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Gelman-Rubin Statistic
The Gelman-Rubin statistic allows a statement about the convergence of Monte Carlo simulations Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi .... Definition J Monte Carlo simulations (chains) are started with different initial values. The samples from the respective burn-in phases are discarded. From the samples x_^,\dots, x_^ (of the j-th simulation), the variance between the chains and the variance in the chains is estimated: :\overline_j=\frac\sum_^L x_i^ Mean value of chain j :\overline_*=\frac\sum_^J \overline_j Mean of the means of all chains :B=\frac\sum_^J (\overline_j-\overline_*)^2 Variance of the means of the chains :W=\frac \sum_^J \left(\frac \sum_^L (x^_i-\overline_j)^2\right) Averaged variances of the individual chains across all chains An estimate of the Gel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Simulations
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical 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: optimization, numerical integration, and generating draws from a probability distribution. In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimation Theory
Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An ''estimator'' attempts to approximate the unknown parameters using the measurements. In estimation theory, two approaches are generally considered: * The probabilistic approach (described in this article) assumes that the measured data is random with probability distribution dependent on the parameters of interest * The set-membership approach assumes that the measured data vector belongs to a set which depends on the parameter vector. Examples For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the parameter sought; the estimate is based on a small random sample of voters. Alternatively, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
