Variance additivity

When measuring the recorded peak variance we must keep in mind that the observed width of a peak consists of two parts the width due to the distribution process in the column and the undesirable dispersion outside the column (injector, connecting tubing, detector, everything that comes from the instrument). From the theorem of variance additivity of independent processes we can write... 

Dose and Guiochon [1] have demonstrated, however, that the rule of variance additivity does not apply in nonlinear chromatography, as the convolution of the... 

Shift-Variant (-Invariant) Convolution Convolutions in which the same spreading function is applied to every time element or data. This is the case in linear chromatography where the spreading caused by each plate on the passing distribution is the same. In nonlinear chromatography, the effect of each plate on the profile depends on the concentration in that plate. The convolution is said to be shift-variant. As a consequence the rules of variance addition do not apply. 

The F statistic describes the distribution of the ratios of variances of two sets of samples. It requires three table labels the probability level and the two degrees of freedom. Since the F distribution requires a three-dimensional table which is effectively unknown, the F tables are presented as large sets of two-dimensional tables. The F distribution in Table 2.29 has the different numbers of degrees of freedom for the denominator variance placed along the vertical axis, while in each table the two horizontal axes represent the numerator degrees of freedom and the probability level. Only two probability levels are given in Table 2.29 the upper 5% points (F0 95) and the upper 1% points (Fq 99). More extensive tables of statistics will list additional probability levels, and they should be consulted when needed. 

Precision is a measure of the spread of data about a central value and may be expressed as the range, the standard deviation, or the variance. Precision is commonly divided into two categories repeatability and reproducibility. Repeatability is the precision obtained when all measurements are made by the same analyst during a single period of laboratory work, using the same solutions and equipment. Reproducibility, on the other hand, is the precision obtained under any other set of conditions, including that between analysts, or between laboratory sessions for a single analyst. Since reproducibility includes additional sources of variability, the reproducibility of an analysis can be no better than its repeatability. 

The following texts provide additional information about ANOVA calculations, including discussions of two-way analysis of variance. Graham, R. C. Data Analysis for the Chemical Sciences. VCH Publishers New York, 1993. 

With the addition of increasing amounts of electrolyte this variance decreases and an approximate linear relationship between internal and external pH exists in a 1 Af electrolyte solution. The cell-0 concentration is dependent on the internal pH, and the rate of reaction of a fiber-reactive dye is a function of cell-0 (6,16). Thus the higher the concentration of cell-0 the more rapid the reaction and the greater the number of potential dye fixation sites. 

We thus get the values of a and h with maximum likelihood as well as the variances of a and h Using the value of yj for this a and h, we can also calculate the goodness of fit, P In addition, the linear correlation coefficient / is related by... 

If the normalized method is used in addition, the value of Sjj is 3.8314 X 10 /<3 , where <3 is the variance of the measurement of y. The values of a and h are, of course, the same. The variances of a and h are <3 = 0.2532C , cf = 2.610 X 10" <3 . The correlation coefficient is 0.996390, which indicates that there is a positive correlation between x and y. The small value of the variance for h indicates that this parameter is determined very well by the data. The residuals show no particular pattern, and the predictions are plotted along with the data in Fig. 3-58. If the variance of the measurements of y is known through repeated measurements, then the variance of the parameters can be made absolute. 

The direct-labor-cost variance can, if necessary, be broken down into a direc t-labor-idle-time variance in addition to the direct-wage-rate and direct-labor-efficiency variances. The direc t-labor-idle-time variance is simply the number of idle labor-hours in the period multiplied by the standard wage rate. This is rarely relevant to the conditions existing in process plants except when maintenance is involved. 

Work done by Wiesner [6] is a much more accurate approach. The subject has also been reported on more recently by Simon and Bulskamper [71. They generally agree with Wiesner that the variance of performance with Reynolds number was more true at low value that at high values. The additional influence above a Reynolds number of 10 is not much. It would appear that if a very close guarantee depended on the Reynolds number to get the compressor within the acceptance range (if the Reynolds number was high to begin with), the vendor would be rather desperate. 

