Least squares extrapolations

Figure 3. Computer plot obtained by isochronous interpolation of the experimental light-scattering data (O) of HEC during endocellulase attack. The Langrangian interpolation functions are given by the horizontal curves ana the isochronous interpolated Kc/Re values by A. The quadratic least squares extrapolations to zero angle ( ) are given by the...

Fig. 6.1-1 The pKa- values calculated from the data (open circles) and the straight line drawn through them by least-squares, extrapolated (thin colored line) to yield the infinite dilution value.

Figure 5.9. Least-squares extrapolation or peak temperatures to zero heating rate (64j.

Examination of Table 6-1 reveals how the weighting treatment takes into account the reliability of the data. The intermediate point, which has the poorest precision, is severely discounted in the least-squares fit. The most interesting features of Table 6-2 are the large uncertainties in the estimates of A and E. These would be reduced if more data points covering a wider temperature range were available nevertheless it is common to find the uncertainty in to be comparable to RT. The uncertainty of A is a consequence of the extrapolation to 1/7" = 0, which, in effect, is how In A is determined. In this example, the data cover the range 0.003 23 to 0.003 41 in 1/r, and the extrapolation is from 0.003 23 to zero thus about 95% of the line constitutes an extrapolation over unstudied tempertures. Estimates of A and E are correlated, as are their uncertainties. ... 

Figure 13-8. Typical results for interna, plroiocniission measurements on an ITO/ OPPV/Ca diode at various applied voltages. Tire lines are least-square tils and their extrapolation yields the barrier height for that applied voltage. Inset shows the experimental setup. Reproduced with permission from IIIV1-...

Figure 13-9. Timc-intcgralcd light output versus pulse length at various applied voltages. The lines arc least-square fils and their extrapolation yields the lime delay t,. Inset shows the liming between the application of a voltage pulse and the observed eleclroluniineseencc. Reproduced with permission from I22. Copyright 1998 by the American Physical Society.

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

A least squares linear regression has been applied to the data pertaining to the p-phase, yielding the values of 1.745 and 0.005166 for the intercept log B and slope Sp, respectively. Using these results, we can compute the extrapolated plasma concentrations between 0 and 20 minutes. From the latter, we subtract the observed concentrations C which yields the concentrations of the a-phase C ... 

These data are also shown in the semilogarithmic plot in Fig. 39.13a, which clearly shows two distinct phases. A straight line has been fitted by least-squares regression through the data starting from the observation at 90 minutes down to the last one. This yields the values of 1.086 and -0.001380 for the intercept log and slope ip, respectively. From these results we have computed the extrapolated p-phase values between 2 and 60 minutes. These have been subtracted from the experimental Cp values in order to yield the a-phase concentrations C ... 

Fig. 39.15. Area under a plasma concentration curve AUC as the sum of a truncated and an extrapolated part. The former is obtained by numerical integration (e.g. trapezium rule) between times 0 and T, the latter is computed from the parameters of a least squares fit to the exponentially decaying part of the curve (P-phase).

Conversion of experimental dose/response data into a form suitable for extrapolation of human risk using least squares or, more usually, maximum likelihood curve fits. 

The theta (0) conditions for the homopolymers and the random copolymers were determined in binary mixtures of CCl and CyHw at 25°. The cloud-point titration technique of Elias (5) as moaified by Cornet and van Ballegooijen (6) was employed. The volume fraction of non-solvent at the cloud-point was plotted against the polymer concentration on a semilogarithmic basis and extrapolation to C2 = 1 made by least squares analysis of the straight line plot. Use of concentration rather than polymer volume fraction, as is required theoretically (6, 7 ), produces little error of the extrapolated value since the polymers have densities close to unity. 

Best fit plot of data from Table 2.7 obtained by least squares regression analysis. (Important note This graph implies a straight line relationship down to zero concentration. It is, however, unsafe to use the extrapolated portion as there are no experimental data for this part of the curve). 

