Monte Carlo Error Simulation
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that we use for the average. A possible measure of the error monte carlo error analysis is the ``variance'' defined by: (269) where and The
Monte Carlo Standard Error
``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 Standard Error Definition
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
Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; http://www.northeastern.edu/afeiguin/phys5870/phys5870/node71.html it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Required number of simulations for Monte Carlo analysis up vote 8 down vote favorite 1 My question is about the required number of simulations for Monte Carlo analysis method. As http://stats.stackexchange.com/questions/95779/required-number-of-simulations-for-monte-carlo-analysis far as I see the required number of simulations for any allowed percentage error $E$ (e.g., 5) is $$ n = \left\{\frac{100 \cdot z_c \cdot \text{std}(x)}{E \cdot \text{mean}(x)} \right\}^2 , $$ where $\text{std}(x)$ is the standard deviation of the resulting sampling, and $z_c$ is the confidence level coefficient (e.g., for 95% it is 1.96). So in this way it is possible to check that the resulting mean and standard deviation of $n$ simulations represent actual mean and standard deviation with 95% confidence level. In my case I run the simualtion 7500 times, and compute moving means and standard deviations for each set of 100 sampling out of the 7500 simulations. The required number of simulation I obtain is always less than 100, but % error of mean and std compare to mean and std of entire results is not always less than 5%. In most cases the % error of mean is less than 5% but the error of std goes up to 30%. What is the best way to determine number of required simulation without know
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