更改

在其他问题中，目标是从满足非线性发展方程的概率分布序列生成图。这些概率分布流总是可以解释为马尔可夫过程的随机状态的分布，其转移概率依赖于当前随机状态的分布(见麦肯-弗拉索夫 McKean-Vlasov过程，非线性滤波方程)。在其他情况下，我们给出了采样复杂度不断增加的概率分布流(如时间范围不断增加的路径空间模型，与温度参数降低有联系的'''玻尔兹曼—吉布斯 Boltzmann-Gibbs'''测度，以及许多其他例子)。这些模型也可以看作是一个非线性马尔可夫链的随机状态规律的演化。模拟这些复杂非线性马尔可夫过程的一个自然的方法是对过程的多个副本进行抽样，用抽样的经验测度替代演化方程中未知的随机状态分布。与传统的蒙特卡罗和 MCMC 方法相比，这些平均场粒子技术依赖于连续的相互作用样本。平均场一词反映了每个样本(也就是粒子、个体、步行者、媒介、生物或表现型)与过程的经验测量相互作用的事实。当系统的大小趋近于无穷时，这些随机经验测度收敛于非线性马尔可夫链随机状态的确定性分布，从而使粒子之间的统计相互作用消失。

在其他问题中，目标是从满足非线性发展方程的概率分布序列生成图。这些概率分布流总是可以解释为马尔可夫过程的随机状态的分布，其转移概率依赖于当前随机状态的分布(见麦肯-弗拉索夫 McKean-Vlasov过程，非线性滤波方程)。在其他情况下，我们给出了采样复杂度不断增加的概率分布流(如时间范围不断增加的路径空间模型，与温度参数降低有联系的'''玻尔兹曼—吉布斯 Boltzmann-Gibbs'''测度，以及许多其他例子)。这些模型也可以看作是一个非线性马尔可夫链的随机状态规律的演化。模拟这些复杂非线性马尔可夫过程的一个自然的方法是对过程的多个副本进行抽样，用抽样的经验测度替代演化方程中未知的随机状态分布。与传统的蒙特卡罗和 MCMC 方法相比，这些平均场粒子技术依赖于连续的相互作用样本。平均场一词反映了每个样本(也就是粒子、个体、步行者、媒介、生物或表现型)与过程的经验测量相互作用的事实。当系统的大小趋近于无穷时，这些随机经验测度收敛于非线性马尔可夫链随机状态的确定性分布，从而使粒子之间的统计相互作用消失。

== Overview 概述 ==

== Overview 概述 ==

量子蒙特卡罗方法，更具体地说，扩散蒙特卡罗方法也可以解释为费曼—卡茨路径积分的平均场粒子蒙特卡罗近似。量子蒙特卡罗方法的起源通常归功于'''恩里科·费米Enrico Fermi'''和'''罗伯特·里希特迈耶 Robert Richtmyer'''于1948年开发了中子链式反应的平均场粒子解释，但是用于估计量子系统的基态能量(在简化矩阵模型中)的第一个类启发式和遗传型粒子算法(也称为重取样或重构蒙特卡洛方法)则是由杰克·H·海瑟林顿在1984年提出。在分子化学中，使用遗传类启发式的粒子方法(又名删减和富集策略)可以追溯到1955年—'''马歇尔·罗森布鲁斯 Marshall Rosenbluth'''和'''阿里安娜·罗森布鲁斯Arianna Rosenbluth'''的开创性工作。

量子蒙特卡罗方法，更具体地说，扩散蒙特卡罗方法也可以解释为费曼—卡茨路径积分的平均场粒子蒙特卡罗近似。量子蒙特卡罗方法的起源通常归功于'''恩里科·费米Enrico Fermi'''和'''罗伯特·里希特迈耶 Robert Richtmyer'''于1948年开发了中子链式反应的平均场粒子解释，但是用于估计量子系统的基态能量(在简化矩阵模型中)的第一个类启发式和遗传型粒子算法(也称为重取样或重构蒙特卡洛方法)则是由杰克·H·海瑟林顿在1984年提出。在分子化学中，使用遗传类启发式的粒子方法(又名删减和富集策略)可以追溯到1955年—'''马歇尔·罗森布鲁斯 Marshall Rosenbluth'''和'''阿里安娜·罗森布鲁斯Arianna Rosenbluth'''的开创性工作。

