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Simulation and the Monte Carlo Method
Abstract
Simulation and the Monte Carlo Method, Second Edition reflects the latest developments in the field and presents a fully updated and comprehensive account of the major topics that have emerged in Monte Carlo simulation since the publication of the classic First Edition over twenty-five years ago. While maintaining its accessible and intuitive approach, this revised edition features a wealth of up-to-date information that facilitates a deeper understanding of problem solving across a wide array of subject areas, such as engineering, statistics, computer science, mathematics, and the physical and life sciences." "Requiring only a basic, introductory knowledge of probability and statistics, Simulation and the Monte Carlo Method, Second Edition is an excellent text for upper-undergraduate and beginning graduate courses in simulation and Monte Carlo techniques. The book also serves as a valuable reference for professionals who would like to achieve a more formal understanding of the Monte Carlo method.
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“Convergence to low variability in terms of mean and standard deviations is reached at M=103, where both metrics stabilized with minimal variation beyond that point and correspond to the plateau of the coefficient of variation [40].”
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Recovering clipped OFDM symbols with Bayesian inference
Declercq, D., & Giannakis, G. B. (2002) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.2000.861898 · Cited by 27
“For a variance parameter, it is computationally e cient to choose as conjugate prior density the Inverse Gamma which can be generated as in [11]:”
MethodFormula-Based Probabilistic InferencePreprint
Gogate, V., & Domingos, P. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1203.3482 · Cited by 26
“Importance sampling (Rubinstein, 1981) is a general scheme which can be used to approximate any quantity such as ZM which can be expressed as a sum of a function over a domain.”
BackgroundMethodInfluentialOptimizing upside variability and antifragility in renewable energy system design
Coppitters, D., & Contino, F. (2023) · Scientific Reports · doi:10.1038/s41598-023-36379-8 · Cited by 26
Jiang, C., Yin, S., Yao, Z., et al. (2024) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2024.129802 · Cited by 26
Use of Flow Meters for Managing Water Supply Networks
Janković-Nišić, B., Maksimović, Č., Butler, D., et al. (2004) · Journal of Water Resources Planning and Management · doi:10.1061/(asce)0733-9496(2004)130:2(171 · Cited by 26
Amicarelli, V., Lombardi, M., Varese, E., & Bux, C. (2023) · Environmental Impact Assessment Review · doi:10.1016/j.eiar.2023.107101 · Cited by 26
Zhou, T., Guo, T., Dong, Y., et al. (2023) · Computers & Structures · doi:10.1016/j.compstruc.2023.107117 · Cited by 26
Dimensionality reduction and visualization of structural reliability problems using polar features
Hurtado, J. E. (2011) · Probabilistic Engineering Mechanics · doi:10.1016/j.probengmech.2011.12.004 · Cited by 26
Sampling-based approach for design optimization in the presence of interval variables
Yoo, D., & Lee, I. (2013) · Structural and Multidisciplinary Optimization · doi:10.1007/s00158-013-0969-7 · Cited by 26
Stochastic Lagrangian trajectory model for drifting objects in the ocean
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Low-variance direct Monte Carlo simulations using importance weights
Al-Mohssen, H. A., & Hadjiconstantinou, N. G. (2010) · ESAIM Mathematical Modelling and Numerical Analysis · doi:10.1051/m2an/2010052 · Cited by 26
“Our framework is based on the general idea of correlated sampling [24] that has been successfully used in a number of diverse fields [14,21].”
A stochastic response surface approach to statistical prediction of mobile robot mobility
Kewlani, G., & Iagnemma, K. D. (2008) · IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) · doi:10.1109/iros.2008.4651187 · Cited by 26
“While for Gaussian variables Hermite polynomials are used, different orthogonal polynomial basis functions are used corresponding to the probability distributions of other non-Gaussian variables.”
MethodExploring Explanations of Subglacial Bedform Sizes Using Statistical Models
Hillier, J. K., Kougioumtzoglou, I. A., Stokes, C. R., et al. (2016) · PLoS ONE · doi:10.1371/journal.pone.0159489 · Cited by 26
“, [95]).”
Pflug, G. C. (2006) · Springer eBooks · doi:10.1007/bfb0042317 · Cited by 26
Verly, G. (1993) · Springer eBooks · doi:10.1007/978-3-642-77859-9_28 · Cited by 26
Estimating principal points of univariate distributions
Tarpey, T. (1997) · Journal of Applied Statistics · doi:10.1080/02664769723503 · Cited by 26
Kang, K., Paul, B. C., & Roy, K. (2006) · ACM Transactions on Design Automation of Electronic Systems · doi:10.1145/1179461.1179464 · Cited by 26
“In order to verify the proposed algorithm, we compared the results with Monte Carlo simulation [Rubinstein 1981].”
