Monte Carlo method

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distance. If instead of the goal being to minimize the total distance traveled to visit each desired destination but rather to minimize the total time needed to reach each destination, this goes beyond conventional optimization since travel time is inherently uncertain (traffic jams, time of day, etc.). As a result, to determine the optimal path a different simulation is required: 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 the travel decisions to identify the best path to follow taking that uncertainty into account.

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a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later , I described the idea to

40: 385:). In other instances, a flow of probability distributions with an increasing level of sampling complexity arise (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain. A natural way to simulate these sophisticated nonlinear Markov processes is to sample multiple copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled 9617: 3002: 1411:. In 1946, nuclear weapons physicists at Los Alamos were investigating neutron diffusion in the core of a nuclear weapon. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam proposed using random experiments. He recounts his inspiration as follows: 2730: 1570:

this field was Genshiro Kitagawa's, on a related "Monte Carlo filter", and the ones by Pierre Del Moral and Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut on particle filters published in the mid-1990s. Particle filters were also developed in signal processing in 1989–1992 by P. Del Moral, J. C. Noyer, G. Rigal, and G. Salut in the LAAS-CNRS in a series of restricted and classified research reports with STCAN (Service Technique des Constructions et Armes Navales), the IT company DIGILOG, and the

1808: 9603: 134: 9641: 2282:, when planning a wireless network, the design must be proven to work for a wide variety of scenarios that depend mainly on the number of users, their locations and the services they want to use. Monte Carlo methods are typically used to generate these users and their states. The network performance is then evaluated and, if results are not satisfactory, the network design goes through an optimization process. 9629: 7321: 2722: 2653:. 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. 1616:, 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). 2629:, 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. Additionally, they can be used to estimate the financial impact of medical interventions. 2665:

in Malaysia. The Monte Carlo simulation utilized previous published National Book publication data and book's price according to book genre in the local market. The Monte Carlo results were used to determine what kind of book genre that Malaysians are fond of and was used to compare book publications

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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 labor prices,

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The first thoughts and attempts I made to practice were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending

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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

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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. Another pioneering article in

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for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring. For example, a comparison of a spreadsheet cost construction model run using traditional "what if" scenarios, and then running the comparison again

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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

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who developed in 1948 a mean-field particle interpretation of neutron-chain reactions, but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due

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Despite its conceptual and algorithmic simplicity, the computational cost associated with a Monte Carlo simulation can be staggeringly high. In general the method requires many samples to get a good approximation, which may incur an arbitrarily large total runtime if the processing time of a single

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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

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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

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Monte Carlo methods are widely used in various fields of science, engineering, and mathematics, such as physics, chemistry, biology, statistics, artificial intelligence, finance, and cryptography. They have also been applied to social sciences, such as sociology, psychology, and political science.

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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

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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

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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, and by Dan Crisan, Pierre Del Moral and Terry Lyons. Further developments in this field were described in 1999 to 2001 by P. Del Moral, A.

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particles, individuals, walkers, agents, creatures, or phenotypes) interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so

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Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. They can also be used to model phenomena with significant uncertainty in inputs, such as calculating the risk of a nuclear power plant

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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

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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

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operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables. Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of

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In this procedure the domain of inputs is the square that circumscribes the quadrant. One can generate random inputs by scattering grains over the square then perform a computation on each input (test whether it falls within the quadrant). Aggregating the results yields our final result, the

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P. Del Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: Particle resolution in filtering and estimation". Studies on: Filtering, optimal control, and maximum likelihood estimation. Convention DRET no. 89.34.553.00.470.75.01. Research report no.4 (210p.), January

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pseudo-random uniform variable from the interval can be used to simulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, but not a Monte Carlo

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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

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Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem, and statistical sampling was used to estimate uncertainties in the simulations. Monte Carlo simulations invert this approach, solving deterministic problems using

2707:) 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. 2966:

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

2380:, 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 1574:(the Laboratory for Analysis and Architecture of Systems) on radar/sonar and GPS signal processing problems. These Sequential Monte Carlo methodologies can be interpreted as an acceptance-rejection sampler equipped with an interacting recycling mechanism. 372:

In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a

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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.

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Monte Carlo methods also have some limitations and challenges, such as the trade-off between accuracy and computational cost, the curse of dimensionality, the reliability of random number generators, and the verification and validation of the results.

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P. Del Moral, G. Rigal, and G. Salut. "Nonlinear and non Gaussian particle filters applied to inertial platform repositioning." LAAS-CNRS, Toulouse, Research Report no. 92207, STCAN/DIGILOG-LAAS/CNRS Convention STCAN no. A.91.77.013, (94p.) September

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When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as normally information on the resolution power of the data is desired. In the general case many parameters are modeled, and an inspection of the

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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

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simulations and averaging the simulations’ results. It has no restrictions on the probability distribution of the inputs to the simulations, requiring only that the inputs are randomly generated and are independent of each other and that

2925:. 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. 1687:

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. Weak correlations between successive samples are also often desirable/necessary.

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P. Del Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: Particle resolution in filtering and estimation: Theoretical results". Convention DRET no. 89.34.553.00.470.75.01, Research report no.3 (123p.), October

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P. Del Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: Particle resolution in filtering and estimation: Experimental results". Convention DRET no. 89.34.553.00.470.75.01, Research report no.2 (54p.), January

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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

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P. Del Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: An unified framework for particle solutions". LAAS-CNRS, Toulouse, Research Report no. 91137, DRET-DIGILOG- LAAS/CNRS contract, April

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to Jack H. Hetherington in 1984. In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with the seminal work of

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P. Del Moral, J.-Ch. Noyer, G. Rigal, and G. Salut. "Particle filters in radar signal processing: detection, estimation and air targets recognition". LAAS-CNRS, Toulouse, Research report no. 92495, December

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failure. Monte Carlo methods are often implemented using computer simulations, and they can provide approximate solutions to problems that are otherwise intractable or too complex to analyze mathematically.

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Carmona, René; Del Moral, Pierre; Hu, Peng; Oudjane, Nadia (2012). "An Introduction to Particle Methods with Financial Applications". In Carmona, René A.; Moral, Pierre Del; Hu, Peng; et al. (eds.).

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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.

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Shojaeefard, M.H.; Khalkhali, A.; Yarmohammadisatri, Sadegh (2017). "An efficient sensitivity analysis method for modified geometry of Macpherson suspension based on Pearson Correlation Coefficient".

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were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

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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

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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

1747:. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10 random numbers. 1639:

of pseudo-random uniform variables from the interval 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

974: 2839: 2765: 1499:-type Monte Carlo methods for estimating particle transmission energies. Mean-field genetic type Monte Carlo methodologies are also used as heuristic natural search algorithms (a.k.a. 96:

Monte Carlo methods have been recognized as one of the most important and influential ideas of the 20th century, and they have enabled many scientific and technological breakthroughs.

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Metropolis, Nicholas; Rosenbluth, Arianna W.; Rosenbluth, Marshall N.; Teller, Augusta H.; Teller, Edward (June 1, 1953). "Equation of State Calculations by Fast Computing Machines".

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required for further postwar development of nuclear weapons, including the design of the H-bomb, though severely limited by the computational tools at the time. Von Neumann,

2872:. 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. 2539:

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

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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

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Int Panis, L.; De Nocker, L.; De Vlieger, I.; Torfs, R. (2001). "Trends and uncertainty in air pollution impacts and external costs of Belgian passenger car traffic".

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shows that the Monte Carlo analysis has a narrower range than the "what if" analysis. This is because the "what if" analysis gives equal weight to all scenarios (see

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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."

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The standards for Monte Carlo experiments in statistics were set by Sawilowsky. In applied statistics, Monte Carlo methods may be used for at least four purposes:

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The approximation is generally poor if only a few points are randomly placed in the whole square. On average, the approximation improves as more points are placed.

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Chen, Shang-Ying; Hsu, Kuo-Chin; Fan, Chia-Ming (March 15, 2021). "Improvement of generalized finite difference method for stochastic subsurface flow modeling".

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Cunha Jr, A.; Nasser, R.; Sampaio, R.; Lopes, H.; Breitman, K. (2014). "Uncertainty quantification through the Monte Carlo method in a cloud computing setting".

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can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

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Bartels, Christian; Karplus, Martin (December 31, 1997). "Probability Distributions for Complex Systems: Adaptive Umbrella Sampling of the Potential Energy".

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Atanassova, E.; Gurov, T.; Karaivanova, A.; Ivanovska, S.; Durchova, M.; Dimitrov, D. (2016). "On the parallelization approaches for Intel MIC architecture".

2913:. The problem is to minimize (or maximize) functions of some vector that often has many dimensions. Many problems can be phrased in this way: for example, a 716:; Driels and Shin observe that “even for sample sizes an order of magnitude lower than the number required, the calculation of that number is quite stable." 2803:, 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 5938:"Increasing Access to Restraining Orders for Low Income Victims of Domestic Violence: A Cost-Benefit Analysis of the Proposed Domestic Abuse Grant Program" 7025:

Int Panis, L.; Rabl, A.; De Nocker, L.; Torfs, R. (2002). Sturm, P. (ed.). "Diesel or Petrol ? An environmental comparison hampered by uncertainty".

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on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics. An earlier pioneering article by

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densities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to the

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problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes:

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characterized by many unknown parameters, many of which are difficult to obtain experimentally. Monte Carlo simulation methods do not always require

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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

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Caffarel, Michel; Ceperley, David; Kalos, Malvin (1993). "Comment on Feynman–Kac Path-Integral Calculation of the Ground-State Energies of Atoms".

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Baeurle, Stephan A. (2009). "Multiscale modeling of polymer materials using field-theoretic methodologies: A survey about recent developments".

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Hill, Stacy D.; Spall, James C. (2019). "Stationarity and Convergence of the Metropolis-Hastings Algorithm: Insights into Theoretical Aspects".

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We know the expected value exists. The dice throws are randomly distributed and independent of each other. So simple Monte Carlo is applicable:

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Grüne-Yanoff, T., & Weirich, P. (2010). The philosophy and epistemology of simulation: A review, Simulation & Gaming, 41(1), pp. 20-50

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Das, Sonjoy; Spall, James C.; Ghanem, Roger (2010). "Efficient Monte Carlo computation of Fisher information matrix using prior information".

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and many random simulations are used to estimate the long-term potential of each move. A black box simulator represents the opponent's moves.

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be the estimated variance, sometimes called the “sample” variance; it is the variance of the results obtained from a relatively small number

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Wei, J.; Kruis, F.E. (2013). "A GPU-based parallelized Monte-Carlo method for particle coagulation using an acceptance–rejection strategy".

2645:. It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk of 1205: 6903: 2615:

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.

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Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably

1772:), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events". 9393: 5744: 2580:

detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of

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Suppose we want to know how many times we should expect to throw three eight-sided dice for the total of the dice throws to be at least

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Milik, M.; Skolnick, J. (January 1993). "Insertion of peptide chains into lipid membranes: an off-lattice Monte Carlo dynamics model".

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Hill, R.; Healy, B.; Holloway, L.; Kuncic, Z.; Thwaites, D.; Baldock, C. (March 2014). "Advances in kilovoltage x-ray beam dosimetry".

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What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are

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The theory of more sophisticated mean-field type particle Monte Carlo methods had certainly started by the mid-1960s, with the work of

361:. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. By the 6441:

C. Forastero and L. Zamora and D. Guirado and A. Lallena (2010). "A Monte Carlo tool to simulate breast cancer screening programmes".

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Liu, Jun S.; Liang, Faming; Wong, Wing Hung (March 1, 2000). "The Multiple-Try Method and Local Optimization in Metropolis Sampling".

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In general, the Monte Carlo methods are used in mathematics to solve various problems by generating suitable random numbers (see also

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for sampling and computing the posterior distribution of a signal process given some noisy and partial observations using interacting

1503:) in evolutionary computing. The origins of these mean-field computational techniques can be traced to 1950 and 1954 with the work of 5975:"Perbandingan Penerbitan dan Harga Buku Mengikut Genre di Malaysia dan Jepun Menggunakan Data Akses Terbuka dan Simulasi Monte Carlo" 2087: 6486:

Golden, Leslie M. (1979). "The Effect of Surface Roughness on the Transmission of Microwave Radiation Through a Planetary Surface".

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algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given

8791: 5877:"A Monte Carlo simulation approach for estimating the health and economic impact of interventions provided at a student-run clinic" 2784:. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an 2353:, or for studying biological systems such as genomes, proteins, or membranes. The systems can be studied in the coarse-grained or 1665: 9230: 3061: 2917:

program could be seen as trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In the

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Gordon, N.J.; Salmond, D.J.; Smith, A.F.M. (April 1993). "Novel approach to nonlinear/non-Gaussian Bayesian state estimation".

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Crisan, Dan; Gaines, Jessica; Lyons, Terry (1998). "Convergence of a branching particle method to the solution of the Zakai".

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frameworks depending on the desired accuracy. Computer simulations allow monitoring of the local environment of a particular

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evolution and microwave radiation transmission through a rough planetary surface. Monte Carlo methods are also used in the

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is more recent. It was in 1993, that Gordon et al., published in their seminal work the first application of a Monte Carlo

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invented the modern version of the Markov Chain Monte Carlo method while he was working on nuclear weapons projects at the

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Del Moral, Pierre; Guionnet, Alice (1999). "On the stability of Measure Valued Processes with Applications to filtering".

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Popular exposition of the Monte Carlo Method was conducted by McCracken. The method's general philosophy was discussed by

9692: 8257: 7405: 7325: 2066: 358: 245:, which are far quicker to use than the tables of random numbers that had been previously used for statistical sampling. 17: 6575: 6182:

Elishakoff, I., (2003) Notes on Philosophy of the Monte Carlo Method, International Applied Mechanics, 39(7), pp.753-762

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Lin, Y.; Wang, F.; Liu, B. (2018). "Random number generators for large-scale parallel Monte Carlo simulations on FPGA".

2963: 2192:. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example, 7166: 9040: 8932: 7281: 6620: 6029:

MEZEI, M (December 31, 1986). "Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias".

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Chaslot, Guillaume M. J. -B; Winands, Mark H. M.; Van Den Herik, H. Jaap (2008). "Parallel Monte-Carlo Tree Search".

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convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.

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Möller, W.; Eckstein, W. (March 1, 1984). "Tridyn — A TRIM simulation code including dynamic composition changes".

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twice—or more frequently—for the efficiency of not having to track which permutations have already been selected).

2420:, infinite sample size and infinitesimally small treatment effect), real data often do not have such distributions. 1695:

the (pseudo-random) number generator has certain characteristics (e.g. a long "period" before the sequence repeats)

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first experimented with the Monte Carlo method while studying neutron diffusion, but he did not publish this work.

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Kroese, D. P.; Brereton, T.; Taimre, T.; Botev, Z. I. (2014). "Why the Monte Carlo method is so important today".

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Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An approximate

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exists), but does not have a formula available to compute it. The simple Monte Carlo method gives an estimate for

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In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the

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An alternate formula can be used in the special case where all simulation results are bounded above and below.

