Variances, summation

Having established that a finite volume of sample causes peak dispersion and that it is highly desirable to limit that dispersion to a level that does not impair the performance of the column, the maximum sample volume that can be tolerated can be evaluated by employing the principle of the summation of variances. Let a volume (Vi) be injected onto a column. This sample volume (Vi) will be dispersed on the front of the column in the form of a rectangular distribution. The eluted peak will have an overall variance that consists of that produced by the column and other parts of the mobile phase conduit system plus that due to the dispersion from the finite sample volume. For convenience, the dispersion contributed by parts of the mobile phase system, other than the column (except for that from the finite sample volume), will be considered negligible. In most well-designed chromatographic systems, this will be true, particularly for well-packed GC and LC columns. However, for open tubular columns in GC, and possibly microbore columns in LC, where peak volumes can be extremely small, this may not necessarily be true, and other extra-column dispersion sources may need to be taken into account. It is now possible to apply the principle of the summation of variances to the effect of sample volume. 

Equation (1) is the algebraic expression of the principle of the summation of variances. If the individual dispersion processes that take place in a column can be identified, and the variance that results... 

Before progressing to the Rate Theory Equation, an interesting and practical example of the use of the summation of variances is the determination of the maximum sample volume that can be placed on a column. This is important because excessive sample volume broadens the peak and reduces the resolution. It is therefore important to be able to choose a sample volume that is as large as possible to provide maximum sensitivity but, at the same time insufficient, to affect the overall resolution. 

Let a volume (Vi) be injected onto a column resulting in a rectangular distribution of sample at the front of the column. According to the principle of the Summation of Variances, the variance of the final peak will be the sum of the variances of the sample volume plus the normal variance of a peak for a small sample. 

To determine the band dispersion that results from a significant, but moderate, sample volume overload the summation of variances can be used. However, when the sample volume becomes excessive, the band dispersion that results becomes equivalent to the sample volume itself. In figure 10, two solutes are depicted that are eluted from a column under conditions of no overload. If the dispersion from the excessive sample volume just allows the peaks to touch at the base, the peak separation in milliliters of mobile phase passed through the column will be equivalent to the sample volume (Vi) plus half the base width of both peaks. It is assumed in figure 10 that the efficiency of each peak is the same and in most cases this will be true. If there is some significant difference, an average value of the efficiencies of the two peaks can be taken. 

The component-of-varlance analysis Is based upon the premise that the total variance for a particular population of samples Is composed of the variance from each of the Identified sources of error plus an error term which Is the sample-to-sample variance. The total population variance Is usually unknown therefore. It must be estimated from a set of samples collected from the population. The total variance of this set of samples Is estimated from the summation of the sum of squares (SS) for each of the Identified components of variance plus a residual error or error SS. For example ... 

Uncertainties relating to the determination of accurate quantitative results are not relevant in these experiments. The observed experimental variance of the INAA results is a summation of the variances of homogeneity and the relevant analytical components as shown in Equation (4.8) ... 

For At = 0 the function is equal to the variance a = X2 (equation 5.1), but for At - oo its value approaches zero because of the increasing probability of products of both positive and negative values, the summation of which becoming zero. Normalization of eqn. 5.2 by dividing both members by a yields the correlation function ... 

We now consider how to compute the variance of AT, according to equation 44-68a. Ordinarily we would first discuss converting the summations of finite differences to... 

SUMMATION OF VARIANCE FROM SEVERAL DATA SETS... 

The scalar variance is found by summation of the spectral energies ... 

FIGURE 6.2 Representation of multivariate data by icons, faces, and music for human cluster analysis and classification in a demo example with mass spectra. Mass spectra have first been transformed by modulo-14 summation (see Section 7.4.4) and from the resulting 14 variables, 8 variables with maximum variance have been selected and scaled to integer values between 1 and 5. A, typical pattern for aromatic hydrocarbons B, typical pattern for alkanes C, typical pattern for alkenes 1 and 2, unknowns (2-methyl-heptane and meta-xylene). The 5x8 data matrix has been used to draw faces (by function faces in the R-library Tea-chingDemos ), segment icons (by R-function stars ), and to create small melodies (Varmuza 1986). Both unknowns can be easily assigned to the correct class by all three representations. 