ResultSoares, C. G. (1990) · Marine Structures · doi:10.1016/0951-8339(90)90015-j · Cited by 26
Scaling of fault displacements and implications for the estimation of sub-seismic strain
Pickering, G., Bull, J. M., & Sanderson, D. J. (1996) · Geological Society London Special Publications · doi:10.1144/gsl.sp.1996.099.01.03 · Cited by 26
Hybrid uncertainty propagation based on multi-fidelity surrogate model
Liu, J., Shi, Y., Ding, C., et al. (2024) · Computers & Structures · doi:10.1016/j.compstruc.2023.107267 · Cited by 25
“Monte Carlo simulation (MCS) [15] is a traditional simulation-based method, the results of which are widely used as a reference to verify the accuracy of other methods due to its simplicity and robustness.”
Kaimathuruthy, B. J., Jalón‐Rojas, I., & Sous, D. (2025) · Geoscientific model development · doi:10.5194/gmd-18-7227-2025 · Cited by 24
Simulation of thep-generalized Gaussian distribution
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“Our generalizations are essentially based upon stochastic representations which make use of the p-generalized trigonometric functions in (Richter, 2007), the non-Euclidean geometric measure representation for ln,p-spherical symmetric distribution laws in (Richter, 2009) and which make also use of certain standard dist…”
MethodScalable information inequalities for uncertainty quantification
Katsoulakis, M. A., Rey-Bellet, L., & Wang, J. (2017) · Journal of Computational Physics · doi:10.1016/j.jcp.2017.02.020 · Cited by 25
“For details on the numerical computation of the autocorrelation function vμp we refer to [44].”
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Probabilistic assessment of voltage quality on solar-powered electric vehicle charging station
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Stochastic nonlocal damage analysis by a machine learning approach
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MOCASSIN : Metropolis and kinetic Monte Carlo for solid electrolytes
Eisele, S., & Grieshammer, S. (2020) · Journal of Computational Chemistry · doi:10.1002/jcc.26418 · Cited by 25
Cheng, J., Lü, W., Liu, Z., et al. (2020) · Structural and Multidisciplinary Optimization · doi:10.1007/s00158-020-02762-6 · Cited by 25
“The techniques for stochastic analysis include Monte Carlo simulation (MCS) (Rubinstein and Kroese 2016), Bayesian Monte Carlo method (Shi et al. 2018), and structural sensitivity analysis (Xiao et al. 2016; Ajamy et al. 2014).”
MethodInfluentialFailure assessment of corrosion affected pipeline networks with limited failure data availability
Aryai, V., Baji, H., & Mahmoodian, M. (2021) · Process Safety and Environmental Protection · doi:10.1016/j.psep.2021.11.024 · Cited by 25
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Quantum-Inspired Power System Reliability Assessment
Nikmehr, N., & Zhang, P. (2022) · IEEE Transactions on Power Systems · doi:10.1109/tpwrs.2022.3204393 · Cited by 29
“Compared to the classical MCS technique where the convergence rate is O ( 1 √ S ) with total samples S [29], the authors in [30], [31], [32] have shown that the quadratic speed up with convergence O ( 1 S ) can be achieved by quantum algorithms.”
BackgroundYeh, Y., Lin, W. T., Chiang, W., et al. (2022) · Journal of Membrane Science · doi:10.1016/j.memsci.2022.121334 · Cited by 25
Batou, A., & Soize, C. (2009) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2009.03.018 · Cited by 25
“The stochastic equation (2) is solved using the Monte Carlo simulation method (see [7]).”
MethodProbabilistic analysis of weld cracks in center-cracked tension specimens
Francis, M., & Rahman, S. (2000) · Computers & Structures · doi:10.1016/s0045-7949(99)00127-3 · Cited by 25
“In this study, standard reliability methods, such as FORM/SORM [45±59], and MCS [60] were used to compute these probabilities.”
BackgroundMethodInfluentialDetermination of Point Displacements in the Geodetic Network
Savšek-Safić, S., Ambrožič, T., Stopar, B., et al. (2006) · Journal of Surveying Engineering · doi:10.1061/(asce)0733-9453(2006)132:2(58 · Cited by 25
Ren, X., Yan, J., & Mo, Y. (2018) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2017.2788420 · Cited by 22
“To simulate the performance with high accuracy, we adopt the importance sampling approach [23].”