The use of [[Sequential Monte Carlo method|Sequential Monte Carlo]] in advanced [[signal processing]] and [[Bayesian inference]] is more recent. It was in 1993, that Gordon et al., published in their seminal work<ref>{{Cite journal|title = Novel approach to nonlinear/non-Gaussian Bayesian state estimation |journal =  IEE Proceedings F - Radar and Signal Processing|date = April 1993|issn = 0956-375X|pages = 107–113|volume = 140|issue = 2|first1 = N.J.|last1 = Gordon|first2 = D.J.|last2 = Salmond|first3 = A.F.M.|last3 = Smith|doi=10.1049/ip-f-2.1993.0015|s2cid = 12644877|url = https://semanticscholar.org/paper/65484334a5cd4cabf6e5f7a17f606f07e2acf625}}</ref> the first application of a Monte Carlo [[Resampling (statistics)|resampling]] algorithm in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. We also quote another pioneering article in this field of Genshiro Kitagawa on a related "Monte Carlo filter",<ref>{{cite journal

The use of [[Sequential Monte Carlo method|Sequential Monte Carlo]] in advanced [[signal processing]] and [[Bayesian inference]] is more recent. It was in 1993, that Gordon et al., published in their seminal work<ref>{{Cite journal|title = Novel approach to nonlinear/non-Gaussian Bayesian state estimation |journal =  IEE Proceedings F - Radar and Signal Processing|date = April 1993|issn = 0956-375X|pages = 107–113|volume = 140|issue = 2|first1 = N.J.|last1 = Gordon|first2 = D.J.|last2 = Salmond|first3 = A.F.M.|last3 = Smith|doi=10.1049/ip-f-2.1993.0015|s2cid = 12644877|url = https://semanticscholar.org/paper/65484334a5cd4cabf6e5f7a17f606f07e2acf625}}</ref> the first application of a Monte Carlo [[Resampling (statistics)|resampling]] algorithm in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. We also quote another pioneering article in this field of Genshiro Kitagawa on a related "Monte Carlo filter",<ref>{{cite journal

在高级信号处理和贝叶斯推断中使用序列蒙特卡罗方法是最近才出现的。1993年，高登等人在他们的开创性工作中发表了蒙特卡罗重采样算法在贝叶斯推论统计学中的首次应用。作者将他们的算法命名为“自举过滤器” ，并证明了与其他过滤方法相比，他们的自举过滤算法不需要任何关于系统状态空间或噪声的假设。此外北川源四郎也进行了”蒙特卡洛过滤器”相关的开创性研究；在1990年代中期，'''皮埃尔·德尔·莫勒尔 Pierre Del Moral'''和'''希米尔康 · 卡瓦略 Himilcon Carvalho'''以及皮埃尔 · 德尔 · 莫勒尔、'''安德烈 · 莫宁 André Monin'''和'''杰拉德 · 萨鲁特 Gérard Salut'''发表了关于粒子过滤器的文章。1989-1992年间，在LAAS-CNRS (系统分析和体系结构实验室)，皮埃尔·德尔·莫勒尔、'''J·C·诺亚 J. C. Noyer'''、'''G·里加尔 G. Rigal''' 和'''杰拉德 · 萨鲁特'''开发了粒子滤波器用于信号处理。他们与STCAN (海军建造和武装服务技术部)、IT公司DIGILOG共同完成了一系列关于雷达/声纳和GPS信号处理问题的限制性和机密性研究报告。这些序列蒙特卡罗方法可以解释为一个接受拒绝采样器配备了相互作用的回收机制。