Any sample placed on to an LC column will have a finite volume, and the variance of the injected sample will contribute directly to the final peak variance that results from the dispersion processes that take place in the column. It follows that the maximum volume of sample that can be placed on the column must be limited, or the column efficiency will be seriously reduced. Consider a volume Vi, injected onto a column, which will form a rectangular distribution on the front of the column. The variance of the peak eluted from the column will be the sum of the variances of the Injected sample plus the normal variance of the eluted peak. The principal of the Summation of Variances will be discussed more extensively in a later chapter, at this time it can be stated that,... 

Individual variances. This is how the Rate Theory provides an equation for the final variance of the peak leaving the column. As an. example the principle of the summation of variances will be applied to extra column dispersion... 

Exact analytical solutions can be obtained for summations of normal distributions. The sum of normal distributions is identically a normal distribution. The mean of the sum is the sum of the means of each input distribution. The variance of the sum is the sum of the variance of the inputs. Any statistic of interest for the output can be estimated by knowing its distribution type and its parameters. For example, for a model output that is a normal distribution with known parameter values, one can estimate the 95th percentile of that output. 

The exact solutions are not valid if any of the model inputs differ from the distribution type that is the basis for the method. For example, the summation of lognormal distributions is not identically normal, and the product of normal distributions is not identically lognormal. However, the Central Limit Theorem implies that the summation of many independent distributions, each of which contributes only a small amount to the variance of the sum, will asymptotically approach normality. Similarly, the product of many independent distributions, each of which has a small variance relative to that of the product, asymptotically approaches lognormality. 

It is assumed that the summation of the noise variables over each of the. y responses can be represented by one variable i = 1,...,. y. Without any loss of generality, we assume that values of the error variables are distributed normally having variance c2 and mean zero, e, A/"(0, a1). Based on these assumptions, one can represent the black box principle in a slightly different way (see Figure 8.1b). Now the measurable response variable, yb can be represented as ... 

Different safety factors may have been used in the derivation of the reference values of the individual substances (RfDA deterministic HI thus sums risk ratios that may reflect different percentile values of a risk probability distribution. Assessment and interpretation of the uncertainty in the HI may be severely hampered by this summation of dissimilar distribution parameters. In a probabilistic risk assessment, the uncertainty in the exposure and reference values is often characterized by lognormal distributions. The ratio of 2 lognormal distributions also is a lognormal distribution. The variance in a quotient of 2 random variables can be approximated as follows (Mood et al. 1974, p 181) ... 

Some of the concepts used in defining confidence limits are extended to the estimation of uncertainty. The uncertainty of an analytical result is a range within which the true value of the analyte concentration is expected to lie, with a given degree of confidence, often 95%. This definition shows that an uncertainty estimate should include the contributions from all the identifiable sources in the measurement process, i.e. including systematic errors as well as the random errors that are described by confidence limits. In principle, uncertainty estimates can be obtained by a painstaking evaluation of each of the steps in an analysis and a summation, in accord with the principle of the additivity of variances (see above) of all the estimated error contributions any systematic errors identified... 

In this introduction, one further important basic concept needs to be discussed as it is used extensively in detector design and in particular, the design of detector connections and detector sensor geometry and that is the principle of the summation of variances. 

The above equation is the algebraic description of the principle of the summation of variances and is fundamentally important. If the individual dispersion processes that are taking place in a column can be identified, and the variance that results from each dispersion determined, then the variance of the final band can be calculated from the sum of all the individual variances. An example of the use of this principle is afforded by the calculation of the maximum extracolumn dispersion that can be tolerated for a particular column. This... 

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