MethodCross-entropy based importance sampling for stochastic simulation models
Cao, Q. D., & Choe, Y. (2019) · Reliability Engineering & System Safety · doi:10.1016/j.ress.2019.106526 · Cited by 25
“nonzero with a suciently large m(0). If the user has no knowledge of potential failure regions, the user can adapt the algorithm by adopting the multilevel CE procedure widely used in the literature [40,41]. Specically, the failure threshold level lcan be adjusted over iterations so that the failure probability estima…”
MethodStochastic response of reinforced concrete buildings using high dimensional model representation
Sahu, D. K., Nishanth, M., Dhir, P. K., et al. (2018) · Engineering Structures · doi:10.1016/j.engstruct.2018.10.083 · Cited by 25
“MCS is a powerful technique which consists of repeated sampling process to obtain the responses but becomes computationally expensive [6] for problems having small probabilities.”
MethodInfluentialA Fast and Effective Sensitivity Calculation Method for Circuit Input Vectors
Xiao, J., Lou, J., & Jiang, J. (2019) · IEEE Transactions on Reliability · doi:10.1109/tr.2019.2897455 · Cited by 25
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Nassar, M., Guizani, L., Nollet, M., et al. (2019) · Engineering Structures · doi:10.1016/j.engstruct.2019.109353 · Cited by 25
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Guo, T., Frangopol, D. M., Han, D., et al. (2010) · Journal of Performance of Constructed Facilities · doi:10.1061/(asce)cf.1943-5509.0000174 · Cited by 25
Variance reduction techniques and quasi-Monte Carlo methods
Wang, X. (2001) · Journal of Computational and Applied Mathematics · doi:10.1016/s0377-0427(00)00331-9 · Cited by 25
The physics of fluid turbulence
Arter, W. (1993) · Computer Physics Communications · doi:10.1016/0010-4655(93)90157-8 · Cited by 25
Simulation-Based Comparison of Existent IAS 19 Accounting Options
Amen, M. (2007) · European Accounting Review · doi:10.1080/09638180701390925 · Cited by 25
“Rubinstein, 1981 ) to analyse the behaviour of the various accounting elements concerned with pension plans.”
BackgroundMethodConfidence of simulation results: put a sensitivity analysis module in your MODEL
Fürbringer, J.-M., & Roulet, C.-A. (1999) · Energy and Buildings · doi:10.1016/s0378-7788(98)00046-2 · Cited by 25
An analysis of emergency physicians' cumulative career risk of HIV infection
Wears, R. L., Vukich, D. J., Winton, C. N., et al. (1991) · Annals of Emergency Medicine · doi:10.1016/s0196-0644(05)80836-4 · Cited by 25
Mamay, C., & Strauss, T. (1991) · IEEE Power Engineering Review · doi:10.1109/mper.1991.90624 · Cited by 25
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Hajivassiliou, V. A. (1993) · The International Journal of High Performance Computing Applications · doi:10.1177/109434209300700305 · Cited by 25
Unbiased Variance Reduction for Global Illumination
Kirk, D. B., & Arvo, J. R. (1994) · Springer eBooks · doi:10.1007/978-3-642-57963-9_5 · Cited by 25
Method To Evaluate Image-Recovery Algorithms Based On Task Performance
Hanson, K. M. (1988) · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · doi:10.1117/12.968649 · Cited by 25
A Threshold for Quantum Advantage in Derivative PricingPreprint
Chakrabarti, S., Krishnakumar, R., Mazzola, G., Stamatopoulos, N., Woerner, S., & Zeng, W. J. (n.d.) · RePEc: Research Papers in Economics · Cited by 27
Drone-based vertical delivery system for high-rise buildings: Multiple drones vs. a single elevator
Ezaki, T., Fujitsuka, K., Imura, N., et al. (2024) · Communications in Transportation Research · doi:10.1016/j.commtr.2024.100130 · Cited by 24
Variance reduction in Monte Carlo capacitance extraction
Batterywala, S. H., & Desai, M. P. (2005) · International Conference on VLSI Design and International Conference on Embedded Systems (VLSID) · doi:10.1109/icvd.2005.169 · Cited by 24
“A detailed survey of them can be obtained from standard texts on Monte Carlo integration [14], [ 15 ], [16], [17].”
