Regression fitted

Fig. 12. The relationship between the mean oceanic residence time, T, yr, and the seawater—cmstal rock partition ratio,, of the elements adapted from Ref. 29. , Pretransition metals I, transition metals , B-metals , nonmetals. Open symbols indicate T-values estimated from sedimentation rates. The sohd line indicates the linear regression fit, and the dashed curves show the Working-Hotelling confidence band at the 0.1% significance level. The horizontal broken line indicates the time required for one stirring revolution of the ocean, T. ...

Application of equation 10 to the experimental D vs. [HSOIJ] data determined at 25°C and both 1 and 2 M acidity yielded straight line plots with slopes indistinguishable from zero and reproduced the Bi values determined in a non-linear regression fit of the data. This result implies no adsorption of PuSO by the resin and justifies use of the simpler data treatment represented by equation 2. A similar analysis of the Th(IV)-HSOiJ system done by Zielen (9) likewise produced results consistent with no adsorption of ThS0 + by Dowex AG50X12 resin. 

It has been shown that PLS regression fits better to the observed activities than principal components regression [53]. The method is non-iterative and, hence, is relatively fast, even in the case of very large matrices. 

We may compare our graphical result with the result obtained from solving for X and y by nonhnear regression fitting of the experimental rate curves to the power law form of eq. (8). Carrying this out using the Excel Solver tool... 

NOTE These results are the predicted values of a nonlinear regression fit of the... 

NOTE The dotted lines at-e used to connect sequential time points. The solid line is a ilnear regression fit of the log-linear terminal elimination phase. 

Here the subscript i denotes th set of the curvilinear coordinates. The local mean and Gaussian curvatures are determined by using nonlinear regression fitting after a number of sections at a given point has been made [this corresponds to different sets of the local coordinates (u, v)]. 

The data lie on a straight line only for Plot (1), the graph of [HI] vs. t. Therefore, the reaction is zero order with respect to HI. The slope of the line = -0.00546 mM-s-1, using a least mean square regression fitting program. However, the slope can be estimated from any two points on the line. If we use the first and last points ... 

Multistranded cables and ribbons were also observed for calcitonin (Bauer et al, 1995). These appeared to contain laterally associated 8-nm-wide fibrils, whereas the diameter of the individual strands within the multistranded cables could not be measured directly from the images. To estimate this diameter, the authors plotted the helical crossover spacing of the cables as a function of their diameter. Using a linear regression fit, the data were extrapolated to zero, yielding a width of 4.1 nm, similar to the width of the single protofibrils (Fig. 2A Bauer et al., 1995). 

Now the functions for doing simple power law-dependent simulations are developed. The zero-shear viscosity, //o. is 1.268 x 10 Pa-s as shown by Fig. 3.22 and the viscosity data in Table 3.6. This holds for all shear rates in the plateau range. For the power law fit, the last six entries in Table 3.6 are used to develop a regression fit, and then the line is extrapolated back to lower shear rates. The regression fit is as follows ... 

Fig. 5.7 Relaxation rate F divided by for PDMS in toluene solution vs. the reduced variable T/ri. Results correspond to NSE measurements =0.1 (filled circle) and DLS for =0.1 (filled triangle), (p=0.07 (empty triangle) and =0.02 (empty diamond). Solid line shows a linear regression fit of the (p=0,l NSE and DLS data. (Reprinted with permission from [186]. Copyright 2001 American Chemical Society)...

Graphically, the elements of b refer to the slope (b) and intercept (bo) of the line of best fit through the observed data points. For our example data in Table 12.1, a graph of the linear regression fit is illustrated in Figure 12.1. Once the b coefficients are estimated, the model error (f) can also be estimated ... 

Figure 12.31 Illustration of a hybrid calibration strategy. (A) Scatter plot of first two PCA scores obtained from a process analytical calibration data set containing both synthesized standards (circles) and actual process samples (triangles). (B) Results of a PLS regression fit to the property of Interest, using all of the calibration samples represented In (A).

TTie classification of kinetic methods proposed by Pardue [18] is adopted in the software philosophy. TTie defined objective of measurement in the system is to obtain the best regression fit to a minimum of 10 data points, taken over either a fixed time (i.e. the maximum time for slow reactions) or variable time (for reactions complete in less than 34 min, which is the maximum practical observation time). In an analytical system generating information at the rate of SO datum points per second, with reactions being monitored for up to 2040 s, effective data-reduction is of prime importance. To reduce this large quantity of analytical data to more manageable proportions, an algorithm was devised to optimize the time-base of the measurements for each individual specimen. 

Fig. 7 Relationship between the reduction peak potential, Epc, of a-K6P2Wi8062 or a-K4SiWi2O40 versus the reduction peak potential, EpcFc+, of ferricinium and the acceptor number of studied solvents. Abbreviations are the same as in Tables 6 and 7. The solid line is the best linear regression fit to all the experimental points, including water as a solvent, (a) a-K6P2Wi8062 correlation coefficient for the solid line 0.974. The correlation coefficient for the best fit to nonaqueous solvents only is 0.994. (b) a-K4SiWi204o correlation coefficient for the solid line 0.983. The correlation coefficient for the best fit to nonaqueous solvents only is 0.995 (taken from Ref 34).