The best fit to the Arrhenius plot can be found by the least squares method (applied to In t or log t) and extrapolated to find the time to the threshold value (tu) at a temperature of interest (Tu). To obtain an estimate of the maximum temperature of use, extrapolate the line to a specified reaction rate or time to reach a threshold value. 20,000 or 100,000 hours duration and 50% change as the threshold value are commonly used for establishing a general maximum temperature of use. 

The PNO extrapolations in Fig. 4.8 and Table 4.6 require localization of the occupied SCF orbitals to ensure size-consistency. In order to preserve this size-consistency for the CBS PNO extrapolations, we have restricted these (Zmax + f)-3 extrapolations to a linear form, Eq. (6.2). The new double extrapolation employs this linear extrapolation of pairs of CBS2/cc-pVnZ calculations and thus is rigorously size-consistent. Note that the nonlinear N-parameter (Zmax + a)-" extrapolations using least-squares fits to more than N cc-pVnZ energies are not size-consistent [53,55],... 

Then the extrapolated value from Figure 17.9 is equal to In a. By a least-squares fit of the most dilute points, which are essentially linear, we obtain a value of 7.3983 + 0.0016. 

A section has been added to Chapter 1 on the distinction between analytic vs. enumerative studies. A section on mixture designs has been added to Chapter 9. A new chapter on the application of linear models and matrix least squares to observational data has been added (Chapter 10). Chapter 13 attempts to give a geometric feel to concepts such as uncertainty, information, orthogonality, rotatability, extrapolation, and rigidity of the design. Finally, Chapter 14 expands on some aspects of factorial-based designs. 

Fig. 18 Top CD spectra. Oligomer 15 (solid line) oligomer 15 in the presence of 100 equiv of (-)-a-pinene (dotted line) and 100 equiv of (+)-a-pinene (dashed line) in a mixed solvent of 40% water in acetonitrile (by volume) at 295 K. [15]=4.2 pmol. Bottom Plot of IniCn for 15 against the solvent composition. The solid line is the least-squares linear fit (correlation coefficient=0.9987), and the dotted line is the extrapolation to 100% water. Error bars are from the nonlinear fitting of the data to...

The procedure starts with dividing the sample into n sub-samples. We spike n-1 sub-samples with the analyte in equidistant steps and measure all n sub-samples. We use least-square regression to calculate the regression line and extrapolate to the intersection, , "... 

Technique 7. In Technique 7, the x-t data are digitized into 70 discrete points. A linear fit is made to three adjacent x-t points, and the slope is taken as the velocity at the midpoint of the line. Then one x-t end point is dropped, a new one is added on the other end, a new linear fit is made, and the velocity is found. This running linear least squares process is repeated until all 70 x-t points have been used. The u-x data are then extrapolated to zero thickness (x = 0) to find the initial shock velocity UsQ. All other analysis is done as in Technique 1. ... 

The absorbance at the band maximum for solutions of lithium, sodium, and barium in ND8 is a linear function of the concentration, and extrapolation of the linear function to zero concentration predicts zero absorbance (Figure 5 and Table II). The molar extinction coefficients as calculated by the least squares method for solutions of lithium, sodium, and barium in ND8 at —70° C. are given in Table II. If it is assumed that barium loses two electrons upon solvation, the molar extinction coefficients of lithium, sodium, and barium solutions are the same to within the estimated error of measurement (4%). The residual absorbance, as calculated from the least squares analysis, in each case is... 

Figure 1. Computer plot of the light-scattering analysis of HEC before enaocellulase hydrolysis. (O) Experimental Kc/R values, (M) extrapolated values (by quadratic least squares analysis). The inverse of the Kc/Re intercept, Mw, equals 1,285,000.

The interpolated values of Kc/Re at constant time are calculated or read from the graph, and then the extrapolation of the isochronous Kc/Re values to zero angle is made in the same manner as mentioned for the Zimm plot, by using quadratic least squares regression analysis. 

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