Fitted y-values

The remaining diagnostic plots shown in Figure 4.17 are the QQ-plot for checking the assumption of normal distribution of the residuals (upper right), the values of the y-variable (response) versus the fitted y values (lower left), and the residuals versus the fitted y values (lower right). The symbols + for outliers were used for the same objects as in the upper left plot. Thus it can be seen that the... 

The adjustable parameters were a, y, and rQ. The best-fit friction factors y for the 43-bp fragment are the same on all four time spans, as shown in Figure 4.9. The best-fit y values are similarly independent of time span for the 69-bp fragment.(109) The hydrodynamic radius a for azimuthal rotation was calculated from the measured friction factor for uniform azimuthal rotation of the entire filament, f = (N+ 1 )y, using the formula of Tirado and Garcia de la Torre,(129)/ii = 3Mlrjna2L(l +<5,), where t is the solvent viscosity, and dM is an end-plate correction, which they tabulate. The same value... 

The hat matrix transforms the vector of measured y values into the vector of fitted y values. The element of H, denoted by is... 

Equation 19 utilizes the Y-residuals, 1) — Y, where 1) are the points on the calculated best-fit line or the fitted 1) values. The appropriate number of degrees of freedom is A — 2 the minus 2 arises from the fact that linear calibration lines are derived from both a slope and an intercept which leads to a loss of two degrees of freedom. 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

Fig, 3.16. The density-dependence of the frequency shift of the Q-branch maximum. The y values for the curves are in the notation of Fig. 3.15. When plotting the experimental data, the cross-section found in the fitting of the density dependence of the width was employed (Fig. 3.15). 

The y and mean y values computed at different theoretical levels are given in Table 1. We can see that the hyperpolarizability increases with the extension of the polymeric chain. It is worth noting that our MNDO values agree with the ab initio ones of Kurtz[15] but do not vary in a parallel direction to the CNDO results of Waite and Papadopoulos[7]. Note that the CNDO values are the closest to the experimental data for butadiene and hexatriene, but these latest data have been used to fit the CNDO parameters. Furthermore, the results... 

Figure 3.9 Apparent value of the dissociation constant (K,) for a labeled inhibitor, I, as a function of the concentration of a second inhibitor, J when measured by equilibrium binding methods. The solid circles represent the behavior expected when compounds I and J bind in a mutually exclusive fashion with one another. The other symbols represent the behavior expected when compounds I and J bind in a nonexclusive, but antagonistic (i.e., noncompetitive, a > 1) fashion, to separate binding sites. The data for mutually exclusive binding were fit to the equation (apparent)K, = A, 1 + ([f ] A",) I and that for nonexclusive binding were fit to the equation (apparent)Kt = ( [J] + Kj / Kj + f[I]/y) ) for y values of 5 (closed triangles), 10 (open squares), 20 (closed squares), and 50 (open circles).

The preferred range for the thickness of the near-wall cell layer is y+ >30. However, this is difficult to achieve in packed tubes. The cells sizes are constrained by the need to fit in between the gaps and/or narrow spaces between particles, so they cannot be too large. This can result in the y+ values being too small for proper application of wall functions. The alternative to use small enough cells to resolve the boundary layer (y+ < 1) increases the computational... 

The best straight line fit is obtained when the sum of the squares of the individual y-axis value deviations (deviations between the plotted y values and the values on the proposed line) is at a minimum. 

The Savitzky-Golay algorithm could readily be adapted for polynomial interpolation. The computations are virtually identical to smoothing. In smoothing, a polynomial is fitted to a range of (x,y)-data pairs arranged around the x-value that needs to be smoothed. For polynomial smoothing, the polynomial is evaluated for a set number of data points around the desired x-value and the computed y-value at that x is the interpolated value. 

Fig. 3.14 Experimental Oeft) and best fit calculated H nmr line shapes of UOjlacacljMe SO in o-C H Clj. Temperature and best fit x values are shown at left and right sides of the figure. Reproduced with permission from Y. Ikeda, H. Tomiyasu and H. Fukutorai, Inorg. Chem. 23, 1356 (1984). (1984) American Chemical Society.

Residual Standard Deviation Another important approach that can be used to evaluate the applicability domain is the degree-of-fit method developed originally by Undberg et al. [40] and modified recently by Cho et al. [6]. According to the original method, the predicted y values are considered to be reliable if the following condihon is met ... 

A statistical algorithm, also known as linear regression analysis, for systems where Y (the random variable) is linearly dependent on another quantity X (the ordinary or controlled variable). The procedure allows one to fit a straight line through points xi, y0, X2,yi), x, ys),..., ( n,yn) where the values jCi are defined before the experiment and y values are obtained experimentally and are subject to random error. The best fit line through such a series of points is called a least squares fit , and the protocol provides measures of the reliability of the data and quality of the fit. 

Y value. The only relevant measure of how well the MLR model performs is provided by the Y variances. Residual Y variance is the variance of the Y residuals. It expresses how much variation remains in the observed response after the modeled part is removed. It is an overall measure of misfit, that is, the error made when fitted... 

The resultant calibration plot is shown in Figure 26. The model equation is, F = 5.759IX — 0.0002 where F, Yhat , is the predicted Y value for a given X from best fit regression line. A plot of the residuals will give an indication of... 

The optimum values of the parameters a, / , y are found by fitting measured values of Cp over a range of temperatures to equation (5.5). Thus, if we know the value of Cp at one temperature, we can evaluate it at another temperature, and thereby determine the effect of that incremental (or decremental) change in temperature, AT, upon Cp, given by ACp. Alternatively, we can use the properties of differentials given in equation (5.4) to evaluate the differential of Cp, dCp, in terms of the differential d T as ... 

If the values for both the solute and the solventvhave the same sign, Equation 11 predicts a linear dependence of x with the concentration of the solution. By a least square fit of this concentration dependence, one can readily obtain of the solute molecule. If the signs of the nonlinearities are opposite but both are real quantities, a concentration dependence study would yield a behavior where the value of the resultant x of the solution decreases and goes to zero at some concentration. In the case when of the solute is complex, Equation 11 yields the signal given by... 

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... 

Figure 6.6. Illustration of the PLS methodology to relate two co-ordinate systems (X and YJ via score vectors (t and u, respectively). The upper left co-ordinate system contains the measurements X and the upper right co-ordinate system contains the external information, Y. The points in the two co-ordinate systems represent the same set of subjects. By fitting a line in each co-ordinate system to the points and then increasing the correlation between the t-scores and the u-score (lower middle plot) by tilting both lines, the PLS solution is obtained. The Y values of a new subject inside the tolerance region in X can be predicted by following the path indicated by the dotted line.

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