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Benov, Dobriyan M. (2016). "The Manhattan Project, the first electronic computer and the Monte Carlo method".

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and power properties of statistics can be calculated for data drawn from classical theoretical distributions (

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Spall, James C. (2005). "Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings".

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Hastings, W. K. (April 1, 1970). "Monte Carlo sampling methods using Markov chains and their applications".

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Uses of Monte Carlo methods require large amounts of random numbers, and their use benefitted greatly from

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sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good

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Being secret, the work of von Neumann and Ulam required a code name. A colleague of von Neumann and Ulam,

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in one pass while minimizing the possibility that accumulated numerical error produces erroneous results:

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Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms

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where Ulam's uncle would borrow money from relatives to gamble. Monte Carlo methods were central to the

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simulations can be run “from scratch,” or, since k simulations have already been done, one can just run

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in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.

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The ratio of the inside-count and the total-sample-count is an estimate of the ratio of the two areas,

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Kitagawa, G. (1996). "Monte carlo filter and smoother for non-Gaussian nonlinear state space models".

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the goal is to minimize distance traveled. There are also applications to engineering design, such as

9082: 8850: 8571: 8496: 8425: 8354: 8274: 8262: 8132: 8120: 8113: 7821: 7542: 2888: 2289:, Monte Carlo simulation is used to compute system-level response given the component-level response. 1914: 1724: 378: 292: 193:

Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1

5827: 5574: 4759: 2695:: a human can be declared unintelligent if their writing cannot be told apart from a generated one. 1537:

path integrals. The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and

9565: 9332: 9195: 8880: 8845: 8809: 8594: 8036: 7945: 7904: 7816: 7507: 7346: 6352: 5608: 5451: 4618: 3897: 3086: 2946: 2876: 2869: 2865: 2704: 2581: 2260: 2125: 1894: 1760: 1729: 1718: 1396:

can be estimated by dropping needles on a floor made of parallel equidistant strips. In the 1930s,

1361:

nature of the algorithm allows this large cost to be reduced (perhaps to a feasible level) through

350: 346: 270: 116: 6612: 4612:(2) (Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996) ed.): 438–495. 2661:

Monte Carlo approach had also been used to simulate the number of book publications based on book

9474: 9087: 9027: 8964: 8602: 8586: 8324: 8186: 8176: 8026: 7940: 7071: 6291:

Markov Chain Monte Carlo Simulations and Their Statistical Analysis (With Web-Based Fortran Code)

4805:"On the stability of interacting processes with applications to filtering and genetic algorithms" 4601: 2904: 2781: 2716: 2475: 2469: 2432: 2286: 1954: 1909: 1864: 1827: 1817: 1765: 1605: 1566: 1358: 1129:< 100 be the desired confidence level, expressed as a percentage. Let every simulation result 296: 123: 66: 6934: 5821:. Springer Proceedings in Mathematics. Vol. 12. Springer Berlin Heidelberg. pp. 3–49. 2625:

at a business unit or corporate level, or other financial valuations. They can be used to model

9512: 9442: 9235: 9172: 8927: 8814: 7811: 7708: 7615: 7494: 7393: 7111: 6347: 5944: 5822: 5764:"Search Modeling and Optimization in USCG's Search and Rescue Optimal Planning System (SAROPS)" 5704: 5569: 5446: 4613: 3818:

McKean, Henry P. (1967). "Propagation of chaos for a class of non-linear parabolic equations".

2909:

Another powerful and very popular application for random numbers in numerical simulation is in

2609: 2365:

is happening for instance. In cases where it is not feasible to conduct a physical experiment,

2101: 1652: 422: 326:

and schedule overruns are routinely better than human intuition or alternative "soft" methods.

6772:; Rosenbluth, Arianna W.; Rosenbluth, Marshall N.; Teller, Augusta H.; Teller, Edward (1953). 2875:

Another class of methods for sampling points in a volume is to simulate random walks over it (

2148:

is usually based on a Monte Carlo approach to select the next colliding atom. In experimental

283:, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see 9537: 9479: 9422: 9248: 9141: 9050: 8776: 8660: 8519: 8511: 8401: 8393: 8208: 8104: 8082: 8041: 8006: 7973: 7919: 7894: 7849: 7788: 7748: 7550: 7373: 7206: 2804: 2772: 2682: 2518: 2346: 2263:

can determine the position of a robot. It is often applied to stochastic filters such as the

2093: 1904: 1799: 1596: 1547: 1526: 1516: 377:

whose transition probabilities depend on the distributions of the current random states (see

276:

In physics-related problems, Monte Carlo methods are useful for simulating systems with many

266: 39: 31: 6600: 3170: 2543:

computations that produce photo-realistic images of virtual 3D models, with applications in

2478:

that is useful for searching for the best move in a game. Possible moves are organized in a

2396:

To compare competing statistics for small samples under realistic data conditions. Although

2384:, making it one of the most physically accurate 3D graphics rendering methods in existence. 9460: 9035: 8984: 8960: 8922: 8840: 8819: 8771: 8650: 8628: 8597: 8506: 8383: 8334: 8252: 8225: 8181: 8137: 7899: 7675: 7555: 7273: 7201:. In Lafferty, J.; Williams, C. K. I.; Shawe-Taylor, J.; Zemel, R. S.; Culotta, A. (eds.). 7034:

Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (1996) .

6966: 6918: 6785: 6682: 6635: 6608: 6497: 6450: 6038: 5936:

Elwart, Liz; Emerson, Nina; Enders, Christina; Fumia, Dani; Murphy, Kevin (December 2006).

5890: 5667: 5301: 5264: 5194: 5132: 5081: 5028: 4955: 4887: 4816: 4491: 4317: 4246: 4200: 4150: 4054: 3967: 3949: 3850: 3709: 3674: 3631: 3586: 3256: 3197: 2959: 2687: 2489:

Starting at root node of the tree, select optimal child nodes until a leaf node is reached.

2442:. This sample then approximates and summarizes all the essential features of the posterior. 2431:(which are often impossible to compute) while being more accurate than critical values for 2226: 2181: 2165: 2030: 1899: 1508: 330: 284: 77: 6091:

Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay (2006). "Sequential Monte Carlo samplers".

3545:"Determining the number of Iterations for Monte Carlo Simulations of Weapon Effectiveness" 2792:

is by no means unusual, since in many physical problems, a "dimension" is equivalent to a

2112:

forms as well as in modeling radiation transport for radiation dosimetry calculations. In

145: 8: 9607: 9532: 9455: 9136: 8900: 8893: 8855: 8763: 8743: 8715: 8448: 8314: 8309: 8299: 8291: 8109: 8070: 7960: 7950: 7859: 7638: 7594: 7512: 7437: 7339: 7227:

Monte Carlo Methods in Global Illumination - Photo-realistic Rendering with Randomization

6821: 6769: 6750: 6601: 6344:

Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92

6206: 4450: 3046: 2850: 2845: 2692: 2540: 2409: 2256: 2208: 2129: 2113: 2045: 1934: 1842: 1543: 1538: 1522: 1473: 1445: 1425: 1382: 315: 44: 6970: 6922: 6789: 6519:

An Introduction to Computer Simulation Methods, Part 2, Applications to Physical Systems

6501: 6454: 6440: 6042: 5894: 5671: 5305: 5268: 5198: 5136: 5085: 5032: 4959: 4891: 4820: 4495: 4321: 4250: 4204: 4154: 3854: 3713: 3678: 3635: 3590: 3260: 3201: 2725:

Monte-Carlo integration works by comparing random points with the value of the function.

9621: 9432: 9286: 9182: 9131: 9007: 8904: 8888: 8865: 8642: 8376: 8359: 8319: 8230: 8125: 8087: 8058: 8018: 7978: 7924: 7841: 7527: 7522: 7134: 6987: 6954: 6890: 6845: 6809: 6719: 6474: 6375: 6330: 6277: 6248: 6218: 6210: 6173:

McCracken, D. D., (1955) The Monte Carlo Method, Scientific American, 192(5), pp. 90-97

6137:

Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control

6118: 6100: 5997: 5913: 5876: 5464: 5317: 5156: 5105: 5049: 5016: 4973: 4945: 4903: 4740: 4699: 4658: 4507: 4413: 4378: 4335: 4237:

Hetherington, Jack H. (1984). "Observations on the statistical iteration of matrices".

3800: 3647: 3621: 3520: 3481: 3463: 3385: 3323: 3280: 3229: 3151: 3041: 3007: 2626: 2454: 2446: 2439: 2381: 2366: 2297: 2279: 2223: 2200:, Monte Carlo methods are applied to analyze correlated and uncorrelated variations in 2121: 2097: 1992: 1785: 1562: 1469: 1433: 1362: 280: 277: 81: 6700:

MacGillivray, H. T.; Dodd, R. J. (1982). "Monte-Carlo simulations of galaxy systems".

6462: 5144: 5093: 5040: 4828: 4804: 4086: 4069: 3873: 3838: 9616: 9527: 9497: 9489: 9309: 9300: 9225: 9156: 9012: 8997: 8972: 8860: 8801: 8667: 8655: 8281: 8198: 8142: 8065: 7909: 7831: 7610: 7484: 7298: 7277: 7254: 7230: 7151: 7096: 7077: 7043: 6992: 6882: 6853: 6801: 6736: 6723: 6686: 6661: 6616: 6581: 6560: 6522: 6509: 6466: 6427: 6392: 6365: 6316: 6294: 6281: 6114: 6050: 6001: 5918: 5840: 5683: 5587: 5348: 5321: 5206: 5148: 5123:

Rogers, D.W.O. (2006). "Fifty years of Monte Carlo simulations for medical physics".

5097: 5054: 4977: 4907: 4744: 4703: 4370: 4216: 4166: 4040: 3995: 3878: 3477: 3431: 3327: 3315: 3272: 3221: 3213: 3001: 2950: 2785: 2780:

points are needed for 100 dimensions—far too many to be computed. This is called the

2576: 2514: 2428: 2362: 2334: 2313: 2293: 2153: 2133: 2035: 2010: 1982: 1669: 1609: 1558: 1496: 386: 366: 6754: 6478: 6407: 6379: 6252: 6122: 5468: 5160: 5109: 4662: 4511: 4382: 4339: 3804: 3651: 3485: 3389: 3284: 3155: 1780:

Monte Carlo methods are especially useful for simulating phenomena with significant

1672:, unpredictability is vital). Many of the most useful techniques use deterministic, 85: 9552: 9507: 9271: 9258: 9151: 9126: 9060: 8992: 8870: 8478: 8371: 8304: 8217: 8164: 7983: 7854: 7648: 7532: 7447: 7414: 7244: 7180: 7138: 7126: 7013: 6982: 6974: 6926: 6894: 6874: 6841: 6837: 6813: 6793: 6711: 6547: 6505: 6488: 6458: 6357: 6269: 6240: 6110: 6073: 6046: 5989: 5908: 5898: 5875:

Arenas, Daniel J.; Lett, Lanair A.; Klusaritz, Heather; Teitelman, Anne M. (2017).

5832: 5675: 5579: 5491: 5456: 5408: 5309: 5272: 5202: 5140: 5089: 5044: 5036: 4963: 4895: 4824: 4730: 4689: 4650: 4623: 4499: 4405: 4362: 4325: 4254: 4208: 4174: 4158: 4114: 4081: 4032: 4022: 3931: 3868: 3858: 3792: 3717: 3682: 3639: 3594: 3473: 3377: 3350: 3311: 3307: 3264: 3233: 3205: 3143: 2650: 2197: 2149: 2145: 2020: 1987: 1869: 1740: 1698:

the (pseudo-random) number generator produces values that pass tests for randomness

1477: 1417: 1357:

sample is high. Although this is a severe limitation in very complex problems, the

679: 181: 5974: 4899: 4031:. Lecture Notes in Mathematics. Vol. 1729. Berlin: Springer. pp. 1–145. 9469: 9213: 9075: 9002: 8677: 8551: 8524: 8501: 8470: 8097: 8092: 8046: 7776: 7427: 7062: 6591: 5903: 5679: 4050: 2942: 2892: 2858: 2301: 2268: 1874: 1530: 382: 362: 319: 234:

If the points are not uniformly distributed, then the approximation will be poor.

69: 8959: 7185: 6978: 5836: 5583: 4719:"A particle approximation of the solution of the Kushner–Stratonovitch equation" 4475:

Carvalho, Himilcon; Del Moral, Pierre; Monin, André; Salut, Gérard (July 1997).

4212: 4026: 3544: 3454:

Del Moral, P.; Doucet, A.; Jasra, A. (2006). "Sequential Monte Carlo samplers".

1680:

is for the pseudo-random sequence to appear "random enough" in a certain sense.

9418: 9413: 7876: 7806: 7452: 6825: 5495: 4968: 4933: 3935: 3780: 3091: 3036: 2914: 2884: 2662: 2568: 2245: 2237: 2025: 2015: 1959: 1919: 1691:

Sawilowsky lists the characteristics of a high-quality Monte Carlo simulation:

1481: 1404: 1032:

more simulations and add their results into those from the sample simulations:

415: 374: 334: 6273: 5412: 5313: 4654: 4627: 3721: 3686: 3643: 2729: 1807: 1750: 1594:

should be defined. For example, Ripley defines most probabilistic modeling as

1529:

can also be interpreted as a mean-field particle Monte Carlo approximation of

9661: 9575: 9542: 9405: 9366: 9177: 9146: 8610: 8564: 8169: 7871: 7698: 7462: 7457: 7017: 5662:

Lorentz, Richard J. (2011). "Improving Monte–Carlo Tree Search in Havannah".

5399:

Cassey; Smith (2014). "Simulating confidence for the Ellison-Glaeser Index".

4374: 4366: 4258: 4162: 3507: 3381: 3354: 3319: 3276: 3268: 3217: 3076: 2264: 2219: 2215: 2201: 2185: 2156:, understanding their behavior and comparing experimental data to theory. In 2100:, and related applied fields, and have diverse applications from complicated 1500: 1461: 1452:

computer to perform the first fully automated Monte Carlo calculations, of a

1378: 5460: 4306:"Monte-Carlo calculations of the average extension of macromolecular chains" 3970:(1957). "Symbiogenetic evolution processes realized by artificial methods". 2498:

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

9517: 9450: 9427: 9342: 8672: 7968: 7866: 7801: 7743: 7728: 7665: 7620: 6996: 6878: 6857: 6470: 6244: 5993: 5922: 5277: 5252: 5152: 5101: 5058: 4220: 4170: 3882: 3863: 3839:"A class of Markov processes associated with nonlinear parabolic equations" 2800: 2548: 2405: 2397: 2377: 2241: 2205: 2157: 2109: 1744: 1488: 1456:

core, in the spring of 1948. In the 1950s Monte Carlo methods were used at

1397: 1365:

strategies in local processors, clusters, cloud computing, GPU, FPGA, etc.

