Random variance

Wright GW, Simon R. A random variance model for detection of differential gene expression in small microarray experiments. Bioinformatics 2003 19 2448-2455. 

Random variance of increment collection (unit variance) see Variance. 

Random variance of increment collection (unit variance) theoretical variance calculated for a uniformly mixed lot and extrapolated to increment size (ASTM D-2234). 

According to Eq. 4-28 the theoretical semivariogram function has the value y = 0 for / = 0. Semivariograms obtained from experimental data often have a positive value of intersection with the y(/)-axis expressed by C0 (see Fig. 4-5). The point of intersection is named nugget effect or nugget variance. This term was coined in the mining industry and indicates an unexplained random variance which characterizes the microinho-... 

Significance of a taxon to the analysis is not dependent on the absolute size of its count, so that taxa having a small total variance, such as rare taxa, can compete in importance with common taxa, and taxa with a large, random variance will not automatically be selected to the exclusion of others. 

E Var h, e)] = a variance that is the sum of individual bias and random variances, for the production process... 

Equations (2.2), (2.5) and (2.6) for random variance presuppose that all the individual particles within the mixture can move independently of their neighbours. It will be seen in Chapter 5 that for cohesive powder systems the structure of the powder mixture can prevent independent movement of individual particles and that these limiting variance values have to be modified. 

Table lA The calculation of the random variance of a non-ideal mixture... 

By adjusting the slope and offset of the sample spectra to the ideal average spectrum, the chemical information is preserved while the differences between the spectra are minimized. This is not to say that the MSC spectra represent the true spectra of the samples, but that the major source of random variance between them has been removed as much as possible. 

Once significant factors are determined, error reduction is accomplished by discarding factors that describe random variance. Thus... 

Partial least squares regression [16-18] has been employed since the early 1980s and is closely related to PCR and MLR [18]. In fact, PLS can be viewed as a compromise midway between PCR and MLR [19]. In determining the decomposition of R (and consequently removing unwanted random variance), PCR is not influenced by knowledge of the estimated property in the calibration set, c. Only the variance in R is employed to determine the latent variables. Conversely, MLR does not factor R prior to regression all variance correlated to c is employed for estimation. PLS determines each latent variable to simultaneously optimize variance described in R and correlation with c, p. Technically, PLS latent variables are not principal components. The PLS factors are rotations of the PCA PCs for a slightly different optimization criterion. 

Fortunately, however, the technique used here does not depend on the magnitude of the variances, but only on their ratios. If estimates of the magnitudes of the variances are wrong but the ratios are correct, the residuals display the random behavior shown in Figure 3. However, the magnitudes of these deviations are then not consistent with the estimated variances. 

These two methods generate random numbers in the normal distribution with zero me< and unit variance. A number (x) generated from this distribution can be related to i counterpart (x ) from another Gaussian distribution with mean (x ) and variance cr using... 

The average random force over the time step is taken from a Gaussian with a varianc 2mk T y(St). Xj is one of the 3N coordinates at time step i E and R are the relevan components of the frictional and random forces at that time n, is the velocity component. 

Qj is a random number with zero mean and unit variance, chosen independently for each pair of particles and at each time step in the integration. 

Understanding the distribution allows us to calculate the expected values of random variables that are normally and independently distributed. In least squares multiple regression, or in calibration work in general, there is a basic assumption that the error in the response variable is random and normally distributed, with a variance that follows a ) distribution. 

The data on the left were obtained under conditions in which random errors in sampling and the analytical method contribute to the overall variance. The data on the right were obtained in circumstances in which the sampling variance is known to be insignificant. Determine the overall variance and the contributions from sampling and the analytical method. 

This experiment introduces random sampling. The experiment s overall variance is divided into that due to the instrument, that due to sample preparation, and that due to sampling. 

Collaborative testing provides a means for estimating the variability (or reproducibility) among analysts in different labs. If the variability is significant, we can determine that portion due to random errors traceable to the method (Orand) and that due to systematic differences between the analysts (Osys). In the previous two sections we saw how a two-sample collaborative test, or an analysis of variance can be used to estimate Grand and Osys (or oJand and Osys). We have not considered, however, what is a reasonable value for a method s reproducibility. 

The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance G. When the population is not normal but skewed, square probabihties could be substantially in error. 

Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a hnear combination of the measured variables [yj], Y = CL y + Cioyo + Let the random variables yi, yo,. . . have means E yi), E y, . . . and variances G yi), G y, . The variable Y has mean... 

Harnhy, N., The Estimation of the Variance of Samples Withdrawn from a Random Mixture of Multi-Sized Particles, Chem. Eng. No. 214 CE270-71 (1967). 

Because the datay are random, the statistics based on y, S(y), are also random. For all possible data y (usually simulated) that can be predicted from H, calculate p(S(ysim) H), the probability distribution of the statistic S on simulated data y ii given the truth of the hypothesis H. If H is the statement that 6 = 0, then y i might be generated by averaging samples of size N (a characteristic of the actual data) with variance G- = G- (yacmai) (yet another characteristic of the data). 

Another consideration when using the approach is the assumption that stress and strength are statistically independent however, in practical applications it is to be expected that this is usually the case (Disney et al., 1968). The random variables in the design are assumed to be independent, linear and near-Normal to be used effectively in the variance equation. A high correlation of the random variables in some way, or the use of non-Normal distributions in the stress governing function are often sources of non-linearity and transformations methods should be considered. 

The random nature of most physieal properties, sueh as dimensions, strength and loads, is well known to statistieians. Engineers too are familiar with the typieal appearanee of sets of tensile strength data in whieh most of the individuals eongregate around mid-range and fewer further out to either side. Statistieians use the mean to identify the loeation of a set of data on the seale of measurement and the variance (or standard deviation) to measure the dispersion about the mean. In a variable x , the symbols used to represent the mean are /i and i for a population and sample respeetively. The symbol for varianee is V. The symbols for standard deviation are cr and. V respeetively, although a is often used for both. In this book we will always use the notation /i for mean and cr for the standard deviation. 

Rather than solve the variance equation for a number of variables directly, this method allows us to simulate the output of the variance, for example the simulated dispersion of a stress variable given that the random variables in the problem can be characterized. 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

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