BackgroundSensitivity analysis for rare events based on Rényi divergence
Dupuis, P. G., Katsoulakis, M. A., Pantazis, Y., & Rey-Bellet, L. (2020) · The Annals of Applied Probability · doi:10.1214/19-aap1468 · Cited by 21
“For example, importance sampling [5, 6, 7] transforms the distribution of random variables in order to make the rare event typical and corrects for bias using the likelihood ratio; interacting particle systems methods [8, 9] use many (dependent) copies of the system to speed the exploration of state space; splitting t…”
MethodCertifiable Risk-Based Engineering Design Optimization
Chaudhuri, A., Krämer, B., Norton, M., Røyset, J. Ø., & Willcox, K. E. (2021) · AIAA Journal · doi:10.2514/1.j060539 · Cited by 27
“Second, MC variance reduction techniques such as importance sampling [31, 32, 33], adaptive importance sampling [34, 35, 36, 37, 38, 39], and multifidelity approaches [40, 41, 42] offer computational advantages.”
BackgroundEnhancing the performance of a bistable energy harvesting device via the cross-entropy method
Cunha, A. (2021) · Nonlinear Dynamics · doi:10.1007/s11071-020-06109-0 · Cited by 24
“As this method has a theoretical guarantee of convergence and evidence in the literature proving its effectiveness [36, 57], the proposed cross-entropy framework for optimization of dynamical systems has the potential to be a very general and robust numerical methodology.”
BackgroundMethodInfluentialWang, J., Lü, Z., & Wang, L. (2022) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.7122 · Cited by 24
“By regularizing α, the unit optimal importance direction eα can be obtained by Equation (7),”
MethodGómez, H. Z., Lagarejos, E., Prado-Velasco, M., Sánchez-Herrero, S., Serna, J., Rueda-Ferreiro, A., Martín‐Suárez, A., Calvo, M. V. M., Pérez‐Blanco, J. S., & Lanao, J. M. (2022) · Frontiers in Pharmacology · doi:10.3389/fphar.2022.977372 · Cited by 23
“If observed concentrations were distributed within the 90% PI, the model prediction capability was deemed to be adequate (Rubinstein and Kroese, 2016).”
MethodKamariotis, A., Sardi, L., Papaioannou, I., Chatzi, E., & Straub, D. (2023) · Data-Centric Engineering · doi:10.1017/dce.2023.13 · Cited by 28
“Based on the generated measurements, the sequence of reference posterior distributions πpos (θ|y1:k) is obtained using the prior distribution as an envelope distribution for rejection sampling [75, 76].”
Examples comparing importance sampling and the Metropolis algorithm
Bassetti, F., & Diaconis, P. W. (2006) · Illinois Journal of Mathematics · doi:10.1215/ijm/1258059470 · Cited by 24
“For sequential importance sampling see [1], for particle ltering see [5], for adaptive importance sampling see [ 20 ].”
BackgroundNiknam, T., Bornapour, M., Ostadi, A., et al. (2013) · Journal of Power Sources · doi:10.1016/j.jpowsour.2013.03.117 · Cited by 24
Post-Processing Accept-Reject Samples: Recycling and Rescaling
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Probabilistic failure analysis of nuclear piping with empirical study of Taiwan's BWR plants
You, J.-S., & Wu, W. (2002) · International Journal of Pressure Vessels and Piping · doi:10.1016/s0308-0161(02)00061-3 · Cited by 24
Generation of classes of robust periodic railway timetables
Odijk, M. A., Romeijn, H. E., & van Maaren, H. (2005) · Computers & Operations Research · doi:10.1016/j.cor.2005.02.004 · Cited by 24
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Optimization with simulation and multiobjective analysis in industrial decision-making: A case study
Moralejo, I. A., Azcárate, C., Mallor, F., et al. (2002) · European Journal of Operational Research · doi:10.1016/s0377-2217(02)00075-9 · Cited by 24
Rohani, M. R., Solanki, K. N., Acar, E., et al. (2010) · International Journal of Vehicle Design · doi:10.1504/ijvd.2010.036839 · Cited by 24
“Simulation-based methods include the direct Monte Carlo Simulation (MCS) (Rubinstein, 1981) and its more advanced variants such as Importance Sampling (Melchers, 1989), Adaptive Importance Sampling (Wu, 1994) and Directional Simulation (Nie and Ellingwood, 2000).”
MethodCollapse fragility curve development using Monte Carlo simulation and artificial neural network
Khojastehfar, E., Aval, S. B. B., Zolfaghari, M. R., et al. (2014) · Proceedings of the Institution of Mechanical Engineers Part O Journal of Risk and Reliability · doi:10.1177/1748006x13518524 · Cited by 24
Dynamic behaviour of a wind turbine gear system with uncertainties
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Gaming and Simulation for Railway Innovation
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“In cases where they deem certain parameters within the system to have a range of values, simulation outcomes are variable and multiple runs will deliver a range of possible system behaviors (Rubinstein, 1981).”
