Variance reduction method

A well-developed statistical inference of the estimators and exists (Rubinstein and Shapiro 1993). That inference aids in the construction of stopping rules, validation analysis, and error bounds for obtained solutions and, furthermore, suggests variance reduction methods that may substantially enhance the rate of convergence of the numerical procedure. For a discussion of this topic and an application to two-stage stochastic programming with recourse, we refer to Shapiro and Homem-de-Mello (1998). 

This simplified description has assumed that the exact physical probabilities are utilized to determine the outcome of every decision when this is done, the resulting simulation is termed an analog simulation. More sophisticated statistical treatments are included in modern computer codes that utilize nonphysical distributions with corrections (in a defined parficle weight) to keep the results of the simulation unbiased these can be shown to improve the efficiency of the simulation. These methods are called "variance reduction" methods, although this is somewhat of a misnomer because many of these methods increase efficiency by saving computer time, not by reducing variance. The exact theory and technique for doing this is beyond the scope of this handbook but is well described in Monte Carlo descriptions such as in Lewis and Miller (1993). 

Theory. PP is also a variable reduction method, very similar to PCA. In fact, PP can be considered a generalization of classical PCA (6,14-18). While in PCA the PCs are determined by maximizing variance, in PP, the latent variables, called the projection pursuit features (PPFs), are obtained by optimizing a given projection index that describes the inhomogeneity of the data, instead of its variance (6,18). In the literature, many PP indices have been described. 

A transformation of the data giving an improved fit may sometimes be obtained by trial and error, but one or both of the following approaches is recommended. One may make a theoretical choice of transformation and test if it gives improved results - a regression that is more significant in analysis of variance, reduction in the number of outliers, residuals normally distributed. Alternatively the Box-Cox transformation method may be used. 

Simulation(s) (Continued) steady-state, 2471-2472 for teamwork training, 934 terminating, 2471 as training technology, 929 and transformability, 320 variance reduction in, 2492-2493 Simulation Dynamics, 2460 Simulation methods, 128 Simulation models, 1630 for client/server (C/S) system evaluation, 728-719... 

There are many successful ways to increase the efficiency of the Monte Carlo method, in particular by decreasing the dispersion, and these are described in several textbooks. Numerous variance reduction techniques, such as splitting/Russian roulette, weight windows, and the exponential transformation method have been proposed to improve the efficiency of Monte Carlo transport calculations. 

Baker, R.S., Larsen, E.W., "A Local Exponential Transform Method for Global Variance Reduction in Monte Carlo Transport Problem", Karlsruhe, 1993, Vol.2. p.725. 

In this paper, in order to decrease the uncertainty of sampling, the variance reduction techniques have been used. Variance reduction techniques are methods that attempt to reduce the variance, i.e., the dispersion associated with the variations, of the parameter being evaluated. 

Different variance reduction strategies can be combined into a single framework so as to leverage advantages associated with each one of them. Thus, for example, the subset simulation method and the Girsanov transformation-based method can be combined in such a way that subset simulations handle the uncertainties associated with parameters, and the Girsanov transformation takes care of the random excitations (Sundar and Manohar 2014b). Performance of few simulation-based variance reduction techniques with respect to few benchmark problems has been documented by Schueller and Pradlwarter (2007). 

The Kalman filter-based dynamic state estimation tools in combination with Monte Carlo simulation methods can be employed to estimate probability of failure in instrumented structures with performance functions encompassing unmeasured system states (Ching and Beck 2007). The variance reduction strategies developed in the context of reliability analysis when applied in conjunction with the dynamic state estimation techniques could be used to determine the updated probability of failure of the structural system. For example, the data-based extreme value analysis and the Girsanov transformation-based method can be used to determine the reliability of existing structures (Radhika and Manohar 2010 Sundar and Manohar 2013). 

The problem of time variant reliability analysis of nonlinear dynamical systems can be studied to a limited extent through analytical methods and more comprehensively with simulation-based strategies. The analytical approximations are based on theory of outcrossing statistics and Markov vector approaches. Monte Carlo simulation-based methods invariably need to be reinforced with suitable variance reduction strategies so that the resulting tools become applicable to tackle... 

Recall that Eq. 14 provided an estimate for the variance Pf. Using the so-called variance reduction techniques, it is possible to reduce this variance and thereby obtain an improved estimate of Pf. Such techniques are called variance reduction techniques (Kalos and Whitlock 2008) and are commonly used while estimating Pf. One such technique is popularly called the conditional expectation method in this method, a control variable is selected and the variance of Pf is reduced by removing the random fluctuations of this control variable which was not conditioned. In another technique, popularly known as the technique of antithetic variates, negative correlation is purposefully induced between successive samples to decrease the variance of the estimated mean value. It is also common to use the technique of antithetic variates in combination with the conditional expectation method (Haidar and Mahadevan 2000). 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The polymer/additive system in combination with the proposed extraction technique determines the preferred solvent. In ASE the solvent must swell but not dissolve the polymer, whereas MAE requires a high dielectric solvent or solvent component. This makes solvent selection for MAE more problematical than for ASE . Therefore, MAE may be the preferred method for a plant laboratory analysing large numbers of similar samples (e.g. nonpolar or polar additives in polyolefins [210]). At variance to ASE , in MAE dissolution of the polymer will not block any transfer lines. Complete dissolution of the sample leads to rapid extractions, the polymer precipitating when the solvent cools. However, partial dissolution and softening of the polymer will result in agglomeration of particles and a reduction in extraction rate. 

Because of the many decisions regarding inclusion or exclusion of studies, different meta-analyses might reach very different conclusions on the same topic. Even after the studies are chosen, there are many other methodologic issues in choosing how to combine means and variances (e.g., what weighting methods should be used). Pooled analysis should report both relative risks and risk reductions as well as absolute risks and risk reductions (Sinclair and Bracken, 1994). 

In contrast to PCA which can be considered as a method for basis rotation, factor analysis is based on a statistical model with certain model assumptions. Like PCA, factor analysis also results in dimension reduction, but while the PCs are just derived by optimizing a statistical criterion (spread, variance), the factors are aimed at having a real meaning and an interpretation. Only a very brief introduction is given here a classical book about factor analysis in chemistry is from Malinowski (2002) many other books on factor analysis are available (Basilevsky 1994 Harman 1976 Johnson and Wichem 2002). 

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