Variance inhomogeneity

The inhomogeneity variance ofnhom determines the sampling error ofmp which can be estimated experimentally and which is crucial for representative sampling1... 

A graphical display of the residuals tells us a lot about our data. They should be normally distributed (top left). If the variances increase with the concentration, we have inhomogeneous variances, called heteroscedasticity (bottom left). The consequences are discussed in the next slide. If we have a linear trend in the residuals, we probably used the wrong approach or we have a calculation error in our procedure (top right). Non-linearity of data deliver the situation described on bottom right, if we nevertheless use the linear function. 

In cases where we have inhomogeneous variances we may reduce the working range or use a regression "" procedure weighted with variances at different concentrations. 

The problem in both methods is the error propagation. If an error exists in the measurement, this error will be submitted to the transformation as well. A second problem arises in the variances. Usually the variances of measurement in TLC are constant within the calibration range. The transformation of data will lead to inhomogeneous variances and this is the reason for unreliable regression analysis. 

Catalyst weakness in the model, which, however, is superimposed by considerably inhomogeneous variances... 

Equation (5), however, would apply only to a perfectly packed column so Van Deemter introduced a constant (2X) to account for the inhomogeneity of real packing (for ideal packing (X) would take the value of 0.5). Consequently, his expression for the multi-path contribution to the total variance per unit length for the column (Hm) is... 

The sampling variance of the material determined at a certain mass and the number of repetitive analyses can be used for the calculation of a sampling constant, K, a homogeneity factor, Hg or a statistical tolerance interval (m A) which will cover at least a 95 % probability at a probability level of r - a = 0.95 to obtain the expected result in the certified range (Pauwels et al. 1994). The value of A is computed as A = k 2R-s, a multiple of Rj, where is the standard deviation of the homogeneity determination,. The value of fe 2 depends on the number of measurements, n, the proportion, P, of the total population to be covered (95 %) and the probability level i - a (0.95). These factors for two-sided tolerance limits for normal distribution fe 2 can be found in various statistical textbooks (Owen 1962). The overall standard deviation S = (s/s/n) as determined from a series of replicate samples of approximately equal masses is composed of the analytical error, R , and an error due to sample inhomogeneity, Rj. As the variances are additive, one can write (Equation 4.2) ... 

The inhomogeneity- and the sampling variance are adequate only with a given risk of error a ojnhom = ofmp... 

Fig. 2.7. Dependence of the total variance on the sample amount characterized by mass merit is the critical sample mass. The statement of homogeneity or inhomogeneity is always related to a given analyte A... 

Figure 5.16. The four-environment model divides a homogeneous flow into four environments, each with its own local concentration vector. The joint concentration PDF is thus represented by four delta functions. The area under each delta function, as well as its location in concentration space, depends on the rates of exchange between environments. Since the same model can be employed for the mixture fraction, the exchange rates can be partially determined by forcing the four-environment model to predict the correct decay rate for the mixture-fraction variance. The extension to inhomogeneous flows is discussed in Section 5.10.

As done below for two examples, expressions can also be derived for the scalar variance starting from the model equations. For the homogeneous flow under consideration, micromixing controls the variance decay rate, and thus y can be chosen to agree with a particular model for the scalar dissipation rate. For inhomogeneous flows, the definitions of G and M(n) must be modified to avoid spurious dissipation (Fox 1998). We will discuss the extension of the model to inhomogeneous flows after looking at two simple examples. 

Comparing (5.377) with (3.105) on p. 85 in the high-Reynolds-number limit (and with e = 0), it can be seen that (5.378) is a spurious dissipation term.149 This model artifact results from the presumed form of the joint composition PDF. Indeed, in a transported PDF description of inhomogeneous scalar mixing, the scalar PDF relaxes to a continuous (Gaussian) form. Although this relaxation process cannot be represented exactly by a finite number of delta functions, Gs and M1 1 can be chosen to eliminate the spurious dissipation term in the mixture-fraction-variance transport equation.150... 

Note, however, that in the absence of micromixing, )n is constant so that this term will be null. Nevertheless, when micromixing is present, the spurious scalar dissipation term will be non-zero, and thus decrease the scalar variance for inhomogeneous flows. 

The procedure followed above can be used to develop a multi-environment conditional LES model starting from (5.396). In this case, all terms in (5.399) will be conditioned on the filtered velocity and filtered compositions,166 in addition to the residual mixture-fraction vector = - . In the case of a one-component mixture fraction, the latter can be modeled by a presumed beta PDF with mean f and variance (f,2>. LES transport equations must then be added to solve for the mixture-fraction mean and variance. Despite this added complication, all model terms carry over from the original model. The only remaining difficulty is to extend (5.399) to cover inhomogeneous flows.167 As with the conditional-moment closure discussed in Section 5.8 (see (5.316) on p. 215), this extension will be non-trivial, and thus is not attempted here. 

The uncertainty of the calculated spreading factor A(o q) depends upon the accuracy of the inhomogeneity index of the sample and that of the total variance of experimental chromatogram. It may be expressed as... 

Van Deemter also introduced a constant (y)into the Longitudinal Dispersion contribution to variance to account for some packing Inhomogeneity and so the expression for the Diffusion contribution to the variance per unit length of the column became,... 

The homogeneity should be established by testing a representative number of laboratory samples taken at random using either the proposed method of analysis or other appropriate tests such as UV absorption, refractive index, etc. The penalty for inhomogeneity is an increased variance in analytical results that is not due to intrinsic method variability. 

A study by Rasemann et al. demonstrated to what extent mercury concentrations depend on the method of handling soil samples between sampling and chemical analysis for samples from a nonuniformly contaminated site [152], Sample pretreatment contributed substantially to the variance in results and was of the same order as the contribution from sample inhomogeneity. Welz et al. [153] and Baxter [154] have conducted speciation studies on mercury in soils. Lexa and Stulik [155] employed a gold film electrode modified by a film of tri-n-octylphosphinc oxide in a PVC matrix to determine mercury in soils. Concentrations of mercury as low as 0.02 ppm were determined. 

The error of the AAS determination of chromium in river water with a determined mean concentration of 4.8 (ig L 1 is acceptable for this particular environmental purpose. The highest variance percentage arises as a result of centrifugation. The sampling error arising from river inhomogeneity is relatively small in this particular environmental situation. 

For the characterization of the selected test area it is necessary to investigate whether there is significant variation of heavy metal levels within this area. Univariate analysis of variance is used analogously to homogeneity characterization of solids [DANZER and MARX, 1979]. Since potential interactions of the effects between rows (horizontal lines) and columns (vertical lines in the raster screen) are unimportant to the problem of local inhomogeneity as a whole, the model with fixed effects is used for the two-way classification with simple filling. The basic equation of the model, the mathematical fundamentals of which are formulated, e.g., in [WEBER, 1986 LOHSE et al., 1986] (see also Sections 2.3 and 3.3.9), is ... 

Univariate analysis of variance enables detection of feature-specific inhomogeneities within an investigated test area. 

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