354: 323: 262: 6886: 6537: 6361: 4735: 4718: 4694: 4677: 3991:

Feynman–Kac formulae. Genealogical and interacting particle approximations

9560: 9522: 9205: 9106: 8968: 8781: 8748: 8240: 8157: 8152: 7796: 7753: 7733: 7713: 7703: 7472: 6308: 6105: 4119: 3796: 3762: 3760: 3468: 2529: 2505:

Monte Carlo Tree Search has been used successfully to play games such as

2479: 2358: 2105: 2040: 1781: 1507:

on genetic type mutation-selection learning machines and the articles by

1504: 1375: 389:. In contrast with traditional Monte Carlo and MCMC methodologies, these 303: 258: 149: 59: 7130: 6337:. Acta Numerica. Vol. 7. Cambridge University Press. pp. 1–49. 1739:

instruction set, as compared to those derived from algorithms, like the

8406: 7886: 7586: 7517: 7467: 7442: 7362: 6849: 6715: 5705:"Arimaa challenge – comparison study of MCTS versus alpha-beta methods" 5666:. Lecture Notes in Computer Science. Vol. 6515. pp. 105–115. 4417: 4036: 2987: 2544: 2413: 1677: 1613: 1441: 73: 7027:

Mitteilungen Institut für Verbrennungskraftmaschinen und Thermodynamik

6930: 6805: 6797: 6077: 4330: 4305: 3781:"Los Alamos Bets on ENIAC: Nuclear Monte Carlo Simulations, 1947-1948" 3757: 3598: 3225: 3209: 3147: 1307:{\displaystyle n\geq 2(b-a)^{2}\ln(2/(1-(\delta /100)))/\epsilon ^{2}} 8559: 8411: 8031: 7826: 7738: 7723: 7718: 7683: 6657: 6603:

How to Measure Anything: Finding the Value of Intangibles in Business

5568:. Lecture Notes in Computer Science. Vol. 5131. pp. 60–71. 4877: 4760:"Discrete filtering using branching and interacting particle systems" 4503: 3437: 2789: 2506: 2252:

method in combination with highly efficient computational algorithms.

682:– the percent chance that, when the Monte Carlo algorithm completes, 133: 62: 6768: 6676: 5745:"How the Coast Guard Uses Analytics to Search for Those Lost at Sea" 5609:

Monte-Carlo Tree Search in the game of Tantrix: Cosc490 Final Report

5509:

Chaslot, Guillaume; Bakkes, Sander; Szita, Istvan; Spronck, Pieter.

4409: 3187: 1388:

An early variant of the Monte Carlo method was devised to solve the

345:

the 'sample mean') of independent samples of the variable. When the

322:, aircraft design, etc.), Monte Carlo–based predictions of failure, 8075: 7693: 7570: 7565: 7560: 5881: 4950: 3626: 3576: 2667: 2571:

utilizes Monte Carlo methods within its computer modeling software

1534: 393:

techniques rely on sequential interacting samples. The terminology

311: 108:

Monte Carlo methods vary, but tend to follow a particular pattern:

5542: 5368: 5253:"A Scalar optimized parallel implementation of the DSMC technique" 1651:

Convergence of the Monte Carlo simulation can be checked with the

1105: 9580: 9281: 7003: 6632:

The Failure of Risk Management: Why It's Broken and How to Fix It

2510: 1465: 695: 254: 7253:. Philadelphia: Society for Industrial and Applied Mathematics. 7036:

Numerical Recipes in Fortran 77: The Art of Scientific Computing

7033: 6953:

Ojeda, P.; Garcia, M.; Londono, A.; Chen, N.Y. (February 2009).

6017: 3922:

Turing, Alan M. (1950). "Computing machinery and intelligence".

3341:

Spall, J. C. (2003). "Estimation via Markov Chain Monte Carlo".

2474:

Monte Carlo methods have been developed into a technique called

1571: 310:

in business and, in mathematics, evaluation of multidimensional

84:

in Monaco, where the primary developer of the method, physicist

9502: 8483: 8457: 8437: 7688: 7479: 7320: 7291:

Mazhdrakov, Metodi; Benov, Dobriyan; Valkanov, Nikolai (2018).

4070:"A Moran particle system approximation of Feynman–Kac formulae" 2777: 2721: 2575:

in order to calculate the probable locations of vessels during

2572: 2552: 2522: 2161: 1736: 1437: 4136:"Diffusion Monte Carlo Methods with a fixed number of walkers" 2438:

To provide a random sample from the posterior distribution in

733:= 0; run the simulation for the first time, producing result 72:

to obtain numerical results. The underlying concept is to use

7331: 6140: 2671: 1449: 406:

that the statistical interaction between particles vanishes.

349:

of the variable is parameterized, mathematicians often use a

7029:. Heft 81 Vol 1. Technische Universität Graz Austria: 48–54. 5874: 5241:

G. A. Bird, Molecular Gas Dynamics, Clarendon, Oxford (1976)

5017:"GPU-based high-performance computing for radiation therapy" 7422: 7024: 6774:"Equation of State Calculations by Fast Computing Machines" 5563: 4474: 4134:

Assaraf, Roland; Caffarel, Michel; Khelif, Anatole (2000).

3994:. Probability and Its Applications. Springer. p. 575. 2978:

information and data with an arbitrary noise distribution.

2949:

in the model space. This probability distribution combines

2646: 2556: 2326: 307: 302:

Other examples include modeling phenomena with significant

6764:(1987 Special Issue dedicated to Stanislaw Ulam): 125–130. 5791: 3611: 2485:

The Monte Carlo tree search (MCTS) method has four steps:

353:(MCMC) sampler. The central idea is to design a judicious 337:

of some random variable can be approximated by taking the

7038:. Fortran Numerical Recipes. Vol. 1 (2nd ed.). 6408:"Stan Ulam, John von Neumann, and the Monte Carlo method" 5815: 5725: 5380: 3735: 3733: 3731: 3133: 2160:, they are used in such diverse manners as to model both 318:. In application to systems engineering problems (space, 137:

Monte Carlo method applied to approximating the value of

6387:

Doucet, Arnaud; Freitas, Nando de; Gordon, Neil (2001).

6342:

Davenport, J. H. (1992). "Primality testing revisited".

5935: 4304:

Rosenbluth, Marshall N.; Rosenbluth, Arianna W. (1955).

3898:"Estimation of particle transmission by random sampling" 3169:

Hubbard, Douglas; Samuelson, Douglas A. (October 2009).

3115: 7290: 6904:"Monte Carlo sampling of solutions to inverse problems" 5762:

Stone, Lawrence D.; Kratzke, Thomas M.; Frost, John R.

5071: 3820:

Lecture Series in Differential Equations, Catholic Univ

3766: 2320: 2180:

Monte Carlo methods are widely used in engineering for

997:

sufficient sample simulations were done to ensure that

171: 6952: 5511:"Monte-Carlo Tree Search: A New Framework for Game AI" 5374: 3779:

Haigh, Thomas; Priestley, Mark; Rope, Crispin (2014).

3728: 2891:, and interacting type MCMC methodologies such as the 2814: 2740: 1668:

to be useful (although, for some applications such as

5508: 4758:

Crisan, Dan; Del Moral, Pierre; Lyons, Terry (1999).

4484:

IEEE Transactions on Aerospace and Electronic Systems

4477:"Optimal Non-linear Filtering in GPS/INS Integration" 4436:"Non Linear Filtering: Interacting Particle Solution" 4190: 4133: 3952:(1954). "Esempi numerici di processi di evoluzione". 2813: 2739: 2685:

writes about Monte Carlo generators in his 2001 book

1707:

the algorithm used is valid for what is being modeled

1208: 1117:

that is twice the maximum allowed difference between

925: 653: 365:, the stationary distribution is approximated by the 9244:

Autoregressive conditional heteroskedasticity (ARCH)

7203:

Advances in Neural Information Processing Systems 23

6090: 5015:

Jia, Xun; Ziegenhein, Peter; Jiang, Steve B (2014).

4934:"Radio-flaring Ultracool Dwarf Population Synthesis" 4757: 4303: 3745: 3453: 2997: 2492:

Expand the leaf node and choose one of its children.

2463: 2244:, where the Boltzmann equation is solved for finite 1495:

and Herman Kahn, published in 1951, used mean-field

6554: 6424:

Monte Carlo: Concepts, Algorithms, and Applications

6386: 4274:"Note on census-taking in Monte Carlo calculations" 3052:

List of software for Monte Carlo molecular modeling

1759:By contrast, Monte Carlo simulations sample from a 1701:

there are enough samples to ensure accurate results

454:; more formally, it will be the case that, for any 8706: 7148:Statistics via Monte Carlo Simulation with Fortran 6901: 6645:Judgement under Uncertainty: Heuristics and Biases 6313:The Monte Carlo Method in Condensed Matter Physics 6151: 6093:Journal of the Royal Statistical Society, Series B 5929: 5014: 4996: 4984: 4678:"Nonlinear filtering and measure-valued processes" 4352: 3778: 3561: 3456:Journal of the Royal Statistical Society, Series B 3442:Monographs on Statistics & Applied Probability 2833: 2759: 2345:Monte Carlo methods are used in various fields of 2275:(simultaneous localization and mapping) algorithm. 1788:degrees of freedom. Areas of application include: 1743:, in Monte Carlo simulations of radio flares from 1306: 968: 7145: 7095:(2nd ed.). New York: John Wiley & Sons. 6674: 5859: 5439:Journal of Computational and Graphical Statistics 5425: 4802: 4783: 4640: 4398:Journal of Computational and Graphical Statistics 3433:Mean field simulation for Monte Carlo integration 2427:that are more efficient than exact tests such as 1751:Monte Carlo simulation versus "what if" scenarios 1658: 9659: 7294:The Monte Carlo Method. Engineering Applications 7146:Sawilowsky, Shlomo S.; Fahoome, Gail C. (2003). 7090: 6699: 5860:Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). 5219: 3543:Driels, Morris R.; Shin, Young S. (April 2004). 3168: 190:scatter a given number of points over the square 176:can be approximated using a Monte Carlo method: 8792:Multivariate adaptive regression splines (MARS) 6830:Journal of the American Statistical Association 6675:Kroese, D. P.; Taimre, T.; Botev, Z.I. (2011). 6642: 4355:IEE Proceedings F - Radar and Signal Processing 3300:Journal of the American Statistical Association 1106:A formula when simulations' results are bounded 88:, was inspired by his uncle's gambling habits. 6820: 6535: 6516: 6063: 5481: 5337:Climate Change 2013 The Physical Science Basis 5184: 4271: 4067: 4020: 3895: 2898: 2495:Play a simulated game starting with that node. 2124:, and Monte Carlo methods are used to compute 1756:variable are chosen and the results recorded. 7347: 7224: 7172:Journal of Modern Applied Statistical Methods 7069: 6864: 6652:Kalos, Malvin H.; Whitlock, Paula A. (2008). 6651: 6536:Grinstead, Charles; Snell, J. Laurie (1997). 5761: 5731: 5386: 4866: 4851: 4849: 4272:Fermi, Enrique; Richtmyer, Robert D. (1948). 3121: 2222:, Monte Carlo methods underpin the design of 2152:, Monte Carlo methods are used for designing 2067: 1643:of the behavior of repeatedly tossing a coin. 7193: 6902:Mosegaard, Klaus; Tarantola, Albert (1995). 6555:Hammersley, J. M.; Handscomb, D. C. (1975). 5484:Computational Statistics & Data Analysis 5250: 4236: 3297: 1770:quantifying uncertainty in corporate finance 969:{\displaystyle n\geq s^{2}/(z\epsilon )^{2}} 5398: 4803:Del Moral, Pierre; Guionnet, Alice (2001). 4716: 4675: 4074:Stochastic Processes and Their Applications 3405: 1420:, and we began to plan actual calculations. 7392: 7354: 7340: 7164: 6749: 6389:Sequential Monte Carlo methods in practice 5291: 4855: 4846: 4094: 4068:Del Moral, Pierre; Miclo, Laurent (2000). 3966: 3948: 3896:Herman, Kahn; Theodore, Harris E. (1951). 3739: 3699: 3549:Naval Postgraduate School Technical Report 3542: 3067:Monte Carlo methods for electron transport 2834:{\displaystyle \scriptstyle 1/{\sqrt {N}}} 2760:{\displaystyle \scriptstyle 1/{\sqrt {N}}} 2092:Monte Carlo methods are very important in 2074: 2060: 1806: 1464:, and became popularized in the fields of 369:of the random states of the MCMC sampler. 8005: 7243: 7184: 7091:Rubinstein, R. Y.; Kroese, D. P. (2007). 6986: 6755:"The beginning of the Monte Carlo method" 6351: 6341: 6335:Monte Carlo and quasi-Monte Carlo methods 6162: 6104: 5912: 5902: 5826: 5573: 5450: 5276: 5048: 4967: 4949: 4920: 4734: 4693: 4617: 4599: 4433: 4429: 4427: 4329: 4184: 4127: 4118: 4100: 4085: 4016: 4014: 3987: 3872: 3862: 3625: 3467: 3429: 3367: 2327:Intergovernmental Panel on Climate Change 2088:Monte Carlo method in statistical physics 900:Note that, when the algorithm completes, 152:. Given that the ratio of their areas is 7109: 6730: 6573: 6405: 6329: 6211:"Metropolis, Monte Carlo and the MANIAC" 6205: 6013: 6011: 5624: 4395: 3751: 3664: 3562:Shonkwiler, R. W.; Mendivil, F. (2009). 3509:Monte Carlo Theory, Methods and Examples 3246: 2728: 2720: 2340: 2128:of simple particle and polymer systems. 1710:it simulates the phenomenon in question. 701:corresponding to that confidence level. 230:There are two important considerations: 132: 38: 7112:"Risk Analysis in Investment Appraisal" 7006:International Journal of Vehicle Design 6629: 6598: 6577:Practical Guide to Computer Simulations 6421: 6259: 5973:Dahlan, Hadi Akbar (October 29, 2021). 5661: 5173: 3983: 3981: 3785:IEEE Annals of the History of Computing 3425: 3423: 3421: 3401: 3399: 3062:Monte Carlo method for photon transport 2587: 214:. Multiply the result by 4 to estimate 27:Probabilistic problem-solving algorithm 14: 9660: 9318:Kaplan–Meier estimator (product limit) 7196:"Monte-Carlo Planning in Large POMDPs" 7056: 6517:Gould, Harvey; Tobochnik, Jan (1988). 6485: 6307: 5972: 5627:"Monte-Carlo Planning in Large POMDPs" 5230: 5122: 4841: 4835: 4424: 4289:Declassified report Los Alamos Archive 4232: 4230: 4061: 4011: 3921: 3836: 3817: 2844:A refinement of this method, known as 2602:Monte Carlo methods for option pricing 1351: 253:Monte Carlo methods are often used in 80:in principle. The name comes from the 9391: 8958: 8705: 8004: 7774: 7391: 7335: 7093:Simulation and the Monte Carlo Method 6230: 6028: 6008: 5436: 4931: 4871: 4860: 4723:Probability Theory and Related Fields 4682:Probability Theory and Related Fields 4299: 4297: 3811: 3772: 3767:Mazhdrakov, Benov & Valkanov 2018 3538: 3536: 3501: 3499: 3497: 3495: 3340: 3072:Monte Carlo N-Particle Transport Code 2923:multidisciplinary design optimization 2698: 2534: 2387: 1704:the proper sampling technique is used 409: 402: 342: 306:in inputs such as the calculation of 9628: 9328:Accelerated failure time (AFT) model 7267: 7194:Silver, David; Veness, Joel (2010). 7076:(2nd ed.). New York: Springer. 6293:. Hackensack, NJ: World Scientific. 6288: 6233:Monte Carlo Methods and Applications 5853: 5002: 4990: 4809:Annales de l'Institut Henri Poincaré 4796: 4777: 4006:Series: Probability and Applications 3978: 3555: 3505: 3418: 3396: 2857:, adaptive umbrella sampling or the 2598:Quasi-Monte Carlo methods in finance 2562: 2372: 2321:Climate change and radiative forcing 1791: 1766:triangular probability distributions 9640: 8923:Analysis of variance (ANOVA, anova) 7775: 7270:Risk Analysis, A Quantitative Guide 6066:The Journal of Physical Chemistry B 4767:Markov Processes and Related Fields 4751: 4643:SIAM Journal on Applied Mathematics 4634: 4593: 4443:Markov Processes and Related Fields 4227: 3564:Explorations in Monte Carlo Methods 3515:. Work in progress. pp. 15–36. 2936: 2120:is an alternative to computational 1608:and Monte Carlo statistical tests. 473:Typically, the algorithm to obtain 397:reflects the fact that each of the 359:stationary probability distribution 24: 9018:Cochran–Mantel–Haenszel statistics 7644:Pearson product-moment correlation 6828:(1949). "The Monte Carlo Method". 6643:Kahneman, D.; Tversky, A. (1982). 4932:Route, Matthew (August 10, 2017). 4717:Crisan, Dan; Lyons, Terry (1999). 4676:Crisan, Dan; Lyons, Terry (1997). 4294: 4107:ESAIM Probability & Statistics 3533: 3492: 2964:posterior probability distribution 2656: 712:of “sample” simulations. Choose a 661: 112:Define a domain of possible inputs 25: 9719: 7313: 6262:Journal of Mathematical Chemistry 5606: 5539:"Monte Carlo Tree Search - About" 4021:Del Moral, P.; Miclo, L. (2000). 3905:Natl. Bur. Stand. Appl. Math. Ser 3551:(March 2003 - March 2004): 10–11. 2643:domestic abuse restraining orders 2559:, and cinematic special effects. 2464:Artificial intelligence for games 2329:relies on Monte Carlo methods in 2306:sequential Monte Carlo techniques 1180:. To have confidence of at least 719:The following algorithm computes 654:Determining a sufficiently large 9639: 9627: 9615: 9602: 9601: 9392: 7319: 7167:"You think you've got trivials?" 7070:Robert, C.; Casella, G. (2004). 6733:Stochastic Simulation in Physics 6185: 6176: 6167: 6156: 6145: 6129: 6115:10.1111/j.1467-9868.2006.00553.x 6031:Journal of Computational Physics 5792:"Project Risk Simulation (BETA)" 5702: 5294:Journal of Computational Physics 5257:Journal of Computational Physics 3702:Journal of Computational Physics 3478:10.1111/j.1467-9868.2006.00553.x 3000: 2623:evaluate investments in projects 2606:Stochastic modelling (insurance) 1784:in inputs and systems with many 1764:with Monte Carlo simulation and 1635:Monte Carlo simulation: Drawing 115:Generate inputs randomly from a 76:to solve problems that might be 9277:Least-squares spectral analysis 7073:Monte Carlo Statistical Methods 6678:Handbook of Monte Carlo Methods 6084: 6057: 6022: 5966: 5868: 5862:Handbook of Monte Carlo Methods 5809: 5784: 5755: 5737: 5696: 5655: 5618: 5600: 5557: 5531: 5502: 5475: 5430: 5419: 5392: 5328: 5285: 5251:Dietrich, S.; Boyd, I. (1996). 5244: 5235: 5224: 5213: 5178: 5167: 5125:Physics in Medicine and Biology 5116: 5074:Physics in Medicine and Biology 5065: 5021:Physics in Medicine and Biology 5008: 4925: 4914: 4710: 4669: 4583: 4573: 4563: 4553: 4543: 4533: 4468: 4389: 4346: 4265: 4028:Séminaire de Probabilités XXXIV 3960: 3942: 3915: 3889: 3830: 3693: 3658: 3614:Computer Physics Communications 3605: 3570: 3447: 3190:The Journal of Chemical Physics 3171:"Modeling Without Measurements" 2351:Bayesian inference in phylogeny 1930:Smoothed particle hydrodynamics 1775: 8258:Mean-unbiased minimum-variance 7361: 7229:. VDM Verlag Dr. Mueller e.K. 7225:Szirmay-Kalos, László (2008). 7165:Sawilowsky, Shlomo S. (2003). 7150:. Rochester Hills, MI: JMASM. 7110:Savvides, Savvakis C. (1994). 6842:10.1080/01621459.1949.10483310 6703:Astrophysics and Space Science 6152:Mosegaard & Tarantola 1995 3361: 3334: 3312:10.1080/01621459.2000.10473908 3291: 3240: 3181: 3162: 3127: 2990:and Grüne-Yanoff and Weirich. 2710: 2619:Monte Carlo methods in finance 2594:Monte Carlo methods in finance 2423:To provide implementations of 2175: 2168:that form the basis of modern 2142:binary collision approximation 2118:Monte Carlo molecular modeling 1659:Monte Carlo and random numbers 1585: 1409:Los Alamos National Laboratory 1286: 1283: 1280: 1266: 1257: 1246: 1231: 1218: 957: 947: 414:Suppose one wants to know the 269:, and generating draws from a 248: 243:pseudorandom number generators 13: 1: 9571:Geographic information system 8787:Simultaneous equations models 6647:. Cambridge University Press. 6544:American Mathematical Society 5625:Silver, David; Veness, Joel. 5426:Sawilowsky & Fahoome 2003 4900:10.1080/00423114.2017.1283046 4829:10.1016/s0246-0203(00)01064-5 4606:Annals of Applied Probability 4087:10.1016/S0304-4149(99)00094-0 3525:: CS1 maint: date and year ( 3406:Kolokoltsov, Vassili (2010). 3370:IEEE Control Systems Magazine 3343:IEEE Control Systems Magazine 3103: 3082:Multilevel Monte Carlo method 3027:Direct simulation Monte Carlo 2981: 2945:leads to the definition of a 2941:Probabilistic formulation of 2881:Metropolis–Hastings algorithm 2855:recursive stratified sampling 2733:Errors reduce by a factor of 2250:direct simulation Monte Carlo 1925:Dissipative particle dynamics 1715:Pseudo-random number sampling 1590:There is no consensus on how 1527:diffusion Monte Carlo methods 678:| > 0. Choose the desired 547: 450:that is arbitrarily close to 442:exists. A sufficiently large 333:, integrals described by the 8754:Coefficient of determination 8365:Uniformly most powerful test 6510:10.1016/0019-1035(79)90199-4 6051:10.1016/0021-9991(87)90054-4 5904:10.1371/journal.pone.0189718 5819:Numerical Methods in Finance 5680:10.1007/978-3-642-17928-0_10 5220:MacGillivray & Dodd 1982 5207:10.1016/0168-583X(84)90321-5 4449:(4): 555–580. Archived from 3667:Chemical Engineering Science 3108: 3097:Temporal difference learning 2879:). Such methods include the 2331:probability density function 2271:that forms the heart of the 2198:microelectronics engineering 1513:Institute for Advanced Study 383:nonlinear filtering equation 289:interacting particle systems 7: 9708:Risk analysis methodologies 9323:Proportional hazards models 9267:Spectral density estimation 9249:Vector autoregression (VAR) 8683:Maximum posterior estimator 7915:Randomized controlled trial 6979:10.1529/biophysj.107.125369 6778:Journal of Chemical Physics 6731:MacKeown, P. Kevin (1997). 