在高级信号处理和贝叶斯推断中使用序列蒙特卡罗方法是最近才出现的。1993年，高登等人在他们的开创性工作中发表了蒙特卡罗重采样算法在贝叶斯推论统计学中的首次应用。作者将他们的算法命名为“自举过滤器” ，并证明了与其他过滤方法相比，他们的自举过滤算法不需要任何关于系统状态空间或噪声的假设。此外北川源四郎也进行了”蒙特卡洛过滤器”相关的开创性研究；在1990年代中期，'''皮埃尔·德尔·莫勒尔 Pierre Del Moral'''和'''希米尔康 · 卡瓦略 Himilcon Carvalho'''以及皮埃尔 · 德尔 · 莫勒尔、'''安德烈 · 莫宁 André Monin'''和'''杰拉德 · 萨鲁特 Gérard Salut'''发表了关于粒子过滤器的文章。1989-1992年间，在LAAS-CNRS (系统分析和体系结构实验室)，皮埃尔·德尔·莫勒尔、'''J·C·诺亚 J. C. Noyer'''、'''G·里加尔 G. Rigal''' 和'''杰拉德 · 萨鲁特'''开发了粒子滤波器用于信号处理。他们与STCAN (海军建造和武装服务技术部)、IT公司DIGILOG共同完成了一系列关于雷达/声纳和GPS信号处理问题的限制性和机密性研究报告。这些序列蒙特卡罗方法可以解释为一个接受拒绝采样器配备了相互作用的回收机制。

From 1950 to 1996, all the publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms. The mathematical foundations and the first rigorous analysis of these particle algorithms were written by Pierre Del Moral in 1996.<ref name="dm9622"/><ref name=":22">{{cite journal|last1 = Del Moral|first1 = Pierre|title = Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems|journal = Annals of Applied Probability|date = 1998|edition = Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996)|volume = 8|issue = 2|pages = 438–495|url = http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535|doi = 10.1214/aoap/1028903535|citeseerx = 10.1.1.55.5257}}</ref>

From 1950 to 1996, all the publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms. The mathematical foundations and the first rigorous analysis of these particle algorithms were written by Pierre Del Moral in 1996.<ref name="dm9622"/><ref name=":22">{{cite journal|last1 = Del Moral|first1 = Pierre|title = Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems|journal = Annals of Applied Probability|date = 1998|edition = Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996)|volume = 8|issue = 2|pages = 438–495|url = http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535|doi = 10.1214/aoap/1028903535|citeseerx = 10.1.1.55.5257}}</ref>

从1950年到1996年，所有关于顺序蒙特卡罗方法的出版物，包括计算物理和分子化学中引入的删减和重采样蒙特卡罗方法，目前应用于不同的情况的自然和类启发式算法，没有任何一致性证明，也没有讨论估计的偏差和基于谱系和遗传树的算法。皮埃尔 · 德尔 · 莫勒尔在1996年的写作中阐述了关于这些粒子算法的数学基础，并对其第一次进行了严格的分析。

从1950年到1996年，所有关于顺序蒙特卡罗方法的出版物，包括计算物理和分子化学中引入的删减和重采样蒙特卡罗方法，目前应用于不同的情况的自然和类启发式算法，没有任何一致性证明，也没有讨论估计的偏差和基于谱系和遗传树的算法。皮埃尔 · 德尔 · 莫勒尔在1996年的写作中阐述了关于这些粒子算法的数学基础，并对其第一次进行了严格的分析。

