Multiplicity Deviation Factor

This is only a functional architecture to illustrate our idea of early warning system in food supply networks. In practice deviations may have multiple determinant factors, so that we have multiple DACWP to monitor. 

Here, Aq(X), a, /3, and e(k) are a real spectrum, multiplicative scatter factor (amplification factor), additive scatter factor (offset deviation), and noise, respectively. There are several methods to eliminate or reduce the effects of a and /3. Here, we describe three of them. 

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 family of curves obtained, and presented in Figure 43-6, show that, not surprisingly, the controlling parameter of the family of curves is the standard deviation of the noise the maximum value of the multiplication factor occurs at a given fraction of the standard deviation of the energy readings. Successive approximations show that the maximum multiplier of approximately 1.28 occurs when If is approximately 2.11 times sigma, the standard deviation of AEr. 

Figure 43-6 Family of curves of multiplication factor as a function of Er, for different values of the parameter sigma (the noise standard deviation), for Normally distributed error. Values of sigma range from 0.1 to 1.0 for the ten curves shown, (see Color Plate 5)...

When the noise is small the multiplication factor approaches unity, as we would expect. As we have seen for the previous two types of noise we considered, the nonlinearity in the computation of transmittance causes the expected value of the computed transmittance to increase as the energy approaches zero, and then decrease again. For the type of noise we are currently considering, however, the situation is complicated by the truncation of the distribution, as we have discussed, so that when only the tail of the distribution is available (i.e., when the distribution is cut off at +3 standard deviations), the character changes from that seen when most of the distribution is used. 

However, as mentioned previously, gas-diffusion electrodes usually deviate substantially from traditional electrochemical—kinetic behavior, often being limited by multiple rate-determining factors and/or changes in those factors with overpotential or other conditions. In attempting to analyze this type of electrode, one of the most influential experimental techniques to take hold in the solid-state electrochemical literature in the last 35 years is electrochemical impedance spectroscopy (EIS)—also know as a.c. impedance. As illustrated in Figure 6, by measuring the sinusoidal i— response as a function... 

The simple approach is just using the mean value of several determinations of blank samples plus a multiple (factor k) of the standard deviation of these measnrements. With the choice of k we define the level of confidence. 

Analysis of Variance (ANOVA). Keeping in mind that the total variance is the sum of squares of deviations from the grand mean, this mathematical operation allows one to partition variance. ANOVA is therefore a statistical procedure that helps one to learn whether sample means of various factors vary significantly from one another and whether they interact significantly with each other. One-way analysis of variance is used to test the null hypothesis that multiple population means are aU equal. 

The differences are squared to make them all positive otherwise, for a large number of random differences, D simply equals zero. The term wt is an optional weighting factor that reflects the reliability of observation i, thus giving greater influence to the most reliable data. According to principles of statistics, wt should be 1/(standard deviation computed from multiple measurements of the same data point (x , v ). 

This pre-treatment method is also focused on applications in spectroscopy, although it can be applied elsewhere. Like the MSC method, the SNV method performs both an additive and a multiplicative adjustment. However, the correction factors are determined differently. For each sample s spectrum, the offset adjustment is simply the mean of the values over all of the variables, and the multiplicative adjustment is simply the standard deviation of the values over all of the variables ... 

Apparently, the discrepancies detected for the substitution data are largely the consequence of a multiplicity of minor influences operative in the transition state. The deviations are sufficiently diverse in character to require the significance of additional influences on the stability of the transition state. Four other important factors are complexing of the substituent with the electrophilic reagent or catalyst, the involvement of 7r-complex character in the transition state for the reaction, rate effects originating in the rupture of carbon-hydrogen bonds, and differential solvation of the electron-deficient transition states. 

Consistent with these definitions, there are two methods for IDL determination. The first method consists of multiple analyses of a reagent blank, followed by the determination of the standard deviation of the responses at the wavelength of the target analyte. The standard deviation multiplied by a factor of three is the IDL. This calculation defines the IDL as an analyte signal that is statistically greater than the noise. 

The laboratories, however, typically rely on the second method for the IDL determination, which is detailed in the CLP SOW (EPA, 1995c). The second method consists of multiple analyses of a standard solution at a concentration that produces a signal five times over the signal-to-noise level. The standard deviation of the measurements is multiplied by a factor of three to produce the IDL. This method assumes that the level of signal-to-noise is known, and this information is usually available from the instrument manufacturer. 

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