6539:Introduction to Probability 6521:. Reading: Addison-Wesley. 6463:10.1088/0031-9155/55/17/021 5837:10.1007/978-3-642-25746-9_1 5584:10.1007/978-3-540-87608-3_6 5145:10.1088/0031-9155/51/13/R17 5094:10.1088/0031-9155/59/6/R183 5041:10.1088/0031-9155/59/4/R151 4213:10.1103/physrevlett.71.2159 3057:Mean-field particle methods 3017:Auxiliary field Monte Carlo 2993: 2971:distribution is available. 2899:Simulation and optimization 2310:mean-field particle methods 2138:radiation materials science 1460:for the development of the 1428:, suggested using the name 1061:run the simulation for the 917:is sufficiently large when 783:run the simulation for the 577:throw the three dice until 498:run the simulation for the 103: 10: 9724: 9693:Statistical approximations 9083:Multivariate distributions 7503:Average absolute deviation 7040:Cambridge University Press 6198: 6141:http://www.jhuapl.edu/ISSO 5496:10.1016/j.csda.2009.09.018 5401:Journal of Urban Economics 5345:Cambridge University Press 4600:Del Moral, Pierre (1998). 4434:Del Moral, Pierre (1996). 4101:Del Moral, Pierre (2003). 3988:Del Moral, Pierre (2004). 3843:Proc. Natl. Acad. Sci. USA 3579:AIP Conference Proceedings 3430:Del Moral, Pierre (2013). 3412:Cambridge University Press 3408:Nonlinear Markov processes 3032:Dynamic Monte Carlo method 3022:Biology Monte Carlo method 2930:traveling salesman problem 2919:traveling salesman problem 2902: 2714: 2691:as a real instance of the 2591: 2527: 2467: 2231:quantitative risk analysis 2126:statistical field theories 2104:calculations to designing 2085: 1860:Morse/Long-range potential 1448:and others programmed the 1368: 146:quadrant (circular sector) 126:computation of the outputs 29: 9597: 9551: 9488: 9441: 9404: 9400: 9387: 9359: 9341: 9308: 9299: 9257: 9204: 9165: 9114: 9105: 9071:Structural equation model 9026: 8983: 8979: 8954: 8913: 8879: 8833: 8800: 8762: 8729: 8725: 8701: 8641: 8550: 8469: 8433: 8424: 8407:Score/Lagrange multiplier 8392: 8345: 8290: 8216: 8207: 8017: 8013: 8000: 7959: 7933: 7885: 7840: 7822:Sample size determination 7787: 7783: 7770: 7674: 7629: 7603: 7585: 7541: 7493: 7413: 7404: 7400: 7387: 7369: 7186:10.22237/jmasm/1051748460 7119:Project Appraisal Journal 6630:Hubbard, Douglas (2009). 6599:Hubbard, Douglas (2007). 6274:10.1007/s10910-008-9467-3 5413:10.1016/j.jue.2014.02.005 5387:Milik & Skolnick 1993 5314:10.1016/J.JCP.2020.110002 4938:The Astrophysical Journal 4867:Kalos & Whitlock 2008 4655:10.1137/s0036139996307371 3837:McKean, Henry P. (1966). 3722:10.1016/j.jcp.2018.01.029 3687:10.1016/j.ces.2013.08.008 3644:10.1016/j.cpc.2014.01.006 3122:Kalos & Whitlock 2008 2889:Wang and Landau algorithm 2870:low-discrepancy sequences 1730:quasi-Monte Carlo methods 1725:Low-discrepancy sequences 764:is the mean of the first 446:will produce a value for 47:with a Monte Carlo method 9566:Environmental statistics 9088:Elliptical distributions 8881:Generalized linear model 8810:Simple linear regression 8580:Hodges–Lehmann estimator 8037:Probability distribution 7946:Stochastic approximation 7508:Coefficient of variation 7018:10.1504/IJVD.2001.001963 6406:Eckhardt, Roger (1987). 5864:. John Wiley & Sons. 4969:10.3847/1538-4357/aa7ede 4367:10.1049/ip-f-2.1993.0015 4259:10.1103/PhysRevA.30.2713 4163:10.1103/physreve.61.4566 3936:10.1093/mind/LIX.236.433 3382:10.1109/MCS.2018.2876959 3355:10.1109/MCS.2003.1188770 3087:Quasi-Monte Carlo method 2947:probability distribution 2877:Markov chain Monte Carlo 2866:quasi-Monte Carlo method 2864:A similar approach, the 2705:Random number generation 2677: 2582:probability distribution 2433:asymptotic distributions 2261:Monte Carlo localization 2136:for quantum systems. In 1761:probability distribution 1719:probability distribution 1612:distinguishes between a 1525:, and more specifically 581:is met or first exceeded 357:model with a prescribed 351:Markov chain Monte Carlo 347:probability distribution 271:probability distribution 144:For example, consider a 117:probability distribution 30:Not to be confused with 9226:Cross-correlation (XCF) 8834:Non-standard predictors 8268:Lehmann–Scheffé theorem 7941:Adaptive clinical trial 7297:. ACMO Academic Press. 6574:Hartmann, A.K. (2009). 6422:Fishman, G. S. (1995). 6289:Berg, Bernd A. (2004). 5461:10.1198/106186005X78800 4880:Vehicle System Dynamics 4628:10.1214/aoap/1028903535 2905:Stochastic optimization 2807:, this method displays 2782:curse of dimensionality 2717:Monte Carlo integration 2476:Monte-Carlo tree search 2470:Monte Carlo tree search 2287:reliability engineering 1865:Lennard-Jones potential 1619:Here are the examples: 1606:Monte Carlo integration 1582:Guionnet and L. Miclo. 1390:Buffon's needle problem 1359:embarrassingly parallel 787:time, producing result 379:McKean–Vlasov processes 297:kinetic models of gases 293:McKean–Vlasov processes 58:, are a broad class of 56:Monte Carlo experiments 43:The approximation of a 9622:Mathematics portal 9443:Engineering statistics 9351:Nelson–Aalen estimator 8928:Analysis of covariance 8815:Ordinary least squares 8739:Pearson product-moment 8143:Statistical functional 8054:Empirical distribution 7887:Controlled experiments 7616:Frequency distribution 7394:Descriptive statistics 7250:Inverse Problem Theory 7057:Ripley, B. D. (1987). 6879:10.1002/prot.340150104 6735:. New York: Springer. 6426:. New York: Springer. 6391:. New York: Springer. 6315:. New York: Springer. 6245:10.1515/mcma-2016-0102 6139:, Wiley, Hoboken, NJ. 5994:10.21315/km2021.39.2.8 5945:State Bar of Wisconsin 5278:10.1006/jcph.1996.0141 4786:C. R. Acad. Sci. Paris 3864:10.1073/pnas.56.6.1907 3269:10.1093/biomet/57.1.97 2911:numerical optimization 2893:sequential Monte Carlo 2835: 2768: 2761: 2726: 2632: 2610:Stochastic asset model 2248:fluid flows using the 2102:quantum chromodynamics 1653:Gelman-Rubin statistic 1641:Monte Carlo simulation 1555:Sequential Monte Carlo 1544:Marshall N. Rosenbluth 1432:, which refers to the 1422: 1308: 970: 591:= the number of throws 141: 48: 9703:Randomized algorithms 9698:Stochastic simulation 9683:Computational physics 9678:Statistical mechanics 9538:Population statistics 9480:System identification 9214:Autocorrelation (ACF) 9142:Exponential smoothing 9056:Discriminant analysis 9051:Canonical correlation 8915:Partition of variance 8777:Regression validation 8621:(Jonckheere–Terpstra) 8520:Likelihood-ratio test 8209:Frequentist inference 8121:Location–scale family 8042:Sampling distribution 8007:Statistical inference 7974:Cross-sectional study 7961:Observational studies 7920:Randomized experiment 7749:Stem-and-leaf display 7551:Central limit theorem 7274:John Wiley & Sons 7059:Stochastic Simulation 6683:John Wiley & Sons 6636:John Wiley & Sons 6609:John Wiley & Sons 6594:on February 11, 2009. 6362:10.1145/143242.143290 6135:Spall, J. C. (2003), 5518:Sander.landofsand.com 5347:. 2013. p. 697. 4736:10.1007/s004400050249 4695:10.1007/s004400050131 3968:Barricelli, Nils Aall 3950:Barricelli, Nils Aall 3506:Owen, Art B. (2013). 3436:. Chapman & Hall/ 2836: 2805:central limit theorem 2773:numerical integration 2762: 2732: 2724: 2683:Nassim Nicholas Taleb 2555:, computer generated 2347:computational biology 2341:Computational biology 2242:rarefied gas dynamics 2094:computational physics 1800:Computational physics 1685:uniformly distributed 1597:stochastic simulation 1548:Arianna W. Rosenbluth 1517:Princeton, New Jersey 1413: 1309: 971: 267:numerical integration 136: 129:Aggregate the results 42: 32:Monte Carlo algorithm 9461:Probabilistic design 9046:Principal components 8889:Exponential families 8841:Nonlinear regression 8820:General linear model 8782:Mixed effects models 8772:Errors and residuals 8749:Confounding variable 8651:Bayesian probability 8629:Van der Waerden test 8619:Ordered alternative 8384:Multiple comparisons 8263:Rao–Blackwellization 8226:Estimating equations 8182:Statistical distance 7900:Factorial experiment 7433:Arithmetic-Geometric 7328:at Wikimedia Commons 7268:Vose, David (2008). 6580:. World Scientific. 6346:. pp. 123–129. 6207:Anderson, Herbert L. 5545:on November 29, 2015 4521:on November 10, 2022 4180:on November 7, 2014. 3797:10.1109/MAHC.2014.40 2960:marginal probability 2811: 2737: 2688:Fooled by Randomness 2588:Finance and business 2182:sensitivity analysis 1965:Metropolis algorithm 1666:truly random numbers 1623:Simulation: Drawing 1509:Nils Aall Barricelli 1206: 1196:/2, use a value for 1065:time, giving result 923: 502:time, giving result 331:law of large numbers 285:cellular Potts model 184:a quadrant within it 180:Draw a square, then 9688:Sampling techniques 9668:Monte Carlo methods 9533:Official statistics 9456:Methods engineering 9137:Seasonal adjustment 8905:Poisson regressions 8825:Bayesian regression 8764:Regression analysis 8744:Partial correlation 8716:Regression analysis 8315:Prediction interval 8310:Likelihood interval 8300:Confidence interval 8292:Interval estimation 8253:Unbiased estimators 8071:Model specification 7951:Up-and-down designs 7639:Partial correlation 7595:Index of dispersion 7513:Interquartile range 7131:10.2139/ssrn.265905 6971:2009BpJ....96.1076O 6923:1995JGR...10012431M 6917:(B7): 12431–12447. 6790:1953JChPh..21.1087M 6654:Monte Carlo Methods 6559:. London: Methuen. 6557:Monte Carlo Methods 6502:1979Icar...38..451G 6455:2010PMB....55.5213F 6043:1987JCoPh..68..237M 5954:on November 6, 2018 5895:2017PLoSO..1289718A 5672:2011LNCS.6515..105L 5664:Computers and Games 5566:Computers and Games 5306:2021JCoPh.42910002C 5269:1996JCoPh.126..328D 5199:1984NIMPB...2..814M 5137:2006PMB....51R.287R 5086:2014PMB....59R.183H 5033:2014PMB....59R.151J 4960:2017ApJ...845...66R 4892:2017VSD....55..827S 4821:2001AIHPB..37..155D 4496:1997ITAES..33..835C 4322:1955JChPh..23..356R 4251:1984PhRvA..30.2713H 4245:(2713): 2713–2719. 4205:1993PhRvL..71.2159C 4155:2000PhRvE..61.4566A 3855:1966PNAS...56.1907M 3714:2018JCoPh.360...93L 3679:2013ChEnS.104..451W 3636:2014CoPhC.185.1355C 3591:2016AIPC.1773g0001A 3261:1970Bimka..57...97H 3202:1953JChPh..21.1087M 3047:Kinetic Monte Carlo 2851:stratified sampling 2846:importance sampling 2693:reverse Turing test 2541:global illumination 2410:Cauchy distribution 2367:thought experiments 2257:autonomous robotics 2209:integrated circuits 2170:weather forecasting 2130:Quantum Monte Carlo 2114:statistical physics 1948:Monte Carlo methods 1604:being reserved for 1523:Quantum Monte Carlo 1489:Henry P. McKean Jr. 1474:operations research 1446:Nicholas Metropolis 1426:Nicholas Metropolis 1403:In the late 1940s, 1383:simulated annealing 1352:Computational costs 1113:Choose a value for 907:is the mean of the 391:mean-field particle 316:boundary conditions 52:Monte Carlo methods 45:normal distribution 18:Monte Carlo methods 9673:Numerical analysis 9553:Spatial statistics 9433:Medical statistics 9333:First hitting time 9287:Whittle likelihood 8938:Degrees of freedom 8933:Multivariate ANOVA 8866:Heteroscedasticity 8678:Bayesian estimator 8643:Bayesian inference 8492:Kolmogorov–Smirnov 8377:Randomization test 8347:Testing hypotheses 8320:Tolerance interval 8231:Maximum likelihood 8126:Exponential family 8059:Density estimation 8019:Statistical theory 7979:Natural experiment 7925:Scientific control 7842:Survey methodology 7528:Standard deviation 7326:Monte Carlo method 6762:Los Alamos Science 6716:10.1007/BF00683346 6415:Los Alamos Science 6219:Los Alamos Science 5751:. January 3, 2014. 