Monte Carlo Error
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that we use for the average. A possible measure of the error monte carlo standard error is the ``variance'' defined by: (269) where and The
Monte Carlo Error Analysis
``standard deviation'' is . However, we should expect that the error decreases with monte carlo error definition the number of points , and the quantity defines by (271) does not. Hence, this cannot be a good measure of the error.
Monte Carlo Integration Error
Imagine that we perform several measurements of the integral, each of them yielding a result . Thes values have been obtained with different sequences of random numbers. According to the central limit theorem, these values whould be normally dstributed around a mean . Suppouse that monte carlo integration algorithm we have a set of of such measurements . A convenient measure of the differences of these measurements is the ``standard deviation of the means'' : (270) where and Although gives us an estimate of the actual error, making additional meaurements is not practical. instead, it can be proven that (271) This relation becomes exact in the limit of a very large number of measurements. Note that this expression implies that the error decreases withthe squere root of the number of trials, meaning that if we want to reduce the error by a factor 10, we need 100 times more points for the average. Subsections Exercise 10.1: One dimensional integration Exercise 10.2: Importance of randomness Next: Exercise 10.1: One dimensional Up: Monte Carlo integration Previous: Simple Monte Carlo integration Adrian E. Feiguin 2009-11-04
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Monte Carlo Integration Example
list Help Journal ListHHS Author ManuscriptsPMC3337209 Am Stat. Author manuscript;
Monte Carlo Error Propagation
available in PMC 2012 Apr 25.Published in final edited form as:Am Stat. 2009 May 1; 63(2): monte carlo standard error definition 155–162. doi: 10.1198/tast.2009.0030PMCID: PMC3337209NIHMSID: NIHMS272824On the Assessment of Monte Carlo Error in Simulation-Based Statistical AnalysesElizabeth Koehler, Biostatistician, Elizabeth Brown, Assistant Professor, and Sebastien J.-P. A. http://www.northeastern.edu/afeiguin/phys5870/phys5870/node71.html Haneuse, Associate Scientific InvestigatorElizabeth Koehler, Department of Biostatistics, Vanderbilt University, Nashville, TN 37232;Contributor Information.Elizabeth Koehler: ude.tlibrednav@relheok.e; Elizabeth Brown: ude.notgnihsaw@bazile; Sebastien J.-P. A. Haneuse: gro.chg@s.esuenah Author information ► Copyright and License information ►Copyright notice and DisclaimerSee other articles in PMC that cite the published article.AbstractStatistical experiments, more commonly referred to as https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3337209/ Monte Carlo or simulation studies, are used to study the behavior of statistical methods and measures under controlled situations. Whereas recent computing and methodological advances have permitted increased efficiency in the simulation process, known as variance reduction, such experiments remain limited by their finite nature and hence are subject to uncertainty; when a simulation is run more than once, different results are obtained. However, virtually no emphasis has been placed on reporting the uncertainty, referred to here as Monte Carlo error, associated with simulation results in the published literature, or on justifying the number of replications used. These deserve broader consideration. Here we present a series of simple and practical methods for estimating Monte Carlo error as well as determining the number of replications required to achieve a desired level of accuracy. The issues and methods are demonstrated with two simple examples, one evaluating operating characteristics of the maximum li
or suggestions for references to include. There's no need to point out busted http://statweb.stanford.edu/~owen/mc/ links (?? in LaTeX) because the computer will catch those for me when it is time to root out the last of them. @book{mcbook,
   author = {Art B. Owen},    year = 2013,    title = {Monte Carlo theory, methods and examples} } Copyright Art Owen, 2009-2013. Contents Introduction monte carlo Simple Monte Carlo Uniform random numbers Non-uniform random numbers Random vectors and objects Processes Other integration methods Variance reduction Importance sampling Advanced variance reduction Markov chain Monte Carlo Gibbs sampler Adaptive and accelerated MCMC Sequential Monte Carlo Quasi-Monte Carlo Lattice rules Randomized quasi-Monte Carlo Chapters 1 and 2 1 Introduction Example: monte carlo error traffic modeling Example: interpoint distances Notation Outline of the book End notes Exercises 2 Simple Monte Carlo Accuracy of simple Monte Carlo Error estimation Safely computing the standard error Estimating probabilities Estimating quantiles Random sample size When Monte Carlo fails Chebychev and Hoeffding intervals End notes Exercises 3 Uniform Random Numbers Random and pseudo-random numbers States, periods, seeds, and streams U(0,1) random variables Inside a random number generator Uniformity measures Statistical tests of random numbers Pairwise independent random numbers End notes Exercises 4 Non-uniform Random Numbers Inverting the CDF Examples of inversion Inversion for the normal distribution Inversion for discrete random variables Numerical inversion Other transformations Acceptance-rejection Gamma random variables Mixtures and automatic generators End notes Exercises 5 Random vectors and objects Generalizations of one-dimensional methods Multivariate normal and t Multinomial Dirichlet Multivariate Poisson and other distributions Copula-marginal sampling Random points on the sphere Random matrices Example: classification error r
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