Line of best fit

Fig. 6. Free energies of hydration calculated, for a series of polar and non-polar solute molecules by extrapolating using (3) from a 1.6 ns trajectory of a softcore cavity in water plotted against values obtained using Thermodynamic Integration. The solid line indicates an ideal one-to-one correspondence. The broken line is a line of best fit through the calculated points.

Fig. 5. Plot of estimated (ASA) from computer simulations against %PPII content (Kelly et al., 2001 A. L. Rucker, M. N. Campbell, and T. P. Creamer, unpublished results). Line of best fit is computed only for filled circles.

The different slopes for the lines of best fit for total reactives and vitrinite + exinite reflects the special petrology of the majority of South African coals used in this study (see Table III). For these coals the reactives contain a high proportion of semi-fusinite in the inertinite. 

So why bother If established and reliable methods exist to solve scientific problems, surely there is no point in using methods from AI that may be slower or less effective The key here is the phrase "if established and reliable methods exist." In many situations, there is no need to turn to AI methods, but not every scientific problem is as straightforward as finding the line of best fit. When more conventional methods are unavailable, or are of insufficient power, an AI method may be just what is needed. 

Draw the line of perfect fit as described above. Each point on the graph is plotted so that it lies away from this line (imprecision) but so that the line of best fit matches the perfect line (accuracy). 

A numerical description of the gradient of the line of best fit using linear... 

Draw and label the axes. The x axis is traditionally where the independent variable is plotted. Draw a line of best fit surrounded by data points. As the line of best fit has a positive slope, both b and r will be positive. However, r will not be +1 as the data points do not lie exactly on the line. In this case r is approximately +0.8. 

This plot is drawn in the same way as the negative plot but now the line of best fit becomes a line of exact fit. Both b and r will now be negative and r will be — 1 as the data points lie exactly on the line. 

Draw and label the axes as before but note that on this plot there is no meaningful line of best fit as the data points are truly random. It is not possible to give a value for b as a line of best fit cannot be generated but the value of r is 0. 

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 7. Plot of AMpyr against vitrinite content (on a wt% hydrogen basis) for bituminous coals with 80-86% C (o) and >86% C (x). The line of best fit to these data for the lower rank subset Is shown.

However, Brown (9) has extrapolated the total number of nuclei back to zero size by using a line of best fit. His plot of N vs t obtained still shows the same trend noted in Figure 5(b). 

MeCN/H,0 mixtures of volume fractions from ip = 0.41 to ip = 0.45 containing 0.1% TFA and butylsilica at temperatures T= 278-338 K. The plots show the experimental data and the lines of best fit according to the quadratic form of the relationship given in eq 7 and the polynomial form of the van t Hoff relationship. b Adapted from Jong, Boysen, and Hearn)24 with permission. 

Fig. 1—The effect of drug loading on tobramycin sulphate release profiles. The solid lines represent the lines of best fit.

Average monthly morning concentrations of radon in air increased with average monthly values of atmospheric stability during the 12 hours before filter collection, but there was appreciable scatter of values about lines of best fit. The squared correlation coefficient was 0.53 for a linear relation between radon concentration and stability. Lower correlations applied to stability values averaged over periods longer or shorter than 12 hours. Precipitation during the 12-hour period before collection decreased radon concentrations. 

The complexes contributing to the least squares line in Figure 44 contain either two-coordinate donor atoms with very little steric hindrance (for example, acetylacetonate, oxalate, dithiocarbamates or xanthates), or tetrahedrally coordinated donor atoms (for example, 1,2-diaminoethane or 1,3-diaminopropane). The agreement between the theoretical curve and the experimental line is very good. The observation that the line of best fit lies 0.5° below the calculated line may be... 

Figure 44 Angle of twist 6 (degrees) and normalized bite b for tris(bipyridyl) and tris(phenanthroline) complexes. Full line, theoretical curve for most stable stereochemistry. Broken line, line of best fit through 141 structurally characterized complexes...

Figure 2. Gel permeation data for polypeptide linear random coils plotted according to the method of Porath (8) M0,555 is plotted vs. Kd1/3. Lines drawn through the data from each column are lines of best fit determined by linear least-squares analysis. Numerical designation for each curve represents the agarose...

In the previous scatterplot, notice that a line of best fit can be created ... 

Lines of best fit for D2 of calculated vs. measured iron content computed from regression analysis at the... 

Figure 2 Determination of irreversible inhibition constants for erythromycin and CYP3A4 microsomes. Microsomes (1 mg/mL microsomal protein) were incubated with erythromycin (0-100 pM) in the presence of NADPH for differing incubation times. The pseudo-first-order rate constant for enzyme inactivation was plotted versus erythromycin concentration to estimate Ki and inact (14.4 pM and 0.045 min-1, respectively). The curve represents the line of best fit. Source From Ref. 32.

Figure 4 Effect of incubation time on the formation of A-desmethyldiltiazem (MA) in HLMs. HLMs (50 pg) were incubated with diltiazem (12.5-1200 pM) and NADPH (1 mM) at 37° for 8 (A), 16 ( ), and 24 ( ) minutes. The dashed line is the line of best fit of the data with the Michaelis-Menten equation. The solid line is the line that represents the predicted MA formation at the corresponding time using instantaneous formation rates. Abbreviations HLMs, human liver microsomes MA, A-desmethyldiltiazem. Source From unpublished data.

Figure 4.1 Measured and estimated (from WSKOWWIN) water solubility (mg/L) of some alkenes. The line of best fit is y= 1.08x- 0.32 (r2 = 0.90).

Figure 4.2 Measured and estimated (from MPBPVP) boiling point (°C) of some aniline derivatives. The line of best fit is y = 0.92x + 22.6 (r2 = 0.83).

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