5732:Szirmay-Kalos 2008 4120:10.1051/ps:2003001 4037:10.1007/BFb0103798 3042:Genetic algorithms 3008:Mathematics portal 2831: 2830: 2769: 2757: 2756: 2727: 2699:Use in mathematics 2621:are often used to 2535:Design and visuals 2455:randomization test 2447:Fisher information 2440:Bayesian inference 2388:Applied statistics 2382:rendering equation 2349:, for example for 2314:empirical measures 2298:Bayesian inference 2280:telecommunications 2229:and contribute to 2224:mineral processing 2132:methods solve the 2122:molecular dynamics 2098:physical chemistry 1993:Molecular dynamics 1563:Bayesian inference 1493:Theodore E. Harris 1470:physical chemistry 1434:Monte Carlo Casino 1363:parallel computing 1304: 966: 410:Simple Monte Carlo 387:empirical measures 367:empirical measures 312:definite integrals 281:degrees of freedom 142: 82:Monte Carlo Casino 49: 9655: 9654: 9593: 9592: 9589: 9588: 9528:National accounts 9498:Actuarial science 9490:Social statistics 9383: 9382: 9379: 9378: 9375: 9374: 9310:Survival function 9295: 9294: 9157:Granger causality 8998:Contingency table 8973:Survival analysis 8950: 8949: 8946: 8945: 8802:Linear regression 8697: 8696: 8693: 8692: 8668:Credible interval 8637: 8636: 8420: 8419: 8236:Method of moments 8105:Parametric family 8066:Statistical model 7996: 7995: 7992: 7991: 7910:Random assignment 7832:Statistical power 7766: 7765: 7762: 7761: 7611:Contingency table 7581: 7580: 7448:Generalized/power 7324:Media related to 7304:978-619-90684-3-4 7260:978-0-89871-572-9 7245:Tarantola, Albert 7236:978-3-8364-7919-6 7157:978-0-9740236-0-1 7102:978-0-470-17793-8 7083:978-0-387-21239-5 7049:978-0-521-43064-7 6940:on March 10, 2021 6931:10.1029/94JB03097 6798:10.1063/1.1699114 6742:978-981-3083-26-4 6692:978-0-470-17793-8 6667:978-3-527-40760-6 6587:978-981-283-415-7 6566:978-0-416-52340-9 6528:978-0-201-16504-3 6449:(17): 5213–5229. 6433:978-0-387-94527-9 6398:978-0-387-95146-1 6371:978-0-89791-489-5 6322:978-0-387-54369-7 6300:978-981-238-935-0 6078:10.1021/jp972280j 6018:Press et al. 1996 5846:978-3-642-25745-2 5689:978-3-642-17927-3 5593:978-3-540-87607-6 5375:Ojeda et al. 2009 5354:978-1-107-66182-0 5131:(13): R287–R301. 4331:10.1063/1.1741967 4046:978-3-540-67314-9 3599:10.1063/1.4964983 3210:10.1063/1.1699114 3148:10.1002/wics.1314 3136:WIREs Comput Stat 2828: 2794:degree of freedom 2786:iterated integral 2754: 2627:project schedules 2577:search and rescue 2563:Search and rescue 2429:permutation tests 2373:Computer graphics 2363:chemical reaction 2335:radiative forcing 2294:signal processing 2184:and quantitative 2134:many-body problem 2084: 2083: 1935:Turbulence models 1915:Lattice Boltzmann 1895:Finite difference 1792:Physical sciences 1670:primality testing 1559:signal processing 1102: 1072: 1059: 897: 877: 858: 847: 832: 821: 795: 781: 769: 754: 686:is indeed within 634:is large enough, 627: 610: 592: 582: 544: 530: 509: 314:with complicated 223:approximation of 16:(Redirected from 9715: 9643: 9642: 9631: 9630: 9620: 9619: 9605: 9604: 9508:Crime statistics 9402: 9401: 9389: 9388: 9306: 9305: 9272:Fourier analysis 9259:Frequency domain 9239: 9186: 9152:Structural break 9112: 9111: 9061:Cluster analysis 9008:Log-linear model 8981: 8980: 8956: 8955: 8897: 8871:Homoscedasticity 8727: 8726: 8703: 8702: 8622: 8614: 8606: 8605:(Kruskal–Wallis) 8590: 8575: 8530:Cross validation 8515: 8497:Anderson–Darling 8444: 8431: 8430: 8402:Likelihood-ratio 8394:Parametric tests 8372:Permutation test 8355:1- & 2-tails 8246:Minimum distance 8218:Point estimation 8214: 8213: 8165:Optimal decision 8116: 8015: 8014: 8002: 8001: 7984:Quasi-experiment 7934:Adaptive designs 7785: 7784: 7772: 7771: 7649:Rank correlation 7411: 7410: 7402: 7401: 7389: 7388: 7356: 7349: 7342: 7333: 7332: 7323: 7308: 7287: 7272:(3rd ed.). 7264: 7240: 7221: 7219: 7217: 7211: 7200: 7190: 7188: 7161: 7142: 7116: 7106: 7087: 7066: 7063:Wiley & Sons 7053: 7030: 7021: 7012:(1–4): 183–194. 7000: 6990: 6965:(3): 1076–1082. 6949: 6947: 6945: 6939: 6933:. Archived from 6908: 6898: 6861: 6836:(247): 335–341. 6817: 6765: 6759: 6746: 6727: 6696: 6671: 6648: 6639: 6626: 6606: 6595: 6590:. Archived from 6570: 6551: 6532: 6513: 6482: 6437: 6418: 6412: 6402: 6383: 6355: 6338: 6326: 6304: 6285: 6256: 6227: 6215: 6192: 6189: 6183: 6180: 6174: 6171: 6165: 6160: 6154: 6149: 6143: 6133: 6127: 6126: 6108: 6106:cond-mat/0212648 6088: 6082: 6081: 6061: 6055: 6054: 6026: 6020: 6015: 6006: 6005: 5979: 5970: 5964: 5963: 5961: 5959: 5953: 5947:. Archived from 5942: 5933: 5927: 5926: 5916: 5906: 5889:(12): e0189718. 5872: 5866: 5865: 5857: 5851: 5850: 5830: 5813: 5807: 5806: 5804: 5802: 5788: 5782: 5781: 5779: 5777: 5768: 5759: 5753: 5752: 5741: 5735: 5729: 5723: 5722: 5720: 5718: 5709: 5700: 5694: 5693: 5659: 5653: 5652: 5650: 5648: 5643:on July 18, 2016 5642: 5636:. Archived from 5631: 5622: 5616: 5615: 5613: 5604: 5598: 5597: 5577: 5561: 5555: 5554: 5552: 5550: 5541:. Archived from 5535: 5529: 5528: 5526: 5524: 5515: 5506: 5500: 5499: 5479: 5473: 5472: 5454: 5434: 5428: 5423: 5417: 5416: 5396: 5390: 5384: 5378: 5372: 5366: 5365: 5363: 5361: 5342: 5332: 5326: 5325: 5289: 5283: 5282: 5280: 5248: 5242: 5239: 5233: 5228: 5222: 5217: 5211: 5210: 5182: 5176: 5171: 5165: 5164: 5120: 5114: 5113: 5080:(6): R183–R231. 5069: 5063: 5062: 5052: 5027:(4): R151–R182. 5012: 5006: 5000: 4994: 4988: 4982: 4981: 4971: 4953: 4929: 4923: 4918: 4912: 4911: 4875: 4869: 4864: 4858: 4853: 4844: 4839: 4833: 4832: 4800: 4794: 4793: 4781: 4775: 4774: 4764: 4755: 4749: 4748: 4738: 4714: 4708: 4707: 4697: 4673: 4667: 4666: 4649:(5): 1568–1590. 4638: 4632: 4631: 4621: 4597: 4591: 4587: 4581: 4577: 4571: 4567: 4561: 4557: 4551: 4547: 4541: 4537: 4531: 4530: 4528: 4526: 4520: 4514:. Archived from 4504:10.1109/7.599254 4481: 4472: 4466: 4465: 4463: 4461: 4456:on March 4, 2016 4455: 4440: 4431: 4422: 4421: 4393: 4387: 4386: 4350: 4344: 4343: 4333: 4301: 4292: 4291: 4278: 4269: 4263: 4262: 4234: 4225: 4224: 4188: 4182: 4181: 4179: 4173:. Archived from 4149:(4): 4566–4575. 4140: 4131: 4125: 4124: 4122: 4098: 4092: 4091: 4089: 4065: 4059: 4058: 4018: 4009: 4008: 3985: 3976: 3975: 3964: 3958: 3957: 3946: 3940: 3939: 3930:(238): 433–460. 3919: 3913: 3912: 3902: 3893: 3887: 3886: 3876: 3866: 3849:(6): 1907–1911. 3834: 3828: 3827: 3815: 3809: 3808: 3776: 3770: 3764: 3755: 3749: 3743: 3737: 3726: 3725: 3697: 3691: 3690: 3662: 3656: 3655: 3629: 3620:(5): 1355–1363. 3609: 3603: 3602: 3574: 3568: 3567: 3559: 3553: 3552: 3540: 3531: 3530: 3524: 3516: 3514: 3503: 3490: 3489: 3471: 3469:cond-mat/0212648 3451: 3445: 3444: 3427: 3416: 3415: 3403: 3394: 3393: 3365: 3359: 3358: 3338: 3332: 3331: 3306:(449): 121–134. 3295: 3289: 3288: 3244: 3238: 3237: 3196:(6): 1087–1092. 3185: 3179: 3178: 3166: 3160: 3159: 3131: 3125: 3119: 3010: 3005: 3004: 2954:defined, etc.). 2943:inverse problems 2937:Inverse problems 2840: 2838: 2837: 2832: 2829: 2824: 2822: 2766: 2764: 2763: 2758: 2755: 2750: 2748: 2651:physical assault 2425:hypothesis tests 2302:particle filters 2240:, in particular 2150:particle physics 2146:ion implantation 2076: 2069: 2062: 1988:Particle-in-cell 1910:Boundary element 1870:Yukawa potential 1833:Particle physics 1823:Electromagnetics 1810: 1796: 1795: 1741:Mersenne Twister 1539:Robert Richtmyer 1478:Rand Corporation 1418:John von Neumann 1395: 1316:For example, if 1313: 1311: 1310: 1305: 1303: 1302: 1293: 1276: 1256: 1239: 1238: 1074: 1060: 1035: 975: 973: 972: 967: 965: 964: 946: 941: 940: 878: 860: 849: 834: 823: 796: 782: 770: 756: 726: 680:confidence level 611: 594: 584: 559: 531: 511: 480: 425:(and knows that 404: 344: 226: 217: 213: 211: 210: 207: 204: 203: 174: 169: 167: 166: 163: 160: 159: 140: 21: 9723: 9722: 9718: 9717: 9716: 9714: 9713: 9712: 9658: 9657: 9656: 9651: 9614: 9585: 9547: 9484: 9470:quality control 9437: 9419:Clinical trials 9396: 9371: 9355: 9343:Hazard function 9337: 9291: 9253: 9237: 9200: 9196:Breusch–Godfrey 9184: 9161: 9101: 9076:Factor analysis 9022: 9003:Graphical model 8975: 8942: 8909: 8895: 8875: 8829: 8796: 8758: 8721: 8720: 8689: 8633: 8620: 8612: 8604: 8588: 8573: 8552:Rank statistics 8546: 8525:Model selection 8513: 8471:Goodness of fit 8465: 8442: 8416: 8388: 8341: 8286: 8275:Median unbiased 8203: 8114: 8047:Order statistic 8009: 7988: 7955: 7929: 7881: 7836: 7779: 7777:Data collection 7758: 7670: 7625: 7599: 7577: 7537: 7489: 7406:Continuous data 7396: 7383: 7365: 7360: 7316: 7311: 7305: 7284: 7261: 7237: 7215: 7213: 7212:on May 25, 2012 7209: 7198: 7158: 7114: 7103: 7084: 7050: 6943: 6941: 6937: 6911:J. Geophys. Res 6906: 6757: 6743: 6693: 6685:. p. 772. 6668: 6623: 6588: 6567: 6529: 6443:Phys. Med. Biol 6434: 6410: 6399: 6372: 6331:Caflisch, R. E. 6323: 6301: 6213: 6201: 6196: 6195: 6190: 6186: 6181: 6177: 6172: 6168: 6161: 6157: 6150: 6146: 6134: 6130: 6089: 6085: 6062: 6058: 6027: 6023: 6016: 6009: 5982:Kajian Malaysia 5977: 5971: 5967: 5957: 5955: 5951: 5940: 5934: 5930: 5873: 5869: 5858: 5854: 5847: 5828:10.1.1.359.7957 5814: 5810: 5800: 5798: 5790: 5789: 5785: 5775: 5773: 5766: 5760: 5756: 5743: 5742: 5738: 5730: 5726: 5716: 5714: 5707: 5701: 5697: 5690: 5660: 5656: 5646: 5644: 5640: 5629: 5623: 5619: 5611: 5605: 5601: 5594: 5575:10.1.1.159.4373 5562: 5558: 5548: 5546: 5537: 5536: 5532: 5522: 5520: 5513: 5507: 5503: 5480: 5476: 5435: 5431: 5424: 5420: 5397: 5393: 5385: 5381: 5373: 5369: 5359: 5357: 5355: 5340: 5334: 5333: 5329: 5290: 5286: 5249: 5245: 5240: 5236: 5229: 5225: 5218: 5214: 5183: 5179: 5172: 5168: 5121: 5117: 5070: 5066: 5013: 5009: 5001: 4997: 4989: 4985: 4930: 4926: 4919: 4915: 4876: 4872: 4865: 4861: 4856:Sawilowsky 2003 4854: 4847: 4840: 4836: 4801: 4797: 4782: 4778: 4762: 4756: 4752: 4715: 4711: 4674: 4670: 4639: 4635: 4598: 4594: 4588: 4584: 4578: 4574: 4568: 4564: 4558: 4554: 4548: 4544: 4538: 4534: 4524: 4522: 4518: 4479: 4473: 4469: 4459: 4457: 4453: 4438: 4432: 4425: 4410:10.2307/1390750 4394: 4390: 4351: 4347: 4302: 4295: 4276: 4270: 4266: 4235: 4228: 4193:Phys. Rev. Lett 4189: 4185: 4177: 4138: 4132: 4128: 4099: 4095: 4066: 4062: 4047: 4019: 4012: 4002: 3986: 3979: 3965: 3961: 3947: 3943: 3920: 3916: 3900: 3894: 3890: 3835: 3831: 3816: 3812: 3777: 3773: 3765: 3758: 3750: 3746: 3740:Metropolis 1987 3738: 3729: 3698: 3694: 3663: 3659: 3610: 3606: 3575: 3571: 3560: 3556: 3541: 3534: 3518: 3517: 3512: 3504: 3493: 3452: 3448: 3440:. p. 626. 