|isbn=978-1-107-66182-0|page=697|url=http://www.climatechange2013.org/images/report/WG1AR5_ALL_FINAL.pdf|accessdate=2 March 2016}}</ref>}}

===Computational biology===

===Computational biology===

Monte Carlo methods are used in various fields of [[computational biology]], for example for [[Bayesian inference in phylogeny]], or for studying biological systems such as genomes, proteins,<ref>{{harvnb|Ojeda|et al.|2009}},</ref> or membranes.<ref>{{harvnb|Milik|Skolnick|1993}}</ref>The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy. Computer simulations allow us to monitor the local environment of a particular [[biomolecule|molecule]] to see if some [[chemical reaction]] is happening for instance. In cases where it is not feasible to conduct a physical experiment, [[thought experiment]]s can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

 

 

Monte Carlo methods are used in various fields of [[computational biology]], for example for [[Bayesian inference in phylogeny]], or for studying biological systems such as genomes, proteins,<ref>{{harvnb|Ojeda|et al.|2009}},</ref> or membranes.<ref>{{harvnb|Milik|Skolnick|1993}}</ref>

Monte Carlo Tree Search has been used successfully to play games such as Go, Tantrix, Battleship, Havannah, and Arimaa.

Monte Carlo methods are used in various fields of computational biology, for example for Bayesian inference in phylogeny, or for studying biological systems such as genomes, proteins, or membranes. The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy. Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical reaction is happening for instance. In cases where it is not feasible to conduct a physical experiment, thought experiments can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

 

 

The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy.  

 

 

Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical reaction is happening for instance. In cases where it is not feasible to conduct a physical experiment, thought experiments can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

计算机模拟使我们能够监测特定分子的局部环境，看看是否正在发生某种化学反应，例如。在无法进行物理实验的情况下，可以进行思维实验(例如: 打破键，在特定位置引入杂质，改变局部/全球结构，或引入外部场)。

计算机模拟使我们能够监测特定分子的局部环境，看看是否正在发生某种化学反应，例如。在无法进行物理实验的情况下，可以进行思维实验(例如: 打破键，在特定位置引入杂质，改变局部/全球结构，或引入外部场)。

#To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the [[Fisher information]] matrix.<ref>{{Cite journal |doi = 10.1198/106186005X78800|title = Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings|journal = Journal of Computational and Graphical Statistics|volume = 14|issue = 4|pages = 889–909|year = 2005|last1 = Spall|first1 = James C.|citeseerx = 10.1.1.142.738|s2cid = 16090098}}</ref><ref>{{Cite journal |doi = 10.1016/j.csda.2009.09.018|title = Efficient Monte Carlo computation of Fisher information matrix using prior information|journal = Computational Statistics & Data Analysis|volume = 54|issue = 2|pages = 272–289|year = 2010|last1 = Das|first1 = Sonjoy|last2 = Spall|first2 = James C.|last3 = Ghanem|first3 = Roger}}</ref>

#To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the [[Fisher information]] matrix.<ref>{{Cite journal |doi = 10.1198/106186005X78800|title = Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings|journal = Journal of Computational and Graphical Statistics|volume = 14|issue = 4|pages = 889–909|year = 2005|last1 = Spall|first1 = James C.|citeseerx = 10.1.1.142.738|s2cid = 16090098}}</ref><ref>{{Cite journal |doi = 10.1016/j.csda.2009.09.018|title = Efficient Monte Carlo computation of Fisher information matrix using prior information|journal = Computational Statistics & Data Analysis|volume = 54|issue = 2|pages = 272–289|year = 2010|last1 = Das|first1 = Sonjoy|last2 = Spall|first2 = James C.|last3 = Ghanem|first3 = Roger}}</ref>

#To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the Fisher information matrix.  提供负对数似然函数的 Hessian 矩阵的有效的随机估计，这些估计的平均值可以形成费雪资讯矩阵的估计。

#To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the Fisher information matrix.  提供负对数似然函数的 Hessian 矩阵的有效的随机估计，这些估计的平均值可以形成费雪资讯矩阵的估计。

Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An approximate randomization test is based on a specified subset of all permutations (which entails potentially enormous housekeeping of which permutations have been considered). The Monte Carlo approach is based on a specified number of randomly drawn permutations (exchanging a minor loss in precision if a permutation is drawn twice—or more frequently—for the efficiency of not having to track which permutations have already been selected).

Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An approximate randomization test is based on a specified subset of all permutations (which entails potentially enormous housekeeping of which permutations have been considered). The Monte Carlo approach is based on a specified number of randomly drawn permutations (exchanging a minor loss in precision if a permutation is drawn twice—or more frequently—for the efficiency of not having to track which permutations have already been selected).

蒙特卡罗方法也是近似随机化和置换检验的折衷。近似随机化测试是基于所有排列的特定子集(这需要潜在的庞大的内务管理，其中排列已被考虑)。蒙特卡罗方法是基于一定数量的随机排列(如果排列被抽取两次或更频繁，精度会有轻微的损失，因为不必追踪哪些排列已经被选择)。

蒙特卡罗方法也是近似随机化和置换检验的折衷。近似随机化测试是基于所有排列的特定子集(这需要潜在的庞大的内务管理，其中排列已被考虑)。蒙特卡罗方法是基于一定数量的随机排列(如果排列被抽取两次或更频繁，精度会有轻微的损失，因为不必追踪哪些排列已经被选择)。

Monte Carlo methods in finance are often used to evaluate investments in projects at a business unit or corporate level, or other financial valuations. They can be used to model project schedules, where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.[https://risk.octigo.pl/] Monte Carlo methods are also used in option pricing, default risk analysis. Additionally, they can be used to estimate the financial impact of medical interventions.

Monte Carlo methods in finance are often used to evaluate investments in projects at a business unit or corporate level, or other financial valuations. They can be used to model project schedules, where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.[https://risk.octigo.pl/] Monte Carlo methods are also used in option pricing, default risk analysis. Additionally, they can be used to estimate the financial impact of medical interventions.

#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.

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 } </math >  

错误减少一个因素 < math > scriptstyle 1/sqrt { n } <nowiki></math ></nowiki>

 

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.

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 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. 100 dimensions is by no means unusual, since in many physical problems, a "dimension" is equivalent to a 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 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. 100 dimensions is by no means unusual, since in many physical problems, a "dimension" is equivalent to a degree of freedom.

确定性数值积分算法在少数维上运行良好，但在函数具有多个变量时会遇到两个问题。首先，随着维数的增加，需要进行的功能评估的数量迅速增加。例如，如果10个评估在一个维度上提供了足够的精确度，那么100个维度需要10个 < sup > 100  点，这太多了以至于无法计算。这就是所谓的维数灾难。其次，多维区域的边界可能非常复杂，因此将问题简化为迭代积分可能是不可行的。100维绝对不是不寻常的，因为在许多物理问题中，一个“维度”等同于一个自由度。

确定性数值积分算法在少数维上运行良好，但在函数具有多个变量时会遇到两个问题。首先，随着维数的增加，需要进行的功能评估的数量迅速增加。例如，如果10个评估在一个维度上提供了足够的精确度，那么100个维度需要10个 < sup > 100  点，这太多了以至于无法计算。这就是所谓的维数灾难。其次，多维区域的边界可能非常复杂，因此将问题简化为迭代积分可能是不可行的。100维绝对不是不寻常的，因为在许多物理问题中，一个“维度”等同于一个自由度。

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably well-behaved, it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the central limit theorem, this method displays <math>\scriptstyle 1/\sqrt{N}</math> convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions. or the VEGAS algorithm.

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably well-behaved, it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the central limit theorem, this method displays <math>\scriptstyle 1/\sqrt{N}</math> convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions. or the VEGAS algorithm.