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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.
The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis.
The Monte Carlo simulation is in fact random experimentations, in the case that, the results of these experiments are not well known.
Monte Carlo simulations are typically characterized by a large number of unknown parameters, many of which are difficult to obtain experimentally.
The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense.
What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are uniformly distributed or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest, and most common ones.
Sawilowsky lists the characteristics of a high quality Monte Carlo simulation: 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.
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 methods. 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 10 7 random numbers.
There are ways of using probabilities that are definitely not Monte Carlo simulations — for example, deterministic modeling using single-point estimates.
Scenarios such as best, worst, or most likely case for each input variable are chosen and the results recorded.
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. 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 a large number of coupled degrees of freedom.
Areas of application include:. Monte Carlo methods are very important in computational physics , physical chemistry , and related applied fields, and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and aerodynamic forms as well as in modeling radiation transport for radiation dosimetry calculations.
In astrophysics , they are used in such diverse manners as to model both galaxy evolution  and microwave radiation transmission through a rough planetary surface.
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. The Intergovernmental Panel on Climate Change relies on Monte Carlo methods in probability density function analysis of radiative forcing.
The PDFs are generated based on uncertainties provided in Table 8. 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 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: 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.
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: Path tracing , occasionally referred to as Monte Carlo ray tracing, renders a 3D scene by randomly tracing samples of possible light paths.
Repeated sampling of any given pixel will eventually cause the average of the samples to converge on the correct solution of the rendering equation , making it one of the most physically accurate 3D graphics rendering methods in existence.
The standards for Monte Carlo experiments in statistics were set by Sawilowsky. 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 have been developed into a technique called Monte-Carlo tree search that is useful for searching for the best move in a game.
Possible moves are organized in a search tree and a large number of random simulations are used to estimate the long-term potential of each 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.
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 games , architecture , design , computer generated films , and cinematic special effects.
Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables. Ultimately this serves as a practical application of probability distribution in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.
Monte Carlo simulation is commonly used to evaluate the risk and uncertainty that would affect the outcome of different decision options. Monte Carlo simulation allows the business risk analyst to incorporate the total effects of uncertainty in variables like sales volume, commodity and labour prices, interest and exchange rates, as well as the effect of distinct risk events like the cancellation of a contract or the change of a tax law.
Monte Carlo methods in finance are often used to evaluate investments in projects at a business unit or corporate level, or to evaluate financial derivatives.
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.
Monte Carlo methods are also used in option pricing, default risk analysis. 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 and 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.
In general, the Monte Carlo methods are used in mathematics to solve various problems by generating suitable random numbers see also Random number generation and observing that fraction of the numbers that obeys some property or properties.
The method is useful for obtaining numerical solutions to problems too complicated to solve analytically.
The most common application of the Monte Carlo method is Monte Carlo integration. 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 10 points are needed for 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.
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 dimensional space, and taking some kind of average of the function values at these points.
A refinement of this method, known as importance sampling in statistics, involves sampling the points randomly, but more frequently where the integrand is large.
To do this precisely one would have to already know the integral, but one can approximate the integral by an integral of a similar function or use adaptive routines such as stratified sampling , recursive stratified sampling , adaptive umbrella sampling   or the VEGAS algorithm.
A similar approach, the quasi-Monte Carlo method , uses low-discrepancy sequences. 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. Another powerful and very popular application for random numbers in numerical simulation is in numerical optimization.
The problem is to minimize or maximize functions of some vector that often has a large number of dimensions. Many problems can be phrased in this way: In the traveling salesman problem the goal is to minimize distance traveled.
There are also applications to engineering design, such as multidisciplinary design optimization. It has been applied with quasi-one-dimensional models to solve particle dynamics problems by efficiently exploring large configuration space.
Reference  is a comprehensive review of many issues related to simulation and optimization. The traveling salesman problem is what is called a conventional optimization problem.
That is, all the facts distances between each destination point needed to determine the optimal path to follow are known with certainty and the goal is to run through the possible travel choices to come up with the one with the lowest total distance.
This goes beyond conventional optimization since travel time is inherently uncertain traffic jams, time of day, etc. As a result, to determine our optimal path we would want to use simulation - optimization to first understand the range of potential times it could take to go from one point to another represented by a probability distribution in this case rather than a specific distance and then optimize our travel decisions to identify the best path to follow taking that uncertainty into account.
Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines prior information with new information obtained by measuring some observable parameters data.
As, in the general case, the theory linking data with model parameters is nonlinear, the posterior probability in the model space may not be easy to describe it may be multimodal, some moments may not be defined, etc.
When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data.
In the general case we may have a large number of model parameters, and an inspection of the marginal probability densities of interest may be impractical, or even useless.
But it is possible to pseudorandomly generate a large collection of models according to the posterior probability distribution and to analyze and display the models in such a way that information on the relative likelihoods of model properties is conveyed to the spectator.
This can be accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formula for the a priori distribution is available.
The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of possibly highly nonlinear inverse problems with complex a priori information and data with an arbitrary noise distribution.
From Wikipedia, the free encyclopedia. Not to be confused with Monte Carlo algorithm. Monte Carlo method in statistical physics. Monte Carlo tree search.
Monte Carlo methods in finance , Quasi-Monte Carlo methods in finance , Monte Carlo methods for option pricing , Stochastic modelling insurance , and Stochastic asset model.
The Journal of Chemical Physics. Journal of the American Statistical Association. Mean field simulation for Monte Carlo integration. The Monte Carlo Method.
Genealogical and interacting particle approximations. Monte Carlo simulation furnishes the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action..
It shows the extreme possibilities—the outcomes of going for broke and for the most conservative decision—along with all possible consequences for middle-of-the-road decisions.
The technique was first used by scientists working on the atom bomb; it was named for Monte Carlo, the Monaco resort town renowned for its casinos.
Since its introduction in World War II, Monte Carlo simulation has been used to model a variety of physical and conceptual systems.
Monte Carlo simulation performs risk analysis by building models of possible results by substituting a range of values—a probability distribution—for any factor that has inherent uncertainty.
It then calculates results over and over, each time using a different set of random values from the probability functions.
Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations before it is complete.
Monte Carlo simulation produces distributions of possible outcome values. By using probability distributions, variables can have different probabilities of different outcomes occurring.
Probability distributions are a much more realistic way of describing uncertainty in variables of a risk analysis.
Values in the middle near the mean are most likely to occur. Examples of variables described by normal distributions include inflation rates and energy prices.
Values are positively skewed, not symmetric like a normal distribution. Examples of variables described by lognormal distributions include real estate property values, stock prices, and oil reserves.
All values have an equal chance of occurring, and the user simply defines the minimum and maximum. Examples of variables that could be uniformly distributed include manufacturing costs or future sales revenues for a new product.
The user defines the minimum, most likely, and maximum values. Values around the most likely are more likely to occur.
Variables that could be described by a triangular distribution include past sales history per unit of time and inventory levels.
The user defines the minimum, most likely, and maximum values, just like the triangular distribution. However values between the most likely and extremes are more likely to occur than the triangular; that is, the extremes are not as emphasized.
An example of the use of a PERT distribution is to describe the duration of a task in a project management model.
The user defines specific values that may occur and the likelihood of each. An example might be the results of a lawsuit: During a Monte Carlo simulation, values are sampled at random from the input probability distributions.
Each set of samples is called an iteration, and the resulting outcome from that sample is recorded.