Variation coefficients

To interpret the solutions obtained from the stochastic models, we propose to investigate their corresponding coefficient of variation Cv. Cv for a set of values is defined as the ratio of the standard deviation to the expected value or mean and is usually expressed as a percentage. It is calculated as  

Statistically, Cv is a measure of reliability, or evaluated from the opposite but equivalent perspective, it is also indicative of the degree of uncertainty. It is alternatively interpreted as the inverse ratio of data to noise in the data in signalprocessing-related applications. Thus, it is apparent that a small value of Cv is desirable as it signifies a small degree of noise or variability (e.g., in a data set) and, hence, reflects low uncertainty. 

In stochastic optimization, Cv can be purposefully employed to investigate, denote, and compare the relative uncertainty in models being studied. In a risk minimization model, as the expected value is reduced, the variability in the expected value (for example, as measured by variance or standard deviation) is reduced. The ratio of this change can be captured and described by Cv. Consequently, a comparison of the relative merit of models in terms of their robustness can be represented by their respective values of Cv, in the sense that a model with a lower Cv is favored since there is less uncertainty associated with it. In fact, Markowitz (1952) advocates that the use of Cy as a measure of risk would equally ensure that the outcome of a decisionmaking process still lies in the set of efficient portfolios for the case of operational investments. 

In a data set of normally distributed demands, if the Cv of demand is given as a case problem parameter, the standard deviation is computed by multiplication of Cv by the deterministic demand. Hence, increasing values of Q result in increasing fluctuations in the demand and this is again undesirable. 

Computation of Cv is based on the objective function of the formulated model. Table 6.1 displays the expressions to compute Cv for the proposed stochastic model formulations. Note that Cv for the deterministic case of each stochastic model should be equal to zero, by virtue of its standard deviation assuming a value of zero since it is based on the expected value solution. 

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The degree of mixing is described as variation coefficient standard deviation, X is fraction of additive, andX is average fraction of... 

Consider now making the variational coefficients in front of the inner basis functions constant, i.e. they are no longer parameters to be determined by the variational principle. The Is-orbital is thus described by a fixed linear combination of say six basis functions. Similarly the remaining four basis functions may be contracted into only two functions, for example by fixing the coefficient in front of the inner three functions. In doing this the number of basis functions to be handled by the variational procedure has been reduced from 10 to three. 

In the relativistic KKR method the trial function inside the MT-sphere is chosen as a linear combination of solutions of the Dirac equation in the center-symmetrical field with variational coefficients C7 (k)... 

X The use of the term coefficient of variation ( variation coefficient ) is not recommended byIUPAC —> IUPAC Orange Book [1997, 2000]... 

A flow injection optical fibre biosensor for choline was also developed55. Choline oxidase (ChOX) was immobilized by physical entrapment in a photo-cross-linkable poly(vinyl alcohol) polymer (PVA-SbQ) after adsorption on weak anion-exchanger beads (DEAE-Sepharose). In this way, the sensing layer was directly created at the surface of the working glassy carbon electrode. The optimization of the reaction conditions and of the physicochemical parameters influencing the FIA biosensor response allows the measurement of choline concentration with a detection limit of 10 pmol. The DEAE-based system also exhibited a good operational stability since 160 repeated measurements of 3 nmol of choline could be performed with a variation coefficient of 4.5%. 

The correlation coefficient does not indicate the quality. If we divide the residual standard deviation by the sensitivity (the slope of the function) we get the process standard deviation and the process variation coefficient respectively. 

The formula for the residual standard deviation is a bit different. This reflects that the degrees of freedom are one less. The slope of the function and therefore also the sensitivity is concentration dependent. So usually the sensitivity in the middle of the working range is reported. We may also calculate the process standard deviation and process variation coefficient in the middle of the working range. 

This slide shows an example taken from an European standard for the analysis of mercury in water. For drinking water the standard states a reproducibility variation coefficient of 30% on a mercury concentration level of 0.8 pg/l. Provided that we can prove that we can perform as described in the standard our expanded measurement uncertainty (95% confidence) is estimated to 60%. 

This slide shows an example from a drinking water PT provided by the University of Stuttgart. At the same level as in the previous example we find a reproducibility variation coefficient of about 20%. So our expanded uncertainty is estimated to 40%. 

The function F(l,2) is in fact the space part of the total wave function, since a non-relativistic two-electron wave function can always be represented by a product of the spin and space parts, both having opposite symmetries with respect to the electrons permutations. Thus, one may skip the spin function and use only the space part of the wave function. The only trace that spin leaves is the definite per-mutational symmetry and sign in Eq.(14) refers to singlet as "+" and to triplet as Xi and yi denote cartesian coordinates of the ith electron. A is commonly known angular projection quantum number and A is equal to 0, 1, and 2 for L, II and A symmetry of the electronic state respectively. The linear variational coefficients c, are found by solving the secular equations. The basis functions i(l,2) which possess 2 symmetry are expressed in elliptic coordinates as ... 