Figure 3.6-4 The Walden plot of the molar conductivity and viscosity data in Tables 3.6-3-3.6-5. The line represents the linear regression fit of the data.

One possible reason of such a discrepancy is that during regression fitting an experimental uncertainty may spread over various parameters, thus leading to somewhat distorted final picture. Probably, a more reliable way to measure similarity/dissimilarity of solvents would be to rely on the direct experimental measurements of the distribution ratios rather than on the derived quantities, that is, LSFER parameters. The present authors employed that approach [15]. 

Figure 8 shows a plot of the concentration of methanol produced by the hydrolysis of SiQAC at pH 4.07 in water and the nonlinear regression curve of equation (18) assuming three consecutive, irreversible first-order reactions. A summary of the observed rate constants at each pH studied is shown in Table 4. Regression fits produced R2 values of better than 0.99 for all the pH values investigated. Plots of the observed values of k, k2, and k3 vs. pH are linear in all cases, with R2 values greater than 0.99, and with slopes of -0.997, -0.992 and -0.999, respectively. The ratio of kt k2 k3 is approximately 20 3 1. 

Step 1 A series of linear regressions of each X- variable to the property of interest is done. Step 2 The single variable that has the best linear regression fit to the property of interest is selected (xt). 

Figure 8.38 Results of a PLS regression fit to the property of interest, using all of the calibration samples represented in Figure 8.37.

This method is probably the simplest of the software-based standardization approaches.73,74 It is applied to each X-variable separately, and requires the analysis of a calibration set of samples on both master and slave instruments. A multivariate calibration model is built using the spectra obtained from the master instrument, and then this model is applied to the spectra of the same samples obtained from the slave instrument. Then, a linear regression of the predicted Y-values obtained from the slave instrument spectra and the known Y-values is performed, and the parameters obtained from this linear regression fit are used to calculate slope and intercept correction factors. In this... 

Fig. 6.21 Semi-logarithmic plots of the absolute value of the current following potential steps solid) and linear regression fits to linear portions (dashed). Potential versus, sum of reduction, and oxidation rate constant kT (a) 0.0 V, 2.33 s (b) 0.323 V,276s 1 and (c) —0.577 V, 2980s. Reproduced with permission from [43]...

Based on the different results obtained for the algal and Microtox tests (Figures 7 and 8), a waste PEEP index value was calculated for each waste and each L/S ratio assessed. Each waste index value was then plotted as a function of the corresponding L/S ratio (Fig. 9) and a simple non-linear regression fit (Power model, y = axb) was applied to predict the ecotoxicological hazard potential of leachate fluxes between L/S 4 and L/S 30 ratios. 

Figure 12. Normalized kinetics vs different energy densities for a para-hexaphenyle film with molecules lying on the substrate fitted with Eq. 5. In inset, logarithmic derivative of the decay at 35 pi per pulse in this film. The linear regression fits the data with 7 o = 1.5 x 1010 s 1, and (3 = 9.7 X 10s s 1. From Ref. [9].

Comparison of V/Al atomic ratios for A120, and V/(Si+Al) atomic ratios for V-loaded gels are presented rrrFig. 8. Overall, and somewhat independent of whether or not the samples have been subjected to H, reduction, these ratios for the gel are approximately 30 s smaller than the values observed for V on Al-O,. The least squares linear regression fits to the V/A120, data extrapolates near the origin whereas the corresponding aaxa for V/gel does not, Fig. 8. Hence, V dispersion on the gel surface, as monitored by XPS, is lower than on Al-O,. The hydrothermal instability of V loaded gel (Table 1) is prooably responsible for the lower V dispersion and for the lower V/Al+Si ratios seen in Fig. 8. 

One can measure the site concentrations in absolute units to 25% by measuring the absorption coefficient and radiative transition probability (which in turn comes from the level lifetime, radiative quantum efficiency and radiative branching ratios) or to 15% by nonlinear regression fitting of relative intensities to total dopant concentration over a range of site distributions. 

Fig. 1. Analysis of the apolar contribution to the dissolution thermodynamics of cyclic dipeptides into water. Each thermodynamic quantity is plotted against the number of apolar hydrogens (aH) (i.e., hydrogens bonded to carbon) (a) AC , (b) AH°, (c) AS0, and (d) AG°. Lines are the linear regression fit of the data. As described in the text, the slope gives the hydrophobic contribution. Data are from Murphy and Gill (1990).

Fig. 3. Linear correlation of AH0 versus ACp at 25°C for protein denaturation for the proteins listed in Table IV. (a) Normalized per number of residues (b) normalized per total buried area. The line in (a) is the linear regression fit with a slope of — 72.4 corresponding to a convergence temperature of 97.4°C. The line in (b) represents the line calculated from the parameters in Table II for convergence at the temperature at which the polar and apolar contributions to AH are equal per unit area. See text for details.

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