Branching type particle methodologies with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons,<ref name=":42">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Gaines|first2 = Jessica|last3 = Lyons|first3 = Terry|title = Convergence of a branching particle method to the solution of the Zakai|journal = SIAM Journal on Applied Mathematics|date = 1998|volume = 58|issue = 5|pages = 1568–1590|doi = 10.1137/s0036139996307371|s2cid = 39982562|url = https://semanticscholar.org/paper/99e8759a243cd0568b0f32cbace2ad0525b16bb6}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = Nonlinear filtering and measure-valued processes|journal = Probability Theory and Related Fields|date = 1997|volume = 109|issue = 2|pages = 217–244|doi = 10.1007/s004400050131|s2cid = 119809371}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = A particle approximation of the solution of the Kushner–Stratonovitch equation|journal = Probability Theory and Related Fields|date = 1999|volume = 115|issue = 4|pages = 549–578|doi = 10.1007/s004400050249|s2cid = 117725141}}</ref> and by Dan Crisan, Pierre Del Moral and Terry Lyons.<ref name=":52">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Del Moral|first2 = Pierre|last3 = Lyons|first3 = Terry|title = Discrete filtering using branching and interacting particle systems|journal = Markov Processes and Related Fields|date = 1999|volume = 5|issue = 3|pages = 293–318|url = http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf}}</ref> Further developments in this field were developed in 2000 by P. Del Moral, A. Guionnet and L. Miclo.<ref name="dmm002" /><ref name="dg99">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of Measure Valued Processes with Applications to filtering|journal = C. R. Acad. Sci. Paris|date = 1999|volume = 39|issue = 1|pages = 429–434}}</ref><ref name="dg01">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of interacting processes with applications to filtering and genetic algorithms|journal = Annales de l'Institut Henri Poincaré|date = 2001|volume = 37|issue = 2|pages = 155–194|url = http://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps|doi = 10.1016/s0246-0203(00)01064-5|bibcode=2001AnIHP..37..155D}}</ref>

Branching type particle methodologies with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons,<ref name=":42">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Gaines|first2 = Jessica|last3 = Lyons|first3 = Terry|title = Convergence of a branching particle method to the solution of the Zakai|journal = SIAM Journal on Applied Mathematics|date = 1998|volume = 58|issue = 5|pages = 1568–1590|doi = 10.1137/s0036139996307371|s2cid = 39982562|url = https://semanticscholar.org/paper/99e8759a243cd0568b0f32cbace2ad0525b16bb6}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = Nonlinear filtering and measure-valued processes|journal = Probability Theory and Related Fields|date = 1997|volume = 109|issue = 2|pages = 217–244|doi = 10.1007/s004400050131|s2cid = 119809371}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = A particle approximation of the solution of the Kushner–Stratonovitch equation|journal = Probability Theory and Related Fields|date = 1999|volume = 115|issue = 4|pages = 549–578|doi = 10.1007/s004400050249|s2cid = 117725141}}</ref> and by Dan Crisan, Pierre Del Moral and Terry Lyons.<ref name=":52">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Del Moral|first2 = Pierre|last3 = Lyons|first3 = Terry|title = Discrete filtering using branching and interacting particle systems|journal = Markov Processes and Related Fields|date = 1999|volume = 5|issue = 3|pages = 293–318|url = http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf}}</ref> Further developments in this field were developed in 2000 by P. Del Moral, A. Guionnet and L. Miclo.<ref name="dmm002" /><ref name="dg99">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of Measure Valued Processes with Applications to filtering|journal = C. R. Acad. Sci. Paris|date = 1999|volume = 39|issue = 1|pages = 429–434}}</ref><ref name="dg01">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of interacting processes with applications to filtering and genetic algorithms|journal = Annales de l'Institut Henri Poincaré|date = 2001|volume = 37|issue = 2|pages = 155–194|url = http://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps|doi = 10.1016/s0246-0203(00)01064-5|bibcode=2001AnIHP..37..155D}}</ref>

20世纪90年代末，'''丹·克里桑 Dan Crisan'''、'''杰西卡·盖恩斯 Jessica Gaines'''和'''特里·利昂斯 Terry Lyons'''，以及丹·克里桑、皮埃尔·德尔·莫勒尔和特里·利昂斯也发展了具有不同种群大小的分支型粒子方法。2000年，皮埃尔·德尔·莫勒尔、'''爱丽丝·吉奥内 A. Guionnet'''和'''洛朗·米克洛 L. Miclo'''进一步发展了这一领域。

20世纪90年代末，'''丹·克里桑 Dan Crisan'''、'''杰西卡·盖恩斯 Jessica Gaines'''和'''特里·利昂斯 Terry Lyons'''，以及丹·克里桑、皮埃尔·德尔·莫勒尔和特里·利昂斯也发展了具有不同种群大小的分支型粒子方法。2000年，皮埃尔·德尔·莫勒尔、'''爱丽丝·吉奥内 A. Guionnet'''和'''洛朗·米克洛 L. Miclo'''进一步发展了这一领域。

