Confidence ellipses for

Figure 6-2. Confidence ellipses for van Laar parameters. Acetone(l)-methanol(2) system at 755 mm Hg (Othmer, 1928).

Figure 1. Optimum binary deviation parameters (k and ji j) and confidence ellipses for the fit of the LHW equation of state to the Nz + Ar VE data of Ref. 11, pp. 166-168...

Figure 6.3 FP sensory map (i.e. score plot from the GPA consensus) with 95% confidence ellipses for each product. Two-digit numbers in product names represent the cocoa content.

Husson, F., Le, S. and Pages, J. (2005). Confidence ellipse for the sensory profiles obtained by principal component analysis. Food Quality and Preference, 16, 245-250. 

Figure 4.2. Variance, covariance, and confidence ellipse for an [A], [B] value of equation (4.5).

Joint confidence ellipse for the ordinate and intercept of the original regression, along with the pairs (intercept, slope) obtained for each of the 24 simulated calibration lines. The triangles, squares and circles correspond to modifications in the lowest (Addi), intermediate (Add2) and highest addition (Add3), respectively. 

Using the ternary tie-line data and the binary VLE data for the miscible binary pairs, the optimum binary parameters are obtained for each ternary of the type 1-2-i for i = 3. .. m. This results in multiple sets of the parameters for the 1-2 binary, since this binary occurs in each of the ternaries containing two liquid phases. To determine a single set of parameters to represent the 1-2 binary system, the values obtained from initial data reduction of each of the ternary systems are plotted with their approximate confidence ellipses. We choose a single optimum set from the intersection of the confidence ellipses. Finally, with the parameters for the 1-2 binary set at their optimum value, the parameters are adjusted for the remaining miscible binary in each ternary, i.e. the parameters for the 2-i binary system in each ternary of the type 1-2-i for i = 3. .. m. This adjustment is made, again, using the ternary tie-line data and binary VLE data. 

The optimum parameters for furfural-benzene are chosen in the region of the overlapping 39% confidence ellipses. The ternary tie-line data were then refit with the optimum furfural-benzene parameters final values of binary parameters were thus obtained for benzene-cyclohexane and for benzene-2,2,4-trimethyl-pentane. Table 4 gives all optimum binary parameters for this quarternary system. 

For the acetone-methanol data of Othmer, the correlation coefficient is -0.678, indicating a moderate degree of correlation between the two van Laar parameters. The elongated confidence ellipses shown in Figure 2 further emphasize this correlation. 

If the parameters were to become increasingly correlated, the confidence ellipses would approach a 45 line and it would become impossible to determine a unique set of parameters. As discussed by Fabrics and Renon (1975), strong correlation is common for nearly ideal solutions whenever the two adjustable parameters represent energy differences. 

Figure 1. Results of Monte Carlo simulations for 1500 pairs of x, y points with a mean of 100, Ganssian errors of 10 (1 o), and four different x-y error correlations (p). Elhpses show 95% confidence hmits for the joint x-y distribution. Note that the ellipses extend farther than the 2o range of either the x or j errors themselves—a non-intuitive characteristic of joint distributions that arises because an x- (or y-) value deviating less than expected permits ay- (orx-) value that deviates more than expected.

Figure 14.22 Hydrogen isotope ratios of CJ6 oand Cis.-o fatty acids in modern horse milk and adipose fat, both from Kazakhstan (a), displaying seasonal influences [ellipses are 68 % (la) confidence intervals], compared (b) with 8 D values recorded for fatty acids from Eneolithic cooking pot sherds from Botai residues assigned as equine fats based on 813C analysis. Confidence ellipses are la and correspond to the modern reference equine fat values (Outram et al. 2009)...

Fig. 21.16. 513C values for the Ci6 o and Ci8 0 fatty acids extracted from the Roman cream, compared with confidence ellipses (la) corresponding to those from modern cow, sheep, and pig adipose fat and sheep and cow butter fat (reference 513C values are adjusted for post-industrial Revolution effects of fossil-fuel burning analytical precision + 0.3%). (Reprinted/redrawn from Nature, 432, 35-36, Copyright 2004, Nature Publishing Group, with permission.)... 

We therefore calculate the coordinates of an arbitrary number of points z, (i = 1,2,...) on the unit circle. This is most easily done by incrementing an arbitrary angle

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Table 4.4. Contours of the 95 percent confidence ellipse of the mean for the Pb isotope composition...

Figure 4.9 The 95 percent confidence ellipse of the mean for the Polynesian data listed in Table 4.3. The horizontal and vertical bars show the 95 percent confidence intervals of the mean calculated independently for each coordinate.

Figure 5. Bivariate plot of chromium and iron base-10 logged concentrations for Dragon Jar paste samples analyzed by INAA. Ellipses represent 90% confidence levels for group membership.

Figure 8. Bivariate plot of iron and lead log base-10 ppm oxide concentrations showing subgroup variation within the mineral and organic-paint groups. Ellipses represent 90% confidence interval for group membership. Unassigned specimens are not shown.

Figure 6. Bivariate plot of log-base 10 values for iron and cerium. The ellipses represent a 90% confidence interval for each group.

Figure 7. Bivariate plot of log-base 10 values of iron and cerium derivedfrom a previous INAA study of early Islamic period glazed ceramics(32). The ellipses represent a 90% confidence interval for each group. Note that the data for Group 2 fall in the same range as the Fe and Ce values for Group 2 in Figure 6.

Figure 4. Plot of zirconium and rubidium base-10 logged concentrations showing the separation cf the Early Postclassic period obsidian excavated from Trinidad de Nosotros. Ellipses represent 90% confidence interval for group...

Figure 2. Plot Ba versus Mn from short-NAA for archaeologically-important obsidian sources in southern Peru. Confidence ellipses at the 95% level are shown for the major (shaded) and minor (not shaded) sources.

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