BackgroundGholaminezhad, I., Jamali, A., & Assimi, H. (2016) · Neural Computing and Applications · doi:10.1007/s00521-016-2392-7 · Cited by 24
Wu, W., Li, Y., Cheng, G., et al. (2019) · Engineering Failure Analysis · doi:10.1016/j.engfailanal.2019.01.080 · Cited by 24
Using causal discovery to analyze emergence in agent-based models
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“A conventional technique for estimating a probability density function of a system’s output response from uncertain input distributions is the Monte Carlo method [11][12].”
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“To generate a set of data distributed with the GR( α , a ) density, theinversetransformmethod[20]wasapplied.”
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Volume estimation by monte carlo methods*
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An asymmetric dimension-adaptive tensor-product method for reliability analysis
Hu, C. K., & Youn, B. D. (2011) · Structural Safety · doi:10.1016/j.strusafe.2011.03.004 · Cited by 24
Eamon, C. D., Thompson, M., & Liu, Z. (2005) · Structural Safety · doi:10.1016/j.strusafe.2005.05.001 · Cited by 24
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“This can for example be done with strati.cation[Rubinstein81]orby usinglow-discrepancyquasi-random sampling[Keller96]. emit photons from diffuse point light() {ne =0 number of emitted photons while (not enough photons) { do { use simple rejection sampling to .nd diffuse photon direction x =…”
BackgroundImportance Sampling of Test Cases in Markovian Software Usage Models
Gutjahr, W. J. (1997) · Probability in the Engineering and Informational Sciences · doi:10.1017/s0269964800004642 · Cited by 24
“This allows a very e cient solution of (11) by means of the well{known Kiefer{Wolfowitz procedure ([8]; see also [10], p.”
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“Instead of directly taking the mean life expectancy for each case, we applied a Monte Carlo approach (Rubinstein, 1981) to account for possible random variation.”
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A Comparative Study of Single-Section Polarization-Mode Dispersion Compensators
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Probabilistic modeling of aquifer heterogeneity using reliability methods
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A Test for Choice between Hyperbolic and Sigmoidal Models of Crop Yield Response to Weed Density
Swinton, S. M., & Lyford, C. P. (1996) · Journal of Agricultural Biological and Environmental Statistics · doi:10.2307/1400562 · Cited by 24
Simulation free reliability analysis: A physics-informed deep learning based approachPreprint
Chakraborty, S. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2005.01302 · Cited by 24
“Monte Carlo simulation (MCS) [3, 4] is perhaps the most straightforward method for reliability analysis.”
MethodFurther analysis of the statistical independence of the NIST SP 800-22 randomness tests
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Markov chain Monte Carlo inversion for the rheology of olivine single crystals
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Measurement-based bearing capacity evaluation for small and medium span bridges
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The Implications of the No-Free-Lunch Theorems for Meta-induction
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Temperature evolution of the phonon dynamics in the Kitaev spin liquid
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“We have furthermore executed 500 emulations for an emulation time of 12 periods with weights randomly chosen between 0 and 31 and verified the obtained solutions with an algorithm from [14], resulting in an accuracy of 88%.”
MethodWang, T., Li, Z., Fan, W., et al. (2021) · Journal of Structural Engineering · doi:10.1061/(asce)st.1943-541x.0003194 · Cited by 23
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“The complete process of MCS can be studied from the literature (Rubinstein and Kroese, 2016).”
BackgroundJäpel, R., & Buyel, J. F. (2022) · Journal of Chromatography A · doi:10.1016/j.chroma.2022.463408 · Cited by 23
Metadata
- Authors
- Rubinstein, R. Y., & Kroese, D. P.
- ORCID
- Kroese, Dirk P.
- Venue
- Wiley Series in Probability and Statistics
- Publisher
- Wiley
- Published
- 2016-11-21
- ISSN
- 1940-6347 (ISSN-L)
- OpenAlex
- W2147632348
- Cited by
- 3,356
- References
- 0
- Citations by year
- 2026: 131 · 2025: 182 · 2024: 168 · 2023: 157 · 2022: 167 · 2021: 162 · 2020: 171 · 2019: 139 · 2018: 118 · 2017: 66 · 2016: 41
- FWCI
- 52.82
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Wiley series in probability and statistics (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- International Linear Collider (submittedVersion)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Simulation Techniques and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Physical Sciences › Mathematics › Numerical Analysis › Mathematical Approximation and Integration
- Keywords
- Monte Carlo simulation, stochastic simulation, random number generation, probability distributions, variance reduction, Markov chain Monte Carlo
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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