Variance-reducing techniques

The goal of variance reduction is to decrease the error of a point estimate, which should lead to a smaller estimated standard error or a narrower confidence interval (see Section 8 for a discussion of measures of error). This section describes the variance reduction technique known as common random numbers (CRN), which is useful for reducing the error in comparing the expected performance of two or more systems. The presentation is based on Nelson (1987). 

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. 

From this fact, a corresponding approximation quality of the failure probability can be deduced, if variance-reduced sampling techniques which rely on the MPP together with the abovementioned response surface techniques are employed. This can be clearly seen from Fig. 7, where the probability of failure obtained with various approximation techniques has been plotted over the threshold value Fq. Importance sampling at the predicted MPP with 30 batches of 10,000 samples has been employed for each failure probability estimate. For global approximations, again polynomials of third degree are necessary in order to obtain accurate predictions. On the other hand, local approximations with linear polynomials lead already to quite accurate results, if only the principle direction is partitioned. 

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). 

In Other words, the assurance of quality by measurement of process impurities in the end product has been replaced by assurance of quality by the removal of variance in the process (by continuous monitoring of a continuous process). Naturally, whether online process analysis is being used as a surrogate for an alternative off-line technique to measure specific analytes or as a monitor to reduce process variance it needs calibration and validation. These stages require measurement of process analytes by a reference off-line technique, usually HPLC, and subsequent demonstration that the resulting calibration model has reliable predictive power. 

In the preceding description of the Mahalanobis distance, the number of coordinates in the distance metric is equal to the number of spectral frequencies. As discussed earlier in the section on principal component analysis, the intensities at many frequencies are dependent, and by using the full spectrum, we fit the noise in addition to the real information. In recent years, Mahalanobis distance has been defined with PCA or PLS scores instead of the spectral frequencies because these techniques eliminate or at least reduce most of the overfitting problem. The overall application of the Mahalanobis distance metric is the same except that the rt intensity values are replaced by the scores from PCA or PLS. An example of a Mahalanobis distance calculation on a set of Raman spectra for 25 carbohydrates is shown in Fig. 5-11. The 25 spectra were first subjected to PCA, and it was found that the first three principal components could account for most of the variance in the spectra. It was first assumed that all 25 spectra belonged to the same class because they were all carbohydrates. However, as shown in the three-dimensional plot in Fig. 5-11, the spectra can be clearly divided into three separate classes, with two of the spectra almost equal distance from each of the three classes. Most of the components in the upper left class in the two-dimensional plot were sugars however, some sugars were found in the other two classes. For unknowns, scores have to be calculated from the principal components and processed in the same way as the spectral intensities. 

The variance estimators s20j) of the estimated effects allow unequal response variances and the use of common pseudorandom numbers. This is a well-known technique used in simulation experiments to reduce the variances of the estimated factor effects (Law and Kelton, 2000). This technique uses the same pseudorandom numbers when simulating the system for different factor combinations, thus creating positive correlations between the responses. Consequently, the variances of the estimated effects are reduced. This technique is similar to blocking in real-world experiments see, for example, Dean and Voss (1999, Chapter 10). 

The vectors of means = (xi, I2,..., x ) and deviations = (ii, S2,. ..,Sp), and matrices of covariances S = (Sij) and correlations R = (tij) can be calculated. For this data matrix, the most used non-supervised methods are Principal Components Analysis (PCA), and/or Factorial Analysis (FA) in an attempt to reduce the dimensions of the data and study the interrelation between variables and observations, and Cluster Analysis (CA) to search for clusters of observations or variables (Krzanowski 1988 Cela 1994 Afifi and Clark 1996). Before applying these techniques, variables are usually first standardised (X, X ) to achieve a mean of 0 and unit variance. 

The main objective of this technique is to reduce the dimensions of data without losing important information, starting with the correlation between variables, to explore the relationship between variables and between observations. The aim is to obtain p new variables (Ti, I2, , Yp), that we will call principal components, which are (1) a normalised linear combination of the original variables (Yi = aijXi fl2,iX2 -I-. .. -I- apjXp J2k< k,i = 1 X (2) uncorrelated ones (cov(T , Yj) = OVt j), (3) with progressively diminishing variances (var(Yi) > var(Y2) >. .. > var Yp) ), and (4) the total variance VT) coincident with that... 

Although both the PCA and the SPM techniques reduce the number of variables, the resulting matrices of PC loadings and variables and the spectral map are still multidimensional. The plot of PC loadings in the first vs. the second principal component has been frequently used for the evaluation of the similarities and differences among the observations. This method takes into consideration only the variance explained in the first two principal components and entirely ignores the impact of variances explained by the other principal com-... 

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