3428: 3419: 3404: 3397: 3366: 3362: 3339: 3335: 3296: 3292: 3245: 3241: 3186: 3182: 3167: 3163: 3132: 3128: 3120: 3116: 3111: 3106: 3101: 3006: 2999: 2996: 2984: 2939: 2907: 2901: 2859:VEGAS algorithm 2823: 2818: 2812: 2809: 2808: 2749: 2744: 2738: 2735: 2734: 2719: 2713: 2701: 2680: 2659: 2657:Library science 2635: 2612: 2590: 2565: 2537: 2532: 2472: 2466: 2390: 2375: 2361:to see if some 2343: 2323: 2308:are a class of 2269:particle filter 2178: 2166:ensemble models 2144:for simulating 2090: 2080: 2051: 2050: 2006: 1998: 1997: 1978: 1970: 1969: 1950: 1940: 1939: 1890: 1880: 1879: 1875:Morse potential 1855: 1845: 1794: 1778: 1753: 1661: 1588: 1393: 1371: 1354: 1298: 1294: 1289: 1272: 1252: 1234: 1230: 1207: 1204: 1203: 1166: 1155: 1148: 1142: 1135: 1108: 1103: 1087: 1070: 1044: 1002: 991: 960: 956: 942: 936: 932: 924: 921: 920: 905: 898: 890: 874: 857: 853: 844: 831: 827: 819: 815: 808: 801: 792: 762: 753: 746: 739: 731: 694:. Let z be the 664: 662:General formula 659: 638:will be within 628: 607: 589: 550: 545: 524: 507: 412: 363:ergodic theorem 320:oil exploration 251: 224: 215: 208: 205: 201: 200: 199: 197: 172: 170:, the value of 164: 161: 157: 156: 155: 153: 148:inscribed in a 138: 119:over the domain 106: 70:random sampling 35: 28: 23: 22: 15: 12: 11: 5: 9721: 9711: 9710: 9705: 9700: 9695: 9690: 9685: 9680: 9675: 9670: 9653: 9652: 9650: 9649: 9637: 9625: 9611: 9598: 9595: 9594: 9591: 9590: 9587: 9586: 9584: 9583: 9578: 9573: 9568: 9563: 9557: 9555: 9549: 9548: 9546: 9545: 9540: 9535: 9530: 9525: 9520: 9515: 9510: 9505: 9500: 9494: 9492: 9486: 9485: 9483: 9482: 9477: 9472: 9463: 9458: 9453: 9447: 9445: 9439: 9438: 9436: 9435: 9430: 9425: 9416: 9414:Bioinformatics 9410: 9408: 9398: 9397: 9385: 9384: 9381: 9380: 9377: 9376: 9373: 9372: 9370: 9369: 9363: 9361: 9357: 9356: 9354: 9353: 9347: 9345: 9339: 9338: 9336: 9335: 9330: 9325: 9320: 9314: 9312: 9303: 9297: 9296: 9293: 9292: 9290: 9289: 9284: 9279: 9274: 9269: 9263: 9261: 9255: 9254: 9252: 9251: 9246: 9241: 9233: 9228: 9223: 9222: 9221: 9219:partial (PACF) 9210: 9208: 9202: 9201: 9199: 9198: 9193: 9188: 9180: 9175: 9169: 9167: 9166:Specific tests 9163: 9162: 9160: 9159: 9154: 9149: 9144: 9139: 9134: 9129: 9124: 9118: 9116: 9109: 9103: 9102: 9100: 9099: 9098: 9097: 9096: 9095: 9080: 9079: 9078: 9068: 9066:Classification 9063: 9058: 9053: 9048: 9043: 9038: 9032: 9030: 9024: 9023: 9021: 9020: 9015: 9013:McNemar's test 9010: 9005: 9000: 8995: 8989: 8987: 8977: 8976: 8952: 8951: 8948: 8947: 8944: 8943: 8941: 8940: 8935: 8930: 8925: 8919: 8917: 8911: 8910: 8908: 8907: 8891: 8885: 8883: 8877: 8876: 8874: 8873: 8868: 8863: 8858: 8853: 8851:Semiparametric 8848: 8843: 8837: 8835: 8831: 8830: 8828: 8827: 8822: 8817: 8812: 8806: 8804: 8798: 8797: 8795: 8794: 8789: 8784: 8779: 8774: 8768: 8766: 8760: 8759: 8757: 8756: 8751: 8746: 8741: 8735: 8733: 8723: 8722: 8719: 8718: 8713: 8707: 8699: 8698: 8695: 8694: 8691: 8690: 8688: 8687: 8686: 8685: 8675: 8670: 8665: 8664: 8663: 8658: 8647: 8645: 8639: 8638: 8635: 8634: 8632: 8631: 8626: 8625: 8624: 8616: 8608: 8592: 8589:(Mann–Whitney) 8584: 8583: 8582: 8569: 8568: 8567: 8556: 8554: 8548: 8547: 8545: 8544: 8543: 8542: 8537: 8532: 8522: 8517: 8514:(Shapiro–Wilk) 8509: 8504: 8499: 8494: 8489: 8481: 8475: 8473: 8467: 8466: 8464: 8463: 8455: 8446: 8434: 8428: 8426:Specific tests 8422: 8421: 8418: 8417: 8415: 8414: 8409: 8404: 8398: 8396: 8390: 8389: 8387: 8386: 8381: 8380: 8379: 8369: 8368: 8367: 8357: 8351: 8349: 8343: 8342: 8340: 8339: 8338: 8337: 8332: 8322: 8317: 8312: 8307: 8302: 8296: 8294: 8288: 8287: 8285: 8284: 8279: 8278: 8277: 8272: 8271: 8270: 8265: 8250: 8249: 8248: 8243: 8238: 8233: 8222: 8220: 8211: 8205: 8204: 8202: 8201: 8196: 8191: 8190: 8189: 8179: 8174: 8173: 8172: 8162: 8161: 8160: 8155: 8150: 8140: 8135: 8130: 8129: 8128: 8123: 8118: 8102: 8101: 8100: 8095: 8090: 8080: 8079: 8078: 8073: 8063: 8062: 8061: 8051: 8050: 8049: 8039: 8034: 8029: 8023: 8021: 8011: 8010: 7998: 7997: 7994: 7993: 7990: 7989: 7987: 7986: 7981: 7976: 7971: 7965: 7963: 7957: 7956: 7954: 7953: 7948: 7943: 7937: 7935: 7931: 7930: 7928: 7927: 7922: 7917: 7912: 7907: 7902: 7897: 7891: 7889: 7883: 7882: 7880: 7879: 7877:Standard error 7874: 7869: 7864: 7863: 7862: 7857: 7846: 7844: 7838: 7837: 7835: 7834: 7829: 7824: 7819: 7814: 7809: 7807:Optimal design 7804: 7799: 7793: 7791: 7781: 7780: 7768: 7767: 7764: 7763: 7760: 7759: 7757: 7756: 7751: 7746: 7741: 7736: 7731: 7726: 7721: 7716: 7711: 7706: 7701: 7696: 7691: 7686: 7680: 7678: 7672: 7671: 7669: 7668: 7663: 7662: 7661: 7656: 7646: 7641: 7635: 7633: 7627: 7626: 7624: 7623: 7618: 7613: 7607: 7605: 7604:Summary tables 7601: 7600: 7598: 7597: 7591: 7589: 7583: 7582: 7579: 7578: 7576: 7575: 7574: 7573: 7568: 7563: 7553: 7547: 7545: 7539: 7538: 7536: 7535: 7530: 7525: 7520: 7515: 7510: 7505: 7499: 7497: 7491: 7490: 7488: 7487: 7482: 7477: 7476: 7475: 7470: 7465: 7460: 7455: 7450: 7445: 7440: 7438:Contraharmonic 7435: 7430: 7419: 7417: 7408: 7398: 7397: 7385: 7384: 7382: 7381: 7376: 7370: 7367: 7366: 7359: 7358: 7351: 7344: 7336: 7330: 7329: 7315: 7314:External links 7312: 7310: 7309: 7303: 7288: 7282: 7265: 7259: 7241: 7235: 7222: 7191: 7179:(1): 218–225. 7162: 7156: 7143: 7107: 7101: 7088: 7082: 7067: 7054: 7048: 7031: 7022: 7001: 6950: 6899: 6862: 6822:Metropolis, N. 6818: 6770:Metropolis, N. 6766: 6751:Metropolis, N. 6747: 6741: 6728: 6710:(2): 419–435. 6697: 6691: 6672: 6666: 6649: 6640: 6627: 6621: 6596: 6586: 6571: 6565: 6552: 6533: 6527: 6514: 6496:(3): 451–455. 6483: 6438: 6432: 6419: 6417:(15): 131–137. 6403: 6397: 6384: 6370: 6353:10.1.1.43.9296 6339: 6327: 6321: 6305: 6299: 6286: 6268:(2): 363–426. 6257: 6228: 6202: 6200: 6197: 6194: 6193: 6184: 6175: 6166: 6163:Tarantola 2005 6155: 6144: 6128: 6099:(3): 411–436. 6083: 6072:(5): 865–880. 6056: 6037:(1): 237–248. 6021: 6007: 5988:(2): 179–202. 5965: 5928: 5867: 5852: 5845: 5808: 5796:risk.octigo.pl 5783: 5754: 5736: 5724: 5695: 5688: 5654: 5634:0.cs.ucl.ac.uk 5617: 5599: 5592: 5556: 5530: 5501: 5490:(2): 272–289. 5474: 5452:10.1.1.142.738 5445:(4): 889–909. 5429: 5418: 5391: 5379: 5367: 5353: 5327: 5284: 5243: 5234: 5223: 5212: 5193:(1): 814–818. 5177: 5166: 5115: 5064: 5007: 4995: 4983: 4924: 4921:Davenport 1992 4913: 4886:(6): 827–852. 4870: 4859: 4845: 4834: 4815:(2): 155–194. 4795: 4776: 4750: 4729:(4): 549–578. 4709: 4688:(2): 217–244. 4668: 4633: 4619:10.1.1.55.5257 4592: 4582: 4572: 4562: 4552: 4542: 4532: 4490:(3): 835–850. 4467: 4423: 4388: 4361:(2): 107–113. 4345: 4316:(2): 356–359. 4293: 4264: 4226: 4183: 4126: 4093: 4080:(2): 193–216. 4060: 4045: 4010: 4000: 3977: 3959: 3941: 3914: 3888: 3829: 3810: 3771: 3769:, p. 250. 3756: 3744: 3727: 3692: 3657: 3604: 3569: 3554: 3532: 3491: 3462:(3): 411–436. 3446: 3417: 3414:. p. 375. 3395: 3360: 3333: 3290: 3239: 3180: 3161: 3142:(6): 386–392. 3126: 3113: 3112: 3110: 3107: 3105: 3102: 3100: 3099: 3094: 3092:Sobol sequence 3089: 3084: 3079: 3074: 3069: 3064: 3059: 3054: 3049: 3044: 3039: 3034: 3029: 3024: 3019: 3013: 3012: 3011: 2995: 2992: 2983: 2980: 2938: 2935: 2915:computer chess 2903:Main article: 2900: 2897: 2885:Gibbs sampling 2827: 2821: 2817: 2771:Deterministic 2753: 2747: 2743: 2715:Main article: 2712: 2709: 2700: 2697: 2679: 2676: 2658: 2655: 2634: 2631: 2589: 2586: 2569:US Coast Guard 2564: 2561: 2536: 2533: 2500: 2499: 2496: 2493: 2490: 2468:Main article: 2465: 2462: 2451: 2450: 2443: 2436: 2421: 2389: 2386: 2374: 2371: 2342: 2339: 2322: 2319: 2318: 2317: 2290: 2283: 2276: 2253: 2246:Knudsen number 2238:fluid dynamics 2234: 2212: 2190:process design 2177: 2174: 2082: 2081: 2079: 2078: 2071: 2064: 2056: 2053: 2052: 2049: 2048: 2043: 2038: 2033: 2028: 2023: 2018: 2013: 2007: 2004: 2003: 2000: 1999: 1996: 1995: 1990: 1985: 1979: 1976: 1975: 1972: 1971: 1968: 1967: 1962: 1960:Gibbs sampling 1957: 1951: 1946: 1945: 1942: 1941: 1938: 1937: 1932: 1927: 1922: 1920:Riemann solver 1917: 1912: 1907: 1905:Finite element 1902: 1897: 1891: 1888:Fluid dynamics 1886: 1885: 1882: 1881: 1878: 1877: 1872: 1867: 1862: 1856: 1853: 1852: 1849: 1848: 1847: 1846: 1840: 1838:Thermodynamics 1835: 1830: 1825: 1820: 1812: 1811: 1803: 1802: 1793: 1790: 1777: 1774: 1752: 1749: 1712: 1711: 1708: 1705: 1702: 1699: 1696: 1660: 1657: 1645: 1644: 1637:a large number 1633: 1629: 1587: 1584: 1482:U.S. Air Force 1454:fission weapon 1405:Stanislaw Ulam 1379:metaheuristics 1370: 1367: 1353: 1350: 1301: 1297: 1292: 1288: 1285: 1282: 1279: 1275: 1271: 1268: 1265: 1262: 1259: 1255: 1251: 1248: 1245: 1242: 1237: 1233: 1229: 1226: 1223: 1220: 1217: 1214: 1211: 1164: 1153: 1146: 1140: 1133: 1107: 1104: 1085: 1068: 1042: 1034: 1000: 989: 963: 959: 955: 952: 949: 945: 939: 935: 931: 928: 903: 888: 872: 855: 851: 842: 829: 825: 817: 813: 806: 799: 790: 760: 751: 744: 737: 729: 725: 663: 660: 658: 652: 605: 587: 558: 549: 546: 522: 505: 479: 416:expected value 411: 408: 375:Markov process 339:empirical mean 335:expected value 250: 247: 239: 238: 235: 220: 219: 194: 191: 185: 131: 130: 127: 120: 113: 105: 102: 86:Stanislaw Ulam 26: 9: 6: 4: 3: 2: 9720: 9709: 9706: 9704: 9701: 9699: 9696: 9694: 9691: 9689: 9686: 9684: 9681: 9679: 9676: 9674: 9671: 9669: 9666: 9665: 9663: 9648: 9647: 9638: 9636: 9635: 9626: 9624: 9623: 9618: 9612: 9610: 9609: 9600: 9599: 9596: 9582: 9579: 9577: 9576:Geostatistics 9574: 9572: 9569: 9567: 9564: 9562: 9559: 9558: 9556: 9554: 9550: 9544: 9543:Psychometrics 9541: 9539: 9536: 9534: 9531: 9529: 9526: 9524: 9521: 9519: 9516: 9514: 9511: 9509: 9506: 9504: 9501: 9499: 9496: 9495: 9493: 9491: 9487: 9481: 9478: 9476: 9473: 9471: 9467: 9464: 9462: 9459: 9457: 9454: 9452: 9449: 9448: 9446: 9444: 9440: 9434: 9431: 9429: 9426: 9424: 9420: 9417: 9415: 9412: 9411: 9409: 9407: 9406:Biostatistics 9403: 9399: 9395: 9390: 9386: 9368: 9367:Log-rank test 9365: 9364: 9362: 9358: 9352: 9349: 9348: 9346: 9344: 9340: 9334: 9331: 9329: 9326: 9324: 9321: 9319: 9316: 9315: 9313: 9311: 9307: 9304: 9302: 9298: 9288: 9285: 9283: 9280: 9278: 9275: 9273: 9270: 9268: 9265: 9264: 9262: 9260: 9256: 9250: 9247: 9245: 9242: 9240: 9238:(Box–Jenkins) 9234: 9232: 9229: 9227: 9224: 9220: 9217: 9216: 9215: 9212: 9211: 9209: 9207: 9203: 9197: 9194: 9192: 9191:Durbin–Watson 9189: 9187: 9181: 9179: 9176: 9174: 9173:Dickey–Fuller 9171: 9170: 9168: 9164: 9158: 9155: 9153: 9150: 9148: 9147:Cointegration 9145: 9143: 9140: 9138: 9135: 9133: 9130: 9128: 9125: 9123: 9122:Decomposition 9120: 9119: 9117: 9113: 9110: 9108: 9104: 9094: 9091: 9090: 9089: 9086: 9085: 9084: 9081: 9077: 9074: 9073: 9072: 9069: 9067: 9064: 9062: 9059: 9057: 9054: 9052: 9049: 9047: 9044: 9042: 9039: 9037: 9034: 9033: 9031: 9029: 9025: 9019: 9016: 9014: 9011: 9009: 9006: 9004: 9001: 8999: 8996: 8994: 8993:Cohen's kappa 8991: 8990: 8988: 8986: 8982: 8978: 8974: 8970: 8966: 8962: 8957: 8953: 8939: 8936: 8934: 8931: 8929: 8926: 8924: 8921: 8920: 8918: 8916: 8912: 8906: 8902: 8898: 8892: 8890: 8887: 8886: 8884: 8882: 8878: 8872: 8869: 8867: 8864: 8862: 8859: 8857: 8854: 8852: 8849: 8847: 8846:Nonparametric 8844: 8842: 8839: 8838: 8836: 8832: 8826: 8823: 8821: 8818: 8816: 8813: 8811: 8808: 8807: 8805: 8803: 8799: 8793: 8790: 8788: 8785: 8783: 8780: 8778: 8775: 8773: 8770: 8769: 8767: 8765: 8761: 8755: 8752: 8750: 8747: 8745: 8742: 8740: 8737: 8736: 8734: 8732: 8728: 8724: 8717: 8714: 8712: 8709: 8708: 8704: 8700: 8684: 8681: 8680: 8679: 8676: 8674: 8671: 8669: 8666: 8662: 8659: 8657: 8654: 8653: 8652: 8649: 8648: 8646: 8644: 8640: 8630: 8627: 8623: 8617: 8615: 8609: 8607: 8601: 8600: 8599: 8596: 8595:Nonparametric 8593: 8591: 8585: 8581: 8578: 8577: 8576: 8570: 8566: 8565:Sample median 8563: 8562: 8561: 8558: 8557: 8555: 8553: 8549: 8541: 8538: 8536: 8533: 8531: 8528: 8527: 8526: 8523: 8521: 8518: 8516: 8510: 8508: 8505: 8503: 8500: 8498: 8495: 8493: 8490: 8488: 8486: 8482: 8480: 8477: 8476: 8474: 8472: 8468: 8462: 8460: 8456: 8454: 8452: 8447: 8445: 8440: 8436: 8435: 8432: 8429: 8427: 8423: 8413: 8410: 8408: 8405: 8403: 8400: 8399: 8397: 8395: 8391: 8385: 8382: 8378: 8375: 8374: 8373: 8370: 8366: 8363: 8362: 8361: 8358: 8356: 8353: 8352: 8350: 8348: 8344: 8336: 8333: 8331: 8328: 8327: 8326: 8323: 8321: 8318: 8316: 8313: 8311: 8308: 8306: 8303: 8301: 8298: 8297: 8295: 8293: 8289: 8283: 8280: 8276: 8273: 8269: 8266: 8264: 8261: 8260: 8259: 8256: 8255: 8254: 8251: 8247: 8244: 8242: 8239: 8237: 8234: 8232: 8229: 8228: 8227: 8224: 8223: 8221: 8219: 8215: 8212: 8210: 8206: 8200: 8197: 8195: 8192: 8188: 8185: 8184: 8183: 8180: 8178: 8175: 8171: 8170:loss function 8168: 8167: 8166: 8163: 8159: 8156: 8154: 8151: 8149: 8146: 8145: 8144: 8141: 8139: 8136: 8134: 8131: 8127: 8124: 8122: 8119: 8117: 8111: 8108: 8107: 8106: 8103: 8099: 8096: 8094: 8091: 8089: 8086: 8085: 8084: 8081: 8077: 8074: 8072: 8069: 8068: 8067: 8064: 8060: 8057: 8056: 8055: 8052: 8048: 8045: 8044: 8043: 8040: 8038: 8035: 8033: 8030: 8028: 8025: 8024: 8022: 8020: 8016: 8012: 8008: 8003: 7999: 7985: 7982: 7980: 7977: 7975: 7972: 7970: 7967: 7966: 7964: 7962: 7958: 7952: 7949: 7947: 7944: 7942: 7939: 7938: 7936: 7932: 7926: 7923: 7921: 7918: 7916: 7913: 7911: 7908: 7906: 7903: 7901: 7898: 7896: 7893: 7892: 7890: 7888: 7884: 7878: 7875: 7873: 7872:Questionnaire 7870: 7868: 7865: 7861: 7858: 7856: 7853: 7852: 7851: 7848: 7847: 7845: 7843: 7839: 7833: 7830: 7828: 7825: 7823: 7820: 7818: 7815: 7813: 7810: 7808: 7805: 7803: 7800: 7798: 7795: 7794: 7792: 7790: 