In Figure 4.6 a number of iso-Mahalanobis distance contours of a three component mixture have been depicted. The square of the variation coefficient (v) is constant. The ellipses drawn at Figure 4.5 are each contour lines with the same probability density value. This means that if a mixture is set to the centre point of one of the ellipses in the figure, than the probability that the composition of the mixture is present inside the drawn ellipse is in all the cases the same. 

Figure 15 Variation coefficient v as a function of the dimensionless mixing time for different L/D ratios. Copper and nickel particles of

Figure 18 Homogenization effect of the extruder/pump combination. Influence of the kinematic flow parameter A on the variation coefficient v at the distribution of iron powder in silicone oil. d=60 mm L/d= 5.23. Source Ref 18, Figure 1.4.1.

Limit of Quantification For the limit of quantification, or limit of determination, definitions and formulas are very similar to those of LOD, except that for LOQ, is taken to be 5, 6, or even 10 [2, 4,15, 56,72, 96]. A value of 10 for means that the %RSD at the limit of quantification is 10%. The LOQ thus corresponds to that concentration or amount of analyte quantifiable with a variation coefficient not higher than 10% [98]. The LOQ is always higher than the LOD and is often taken as a fixed multiple (typically 2) of the detection limit [4]. Also, the determination limit is referred to as the signal 10 times above the noise or background signal, corresponding to a signal-to-noise ratio of 10 1 [72, 85]. 

A CSF sample was analyzed 11-fold. The within-run variation coefficient ranged from 1 to 3.5% with two exceptions tryptophan (5%) and methionine (7%), which partially coeluted. The interassay coefficients of variation were calculated from a series of 11 analyses over a 7-month period. The median CV was 8% only taurine, arginine, and glutamate had CVs slightly in excess of 10%. The recovery of added amino acids to three CSF samples ranged form 83% (taurine) to 101% (isoleucine). Most recoveries were between 90 and 100%. At the lower end of the concentration range for CSF, a level of 1 pmol/1 can be safely detected. 

To check the quality of separation into groups the variation coefficient of qualitative features was estimated. The calculation of coefficient Q(x) is based on the contrast of the number of different events pairs ... 

The variation coefficient of qualitative features reacts only on the objects distribution according to feature gradation. It has maximum value of 1 when frequencies of feature gradation are equal and it has minimum value of 0 when whole aggregate is in one group on this feature. A small coefficient value shows that elements are distributed irregularly on groups. In some cases features with low Q(x) are not taken into account. 

TABLE 1. The variation coefficient of qualitative features for the different zoning methods. ... 

A.C. Apparent Crystallinity T.C. True Crystallinity C.V. Variation Coefficient... 

A large value of variation coefficient indicates that the data are widely spread about the mean. In contrast, if all values for the data points were nearly the same, the variation coefficient would be very small. 

The third period (1986-1994) was marked by a return to the range of 9-13.5%, similar to that in the first period but with a higher interannual variation (coefficient of variation 14.7%). Unlike the situation in the second period, the early years of the third showed a negative correlation between fatness and stock density. It is worth mentioning that sprat from around the island of Zmeiny were almost always fatter than their counterparts in waters around the Cape of Tarkhankut, presumably because of the input of nutrients from the Danube. 

The data on the volume properties of PMS liquids (i.e. coefficient of volumetric expansion, relative volume variation, coefficient of isothermal compressibility) are essential for the performance characteristics of oli-godimethylsiloxanes in hydraulic systems, hydraulic shocks and dampers they allow one to determine the working characteristics of these systems with some brands of PMS liquids at different temperatures and pressures. 

The ratio of the standard deviation of the mean velocity to its value proper, which is called the variation coefficient, almost does not depend on depth (Fig. 3b). In most cases, the fluctuations of the currents are greater than their mean velocities, which is characteristic of sufficiently long observation at... 

The normal to the coast distributions of the variation coefficients of the currents in the upper 150-m and the lower 200-1500-m layers shown in Fig. 5 occurred fundamentally different. In the upper layer, the minimum of the variation coefficient in Figs. 5a-5c coincides with the velocity maximum in Figs. 4a-4d, which confirms the above conclusion on the high stability of the MRC. Meanwhile, in the lower layer, zones of the maximal velocities of mean currents in Figs. 4e-4g coincide with the maximums of the variation coefficient in Fig. 5d. Thus, in the intermediate and deep layers, the MRC is rather instable, which allows some researchers, following Neumann, to seek for an intermediate or deep countercurrent beneath the MRC [17]. 

Coefficient of variation Coefficient of variation Coefficient of variation Coefficient of variation... 

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