==Definitions==

==Definitions==

''A Monte Carlo method simulation is defined as any method that utilizes sequences of random numbers to perform the simulation.  Monte Carlo simulations are applied to many topics including quantum chromodynamics, cancer radiation therapy, traffic flow, stellar evolution and VLSI design.  All these simulations require the use of random numbers and therefore pseudorandom number generators, which makes creating random-like numbers very important.''

A Monte Carlo method simulation is defined as any method that utilizes sequences of random numbers to perform the simulation.  Monte Carlo simulations are applied to many topics including quantum chromodynamics, cancer radiation therapy, traffic flow, stellar evolution and VLSI design.  All these simulations require the use of random numbers and therefore pseudorandom number generators, which makes creating random-like numbers very important.

There is no consensus on how ''Monte Carlo'' should be defined. For example, Ripley<ref name=Ripley>{{harvnb|Ripley|1987}}</ref> defines most probabilistic modeling as ''[[stochastic simulation]]'', with ''Monte Carlo'' being reserved for [[Monte Carlo integration]] and Monte Carlo statistical tests. [[Shlomo Sawilowsky|Sawilowsky]]<ref name=Sawilowsky>{{harvnb|Sawilowsky|2003}}</ref> distinguishes between a [[simulation]], a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). Examples:

There is no consensus on how ''Monte Carlo'' should be defined. For example, Ripley<ref name=Ripley>{{harvnb|Ripley|1987}}</ref> defines most probabilistic modeling as ''[[stochastic simulation]]'', with ''Monte Carlo'' being reserved for [[Monte Carlo integration]] and Monte Carlo statistical tests. [[Shlomo Sawilowsky|Sawilowsky]]<ref name=Sawilowsky>{{harvnb|Sawilowsky|2003}}</ref> distinguishes between a [[simulation]], a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). Examples:

*Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.

*Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.

*Monte Carlo simulation: Drawing <nowiki>''</nowiki>a large number<nowiki>''</nowiki> of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a <nowiki>''</nowiki>Monte Carlo simulation<nowiki>''</nowiki> of the behavior of repeatedly tossing a coin.

*Monte Carlo simulation: Drawing <nowiki>''</nowiki>a large number<nowiki>''</nowiki> of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a <nowiki>''</nowiki>Monte Carlo simulation<nowiki>''</nowiki> of the behavior of repeatedly tossing a coin.

Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."

Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."

A simple example of how a computer would perform a Monte Carlo simulation is the calculation of π.  If a square enclosed a circle and a point were randomly chosen inside the square the point would either lie inside the circle or outside it.  If the process were repeated many times, the ratio of the random points that lie inside the circle to the total number of random points in the square would approximate the ratio of the area of the circle to the area of the square.  From this we can estimate pi, as shown in the Python code below utilizing a SciPy package to generate pseudorandom numbers with the MT19937 algorithm.  Note that this method is a computationally inefficient way to numerically approximate π.

A simple example of how a computer would perform a Monte Carlo simulation is the calculation of π.  If a square enclosed a circle and a point were randomly chosen inside the square the point would either lie inside the circle or outside it.  If the process were repeated many times, the ratio of the random points that lie inside the circle to the total number of random points in the square would approximate the ratio of the area of the circle to the area of the square.  From this we can estimate pi, as shown in the Python code below utilizing a SciPy package to generate pseudorandom numbers with the MT19937 algorithm.  Note that this method is a computationally inefficient way to numerically approximate π.

* In groundwater modeling, Monte Carlo methods are utilized to generate a large number of realizations of heterogeneous parameter field for model uncertainty quantification or parameter inversion.

* In groundwater modeling, Monte Carlo methods are utilized to generate a large number of realizations of heterogeneous parameter field for model uncertainty quantification or parameter inversion.

===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>}}