7786: 7782: 7778: 7773: 7769: 7755: 7752: 7750: 7747: 7745: 7742: 7740: 7737: 7735: 7732: 7730: 7727: 7725: 7722: 7720: 7717: 7715: 7712: 7710: 7707: 7705: 7702: 7700: 7699:Control chart 7697: 7695: 7692: 7690: 7687: 7685: 7682: 7681: 7679: 7677: 7673: 7667: 7664: 7660: 7657: 7655: 7652: 7651: 7650: 7647: 7645: 7642: 7640: 7637: 7636: 7634: 7632: 7628: 7622: 7619: 7617: 7614: 7612: 7609: 7608: 7606: 7602: 7596: 7593: 7592: 7590: 7588: 7584: 7572: 7569: 7567: 7564: 7562: 7559: 7558: 7557: 7554: 7552: 7549: 7548: 7546: 7544: 7540: 7534: 7531: 7529: 7526: 7524: 7521: 7519: 7516: 7514: 7511: 7509: 7506: 7504: 7501: 7500: 7498: 7496: 7492: 7486: 7483: 7481: 7478: 7474: 7471: 7469: 7466: 7464: 7461: 7459: 7456: 7454: 7451: 7449: 7446: 7444: 7441: 7439: 7436: 7434: 7431: 7429: 7426: 7425: 7424: 7421: 7420: 7418: 7416: 7412: 7409: 7407: 7403: 7399: 7395: 7390: 7386: 7380: 7377: 7375: 7372: 7371: 7368: 7364: 7357: 7352: 7350: 7345: 7343: 7338: 7337: 7334: 7327: 7322: 7318: 7317: 7306: 7300: 7296: 7295: 7289: 7285: 7283:9780470512845 7279: 7275: 7271: 7266: 7262: 7256: 7252: 7251: 7246: 7242: 7238: 7232: 7228: 7223: 7208: 7204: 7197: 7192: 7187: 7182: 7178: 7174: 7173: 7168: 7163: 7159: 7153: 7149: 7144: 7140: 7136: 7132: 7128: 7124: 7120: 7113: 7108: 7104: 7098: 7094: 7089: 7085: 7079: 7075: 7074: 7068: 7064: 7060: 7055: 7051: 7045: 7041: 7037: 7032: 7028: 7023: 7019: 7015: 7011: 7007: 7002: 6998: 6994: 6989: 6984: 6980: 6976: 6972: 6968: 6964: 6960: 6956: 6951: 6936: 6932: 6928: 6924: 6920: 6916: 6912: 6905: 6900: 6896: 6892: 6888: 6884: 6880: 6876: 6872: 6868: 6863: 6859: 6855: 6851: 6847: 6843: 6839: 6835: 6831: 6827: 6823: 6819: 6815: 6811: 6807: 6803: 6799: 6795: 6791: 6787: 6783: 6779: 6775: 6771: 6767: 6763: 6756: 6752: 6748: 6744: 6738: 6734: 6729: 6725: 6721: 6717: 6713: 6709: 6705: 6704: 6698: 6694: 6688: 6684: 6680: 6679: 6673: 6669: 6663: 6659: 6655: 6650: 6646: 6641: 6637: 6633: 6628: 6624: 6622:9780470110126 6618: 6614: 6610: 6605: 6604: 6597: 6593: 6589: 6583: 6579: 6578: 6572: 6568: 6562: 6558: 6553: 6549: 6545: 6541: 6540: 6534: 6530: 6524: 6520: 6515: 6511: 6507: 6503: 6499: 6495: 6491: 6490: 6484: 6480: 6476: 6472: 6468: 6464: 6460: 6456: 6452: 6448: 6444: 6439: 6435: 6429: 6425: 6420: 6416: 6409: 6404: 6400: 6394: 6390: 6385: 6381: 6377: 6373: 6367: 6363: 6359: 6354: 6349: 6345: 6340: 6336: 6332: 6328: 6324: 6318: 6314: 6310: 6306: 6302: 6296: 6292: 6287: 6283: 6279: 6275: 6271: 6267: 6263: 6258: 6254: 6250: 6246: 6242: 6238: 6234: 6229: 6225: 6221: 6220: 6212: 6208: 6204: 6203: 6188: 6179: 6170: 6164: 6159: 6153: 6148: 6142: 6138: 6132: 6124: 6120: 6116: 6112: 6107: 6102: 6098: 6094: 6087: 6079: 6075: 6071: 6067: 6060: 6052: 6048: 6044: 6040: 6036: 6032: 6025: 6019: 6014: 6012: 6003: 5999: 5995: 5991: 5987: 5983: 5976: 5969: 5950: 5946: 5939: 5932: 5924: 5920: 5915: 5910: 5905: 5900: 5896: 5892: 5888: 5884: 5883: 5878: 5871: 5863: 5856: 5848: 5842: 5838: 5834: 5829: 5824: 5820: 5812: 5797: 5793: 5787: 5772: 5765: 5758: 5750: 5749:Dice Insights 5746: 5740: 5733: 5728: 5713: 5706: 5703:Jakl, Tomas. 5699: 5691: 5685: 5681: 5677: 5673: 5669: 5665: 5658: 5639: 5635: 5628: 5621: 5610: 5607:Bruns, Pete. 5603: 5595: 5589: 5585: 5581: 5576: 5571: 5567: 5560: 5544: 5540: 5534: 5519: 5512: 5505: 5497: 5493: 5489: 5485: 5478: 5470: 5466: 5462: 5458: 5453: 5448: 5444: 5440: 5433: 5427: 5422: 5414: 5410: 5406: 5402: 5395: 5388: 5383: 5376: 5371: 5356: 5350: 5346: 5339: 5338: 5331: 5323: 5319: 5315: 5311: 5307: 5303: 5299: 5295: 5288: 5279: 5274: 5270: 5266: 5263:(2): 328–42. 5262: 5258: 5254: 5247: 5238: 5232: 5227: 5221: 5216: 5208: 5204: 5200: 5196: 5192: 5188: 5181: 5175: 5170: 5162: 5158: 5154: 5150: 5146: 5142: 5138: 5134: 5130: 5126: 5119: 5111: 5107: 5103: 5099: 5095: 5091: 5087: 5083: 5079: 5075: 5068: 5060: 5056: 5051: 5046: 5042: 5038: 5034: 5030: 5026: 5022: 5018: 5011: 5005:, p. 16. 5004: 4999: 4993:, p. 13. 4992: 4987: 4979: 4975: 4970: 4965: 4961: 4957: 4952: 4947: 4943: 4939: 4935: 4928: 4922: 4917: 4909: 4905: 4901: 4897: 4893: 4889: 4885: 4881: 4874: 4868: 4863: 4857: 4852: 4850: 4843: 4838: 4830: 4826: 4822: 4818: 4814: 4810: 4806: 4799: 4792:(1): 429–434. 4791: 4787: 4780: 4773:(3): 293–318. 4772: 4768: 4761: 4754: 4746: 4742: 4737: 4732: 4728: 4724: 4720: 4713: 4705: 4701: 4696: 4691: 4687: 4683: 4679: 4672: 4664: 4660: 4656: 4652: 4648: 4644: 4637: 4629: 4625: 4620: 4615: 4611: 4607: 4603: 4596: 4586: 4576: 4566: 4556: 4546: 4536: 4517: 4513: 4509: 4505: 4501: 4497: 4493: 4489: 4485: 4478: 4471: 4452: 4448: 4444: 4437: 4430: 4428: 4419: 4415: 4411: 4407: 4403: 4399: 4392: 4384: 4380: 4376: 4372: 4368: 4364: 4360: 4356: 4349: 4341: 4337: 4332: 4327: 4323: 4319: 4315: 4311: 4310:J. Chem. Phys 4307: 4300: 4298: 4290: 4286: 4282: 4275: 4268: 4260: 4256: 4252: 4248: 4244: 4240: 4233: 4231: 4222: 4218: 4214: 4210: 4206: 4202: 4198: 4194: 4187: 4176: 4172: 4168: 4164: 4160: 4156: 4152: 4148: 4144: 4137: 4130: 4121: 4116: 4112: 4108: 4104: 4097: 4088: 4083: 4079: 4075: 4071: 4064: 4056: 4052: 4048: 4042: 4038: 4034: 4030: 4029: 4024: 4017: 4015: 4007: 4003: 4001:9780387202686 3997: 3993: 3992: 3984: 3982: 3973: 3969: 3963: 3955: 3951: 3945: 3937: 3933: 3929: 3925: 3918: 3910: 3906: 3899: 3892: 3884: 3880: 3875: 3870: 3865: 3860: 3856: 3852: 3848: 3844: 3840: 3833: 3825: 3821: 3814: 3806: 3802: 3798: 3794: 3790: 3786: 3782: 3775: 3768: 3763: 3761: 3753: 3752:Eckhardt 1987 3748: 3741: 3736: 3734: 3732: 3723: 3719: 3715: 3711: 3707: 3703: 3696: 3688: 3684: 3680: 3676: 3672: 3668: 3661: 3653: 3649: 3645: 3641: 3637: 3633: 3628: 3623: 3619: 3615: 3608: 3600: 3596: 3592: 3588: 3585:(1): 070001. 3584: 3580: 3573: 3565: 3558: 3550: 3546: 3539: 3537: 3528: 3522: 3511: 3510: 3502: 3500: 3498: 3496: 3487: 3483: 3479: 3475: 3470: 3465: 3461: 3457: 3450: 3443: 3439: 3435: 3434: 3426: 3424: 3422: 3413: 3409: 3402: 3400: 3391: 3387: 3383: 3379: 3375: 3371: 3364: 3356: 3352: 3348: 3344: 3337: 3329: 3325: 3321: 3317: 3313: 3309: 3305: 3301: 3294: 3286: 3282: 3278: 3274: 3270: 3266: 3262: 3258: 3255:(1): 97–109. 3254: 3250: 3243: 3235: 3231: 3227: 3223: 3219: 3215: 3211: 3207: 3203: 3199: 3195: 3191: 3184: 3176: 3172: 3165: 3157: 3153: 3149: 3145: 3141: 3137: 3130: 3123: 3118: 3114: 3098: 3095: 3093: 3090: 3088: 3085: 3083: 3080: 3078: 3077:Morris method 3075: 3073: 3070: 3068: 3065: 3063: 3060: 3058: 3055: 3053: 3050: 3048: 3045: 3043: 3040: 3038: 3035: 3033: 3030: 3028: 3025: 3023: 3020: 3018: 3015: 3014: 3009: 3003: 2998: 2991: 2989: 2979: 2977: 2972: 2970: 2965: 2961: 2955: 2952: 2948: 2944: 2934: 2931: 2926: 2924: 2920: 2916: 2912: 2906: 2896: 2894: 2890: 2886: 2882: 2878: 2873: 2871: 2867: 2862: 2860: 2856: 2852: 2847: 2842: 2825: 2819: 2815: 2806: 2802: 2797: 2795: 2791: 2787: 2783: 2779: 2774: 2751: 2745: 2741: 2731: 2723: 2718: 2708: 2706: 2696: 2694: 2690: 2689: 2684: 2675: 2673: 2669: 2664: 2654: 2652: 2648: 2644: 2640: 2630: 2628: 2624: 2620: 2616: 2611: 2607: 2603: 2599: 2595: 2585: 2583: 2578: 2574: 2570: 2560: 2558: 2554: 2550: 2546: 2542: 2531: 2526: 2524: 2520: 2516: 2512: 2508: 2503: 2497: 2494: 2491: 2488: 2487: 2486: 2483: 2481: 2477: 2471: 2461: 2459: 2456: 2448: 2444: 2441: 2437: 2434: 2430: 2426: 2422: 2419: 2415: 2411: 2407: 2403: 2399: 2395: 2394: 2393: 2385: 2383: 2379: 2370: 2368: 2364: 2360: 2356: 2352: 2348: 2338: 2336: 2332: 2328: 2315: 2311: 2307: 2303: 2299: 2295: 2291: 2288: 2284: 2281: 2277: 2274: 2270: 2266: 2265:Kalman filter 2262: 2258: 2254: 2251: 2247: 2243: 2239: 2235: 2232: 2228: 2225: 2221: 2220:geometallurgy 2217: 2216:geostatistics 2213: 2210: 2207: 2203: 2199: 2195: 2194: 2193: 2191: 2187: 2186:probabilistic 2183: 2173: 2171: 2167: 2163: 2159: 2155: 2151: 2147: 2143: 2139: 2135: 2131: 2127: 2123: 2119: 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model 9183:Q-statistic 9132:Stationarity 9028:Multivariate 8971: / 8967: / 8965:Multivariate 8963: / 8903: / 8899: / 8673:Bayes factor 8572:Signed rank 8484: 8458: 8450: 8438: 8133:Completeness 7969:Cohort study 7867:Opinion poll 7802:Missing data 7789:Study design 7744:Scatter plot 7666:Scatter plot 7659:Spearman's ρ 7621:Grouped data 7293: 7269: 7249: 7226: 7214:. 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E 4142: 4129: 4110: 4106: 4096: 4077: 4073: 4063: 4027: 4005: 3990: 3971: 3962: 3953: 3944: 3927: 3923: 3917: 3908: 3904: 3891: 3846: 3842: 3832: 3823: 3819: 3813: 3791:(3): 42–63. 3788: 3784: 3774: 3747: 3705: 3701: 3695: 3670: 3666: 3660: 3617: 3613: 3607: 3582: 3578: 3572: 3563: 3557: 3548: 3508: 3459: 3455: 3449: 3441: 3432: 3407: 3373: 3369: 3363: 3349:(2): 34–45. 3346: 3342: 3336: 3303: 3299: 3293: 3252: 3248: 3242: 3193: 3189: 3183: 3174: 3164: 3139: 3135: 3129: 3117: 2985: 2975: 2973: 2968: 2956: 2940: 2927: 2908: 2874: 2863: 2843: 2801:well-behaved 2798: 2770: 2702: 2686: 2681: 2660: 2636: 2617: 2613: 2566: 2549:architecture 2538: 2504: 2501: 2484: 2473: 2460: 2452: 2417: 2416:conditions ( 2406:normal curve 2401: 2398:type I error 2391: 2378:Path tracing 2376: 2354: 2344: 2333:analysis of 2324: 2188:analysis in 2179: 2158:astrophysics 2106:heat shields 2091: 1947: 1828:Multiphysics 1779: 1776:Applications 1758: 1754: 1745:brown dwarfs 1734: 1723: 1713: 1690: 1682: 1674:pseudorandom 1662: 1650: 1646: 1640: 1636: 1624: 1618: 1601: 1595: 1591: 1589: 1580: 1576: 1557:in advanced 1552: 1521: 1486: 1429: 1423: 1414: 1402: 1398:Enrico Fermi 1387: 1372: 1355: 1345: 1341: 1337: 1333: 1332:)ln(2/0.01)/ 1329: 1325: 1321: 1320:= 99%, then 1317: 1315: 1202: 1197: 1193: 1189: 1185: 1181: 1177: 1173: 1169: 1162: 1158: 1151: 1144: 1137: 1130: 1126: 1122: 1118: 1114: 1112: 1109: 1098: 1094: 1090: 1083: 1079: 1075: 1066: 1062: 1055: 1051: 1047: 1040: 1036: 1029: 1025: 1021: 1017: 1013: 1009: 1005: 998: 994: 987: 983: 979: 977: 919: 914: 913: 908: 901: 899: 893: 886: 882: 879: 870: 866: 862: 848: 840: 836: 822: 811: 804: 797: 788: 784: 778: 775: 771: 765: 758: 748: 741: 734: 727: 720: 718: 713: 709: 705: 703: 696: 691: 687: 683: 675: 671: 667: 665: 655: 647: 643: 639: 635: 631: 629: 623: 619: 615: 612: 603: 599: 595: 585: 578: 574: 571: 567: 564: 560: 553: 551: 540: 536: 532: 527: 520: 516: 512: 503: 499: 495: 492: 488: 485: 481: 474: 472: 467: 463: 459: 455: 451: 447: 443: 439: 434: 430: 426: 418: 413: 398: 394: 371: 355:Markov chain 328: 301: 275: 263:optimization 259:mathematical 252: 240: 229: 221: 143: 107: 98: 94: 90: 55: 51: 50: 36: 9646:WikiProject 9561:Cartography 9523:Jurimetrics 9475:Reliability 9206:Time domain 9185:(Ljung–Box) 9107:Time-series 8985:Categorical 8969:Time-series 8961:Categorical 8896:(Bernoulli) 8731:Correlation 8711:Correlation 8507:Jarque–Bera 8479:Chi-squared 8241:M-estimator 8194:Asymptotics 8138:Sufficiency 7905:Interaction 7817:Replication 7797:Effect size 7754:Violin plot 7734:Radar chart 7714:Forest plot 7704:Correlogram 7654:Kendall's τ 6944:November 1, 6784:(6): 1087. 6546:. pp. 5776:October 28, 5717:October 28, 5647:October 28, 5523:October 28, 5231:Golden 1979 4842:Ripley 1987 4404:(1): 1–25. 4113:: 171–208. 3673:: 451–459. 3566:. Springer. 3175:OR/MS Today 2711:Integration 2545:video games 2530:Computer Go 2480:search tree 2176:Engineering 2110:aerodynamic 2021:von Neumann 1955:Integration 1782:uncertainty 1678:simulations 1632:simulation. 1628:simulation. 1602:Monte Carlo 1592:Monte Carlo 1586:Definitions 1553:The use of 1505:Alan Turing 1442:simulations 1430:Monte Carlo 1392:, in which 1172:for finite 1125:Let 0 < 1050:; for i = 768:simulations 433:by running 304:uncertainty 249:Application 150:unit square 9662:Categories 9513:Demography 9231:ARMA model 9036:Regression 8613:(Friedman) 8574:(Wilcoxon) 8512:Normality 8502:Lilliefors 8449:Student's 8325:Resampling 8199:Robustness 8187:divergence 8177:Efficiency 8115:(monotone) 8110:Likelihood 8027:Population 7860:Stratified 7812:Population 7631:Dependence 7587:Count data 7518:Percentile 7495:Dispersion 7428:Arithmetic 7363:Statistics 6959:Biophys. 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