Degree of Linearity

The optimal prototypes are those corresponding to the purely linear case, i.e., for a = 1. 

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Corrosion Rate by CBD Somewhat similarly to the Tafel extrapolation method, the corrosion rate is found by intersecting the extrapolation of the linear poi tion of the second cathodic curve with the equihbrium stable corrosion potential. The intersection corrosion current is converted to a corrosion rate (mils penetration per year [mpy], 0.001 in/y) by use of a conversion factor (based upon Faraday s law, the electrochemical equivalent of the metal, its valence and gram atomic weight). For 13 alloys, this conversion factor ranges from 0.42 for nickel to 0.67 for Hastelloy B or C. For a qmck determination, 0.5 is used for most Fe, Cr, Ni, Mo, and Co alloy studies. Generally, the accuracy of the corrosion rate calculation is dependent upon the degree of linearity of the second cathodic curve when it is less than... 

Figure 2.15(a) shows the relationship between and Cp for the component characteristics analysed. Note, there are six points at q = 9, Cp = 0. The correlation coefficient, r, between two sets of variables is a measure of the degree of (linear) association. A correlation coefficient of 1 indicates that the association is deterministic. A negative value indicates an inverse relationship. The data points have a correlation coefficient, r = —0.984. It is evident that the component manufacturing variability risks analysis is satisfactorily modelling the occurrence of manufacturing variability for the components tested. 

An alternative method is to fit the best straight line through the linearized set of data assoeiated with distributional models, for example the Normal and 3-parameter Weibull distributions, and then ealeulate the correlation coejficient, r, for eaeh (Lipson and Sheth, 1973). The eorrelation eoeffieient is a measure of the degree of (linear) assoeiation between two variables, x and y, as given by equation 4.4. 

Sole use of the correlation coefficient (r) alone is not recommended as a means to demonstrate linearity. The correlation coefficient describes the relation between two random parameters, and shows no relevance for the analytical calibration [31]. The correlation coefficient does not indicate the linearity or lack thereof, unless r exceeds 0.999 [8, 32, 33]. If the value of r is less than 0.999, other parameters such as Vxo, Xp value, ANOVA linear testing, etc., should be calculated. Ebel [34] described using the transformation of r (i.e., Vu ) for expressing the degree of linearity, where the acceptance value of (1 — r ) should be less than 0.05. Camag (Muttents) described the sdv parameter (i.e., the relative standard deviation of the calibration curve) for expressing the linearity of a calibration curve for TLC/HPTLC in its CATS software, and can be calculated as follows ... 

The significance of conjugation as a contributor to the substantivity of dyes for cellulose is not always easy to distinguish from the effect of the degree of linearity of the molecule. Almost all direct dye molecules possess flexible chains of aryl nuclei linked by azo or other unsaturated groups. Such structures can readily adopt a near-linear spatial conformation, as... 

Linearity is evaluated by appropriate statistical methods such as the calculation of a regression line by the method of least squares. The linearity results should include the correlation coefficient, y-intercept, slope of the regression line, and residual sum of squares as well as a plot of the data. Also, it is helpful to include an analysis of the deviation of the actual data points for the regression line to evaluate the degree of linearity. 

Among the reported parameters the only one which shows some degree of linear correlation with the redox potential is the acidity parameter t(30), Figure 7. 

The specified range is normally derived from linearity studies and depends on the intended application of the procedure. This is established by confirming that the analytical procedure provides an acceptable degree of linearity, accuracy, and precision within the extremes of the specified range. 

By plotting logio7 , hci + D vs. much a straight line with the slope (H+,cr) is obtained. The degree of linearity should in itself indicate the range of validity of the specific ion interaction approach. Osmotic coefficient data can be treated in an analogous way. 

Table II shows that the observed 29) shifts 8 (taken as positive downfield) show a degree of linear additivity in the methyl-substituted methanes, with a shortfall for neopentane. Table II shows also the division of the shielding into atom-plus-ligand diamagnetic and paramagnetic parts, relative to methane for which era is 295 ppm. The diamagnetic terms were calculated by Flygare s method 25). With each substitution of hydrogen by carbon, aa for the central carbon increases by 28 ppm but o-p increases by about 37 ppm, and the line moves 9 ppm downfield. Analogous relationships have been demonstrated for shielding in methyl-substituted NH3 and NH4+ 30). Table II shows that the shortfall at neopentane is in the paramagnetic term.

Therefore, by rotating a linear polarizer in an arbitrary beam and noting the maximum and minimum transmitted irradiance, the degree of linear polarization can be measured regardless of the value of V. 

Other quantities commonly measured are the degree of linear polarization P of the scattered light for incident unpolarized light... 

Calculated and measured values of P = —Sn/Su, the degree of linear polarization, for several nonspherical particles are shown in Fig. 13.9. The prolate and oblate spheroids, cubes, and irregular quartz particles have made their appearance already (Fig. 13.8) a new addition is NaCl cubes. Also shown are calculations for equivalent spheres. 

Noctilucent cloud particles are now generally believed to be ice, although more by default—no serious competitor is still in the running—than because of direct evidence. The degree of linear polarization of visible light scattered by Rayleigh ellipsoids of ice is nearly independent of shape. This follows from (5.52) and (5.54) if the refractive index is 1.305, then P(90°) is 1.0 for spheres, 0.97 for prolate spheroids, and 0.94 for oblate spheroids. 

Alternate products of a high degree of linearity can be produced by oxonation of linear alpha olefins. By this process, one can obtain alcohols of any desired chain length, including the odd-numbered ones. Except for tridecyl alcohol, odd-numbered alcohols in the plasticizer range are not currently produced in volume at attractive prices in the United States. 

Fig. 9.—F.t.-I.r. Difference, Absorbance Peak-height y (at 1080 cm-1) in Absorbance Units vs. l/(n +1), Where n is the Degree of Linearity in Terms of the Average Numbers of D-Glucopyranosyl Residues Between Branching Residues. (The circles correspond to different dextrans used.) (From Ref. 172.)...

FIGURE 5.6 Partial least-squares regression model showing the correlation between alanine (nmol/g cheese) predicted by GC-FID and FTIR. The model shows a high degree of linear correlation (r-value = 0.99) and a low estimated standard error of prediction (12.70 nmol/g cheese). 

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