To confirm the pertinence of a particular dispersion equation, it is necessary to use extremely precise and accurate data. Such data can only be obtained from carefully designed apparatus that provides minimum extra-column dispersion. In addition, it is necessary to employ columns that have intrinsically large peak volumes so that any residual extra-column dispersion that will contribute to the overall variance is not significant. Such conditions were employed by Katz et al. (E. D. Katz, K. L. Ogan and R. P. W. Scott, J. Chromatogr., 270(1983)51) to determine a large quantity of column dispersion data that overall had an accuracy of better than 3%. The data they obtained are as follows and can be used confidently to evaluate other dispersion equations should they appear in the literature. 

One often wishes to determine if, in a least squares treatment, addition of a new parameter will improve significantly the fit of the data. This is readily achieved by analysis of variance. Since this technique is little known, it will be briefly outlined here. 

The ability of a GC column to theoretically separate a multitude of components is normally defined by the capacity of the column. Component boiling point will be an initial property that determines relative component retention. Superimposed on this primary consideration is then the phase selectivity, which allows solutes of similar boiling point or volatility to be differentiated. In GC X GC, capacity is now defined in terms of the separation space available (11). As shown below, this space is an area determined by (a) the time of the modulation period (defined further below), which corresponds to an elution property on the second column, and (b) the elution time on the first column. In the normal experiment, the fast elution on the second column is conducted almost instantaneously, so will be essentially carried out under isothermal conditions, although the oven is temperature programmed. Thus, compounds will have an approximately constant peak width in the first dimension, but their widths in the second dimension will depend on how long they take to elute on the second column (isothermal conditions mean that later-eluting peaks on 2D are broader). In addition, peaks will have a variance (distribution) in each dimension depending on... 

In molecular doped polymers the variance of the disorder potential that follows from a plot of In p versus T 2 is typically 0.1 eV, comprising contributions from the interaction of a charge carrier with induced as well as with permanent dipoles [64-66]. In molecules that suffer a major structural relaxation after removal or addition of an electron, the polaron contribution to the activation energy has to be taken into account in addition to the (temperature-dependent) disorder effect. In the weak-field limit it gives rise to an extra Boltzmann factor in the expression for p(T). More generally, Marcus-type rates may have to be invoked for the elementary jump process [67]. 

This lack of sharpness of the 1-way F-test on REV s is sometimes seen when there is information spanned by some eigenvectors that is at or below the level of the noise spanned by those eigenvectors. Our data sets are a good example of such data. Here we have a 4 component system that contains some nonlinearities. This means that, to span the information in our data, we should expect to need at least 4 eigenvectors — one for each of the components, plus at least one additional eigenvector to span the additional variance in the data caused by the non-linearity. But the F-test on the reduced eigenvalues only... 

The last example brings out very clearly that knowledge of only the mean and variance of a distribution is often not sufficient to tell us much about the shape of the probability density function. In order to partially alleviate this difficulty, one sometimes tries to specify additional parameters or attributes of the distribution. One of the most important of these is the notion of the modality of the distribution, which is defined to be the number of distinct maxima of the probability density function. The usefulness of this concept is brought out by the observation that a unimodal distribution (such as the gaussian) will tend to have its area concentrated about the location of the maximum, thus guaranteeing that the mean and variance will be fairly reasdnable measures of the center and spread of the distribution. Conversely, if it is known that a distribution is multimodal (has more than one... 

Additive Gaussian Noise Charmed.17 An example of the use of these bounds will now be helpful. Consider a channel for which tire input is an arbitrary real number and the output is the sum of the input and an independent gaussian random variable of variance a3. Thus,... 

In other words, those factors and operations that contribute the most toward the total variance (see additivity of variances, next section) need to be individually repeated for two measurements on the same sample to be independent. Provided the two samples are taken with a sufficiently long delay between them, they can be regarded as giving independent information on the examined system. 

In mathematical terms, using the additivity of variances rule,... 

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