Best “least-square fit

Fig. 14a c. Anisotropy of n.m.r. second moment in P 4GT (specimen B) and the best least squares fit of the anisotropy predicted from (a) Hall-Pass model 6, 18% crystallinity (b) Hall-Pass model 7, 40% crystallinity (c) Yokouchi model, 37% crystallinity. Reproduced from Polymer by permission of the publishers Butterworth Co (Publishers) Ltd. (C)... 

Figure 4. The number of deuterium molecules found in products on Vx clusters produced while attempting to saturate the reaction. The solid lines are plots of D2 V ratios of 1 and 0.5, including all vanadium atoms. The dashed lines are corrected, assuming globular shaped clusters. The best least-squares fit to the data, D2 V = 0.68, is also plotted as a dashed line.

Ihrig and Smith extended their study by running a regression analysis including reaction field terms, dispersion terms and various combinations of the solvent refractive index and dielectric constant. The best least squares fit between VF F and solvent parameters was found with a linear function of the reaction field term and the dispersion term. The reaction field term was found to be approximately three times as important as the dispersion term and the coefficients of the terms were opposite in sign. 

Table 2. Structural Parameters Obtained from Best Least-Squares Fits of the EXAFS for the 20 wt % Pt/C Catalyst Electrode for Various Potentials,...

A structural model is required, and the parameters in the model are adjusted programmatically, by computer, to give the best least squares fit of the whole calculated powder diffraction pattern to the whole observed pattern. Many things besides crystal structure can be and often need be considered in the calculations, e.g., instrumental profiles, preferred orientation, and contributions from background and from other phases. It is possible to refine simultaneously the structures, or at least amounts present, of two or more phases present (7J. 

FIGURE 2 SE threshold as a function of temperature for a GaN thin film grown on SiC. The solid line represents the best least-squares fit to the empirical form I T) = I, exp (T/T0), which yielded T0 170 K. 

It should be noticed also that minimizing the energy is not the only minimizing problem that is of interest in quantum chemistry, and that problems of minimizing electron repulsion integrals, differences between functions, best least-squares fitting, and so on, frequently arise at intermediate stages in established computational schemes. 

Figure 31. Representative data showing OH formation following 236 nm CO2-HI excitation. The ordinates in (a) and (b) are the Qi,(l) and Qi,(6) LIF signals, respectively, while the abscissa is the delay time. The dashed curve is the response function of the laser system. The solid curves are the calculated best fits, assuming a two-parameter description for the time dependence of OH formation the best least-squares fits yielded (a) t, = 0.9 ps and Tj = 1.9 ps, and (b) x, = 0.7 ps and Xj = 1.1 ps (see Table 4). The points at the top are the residuals between the experimental points and the smooth fit. From Ref. 43 with permission of the Journal of Chemical Physics.

Figure 5. A typical fit to the aggregated variance versus the aggregated mean for BRV and HRV time series obtained by West et al. [14]. The points are calculated from the data and the solid curve is the best least-square fit to the data. The upper curve is the fit to the BRV data (slope = 0.86), and the lower curve is the best fit to the HRV data (slope = 0.80). It is evident from these two graphs that the allometric relation given by Eq. (9) does indeed fit both data sets extremely well and lies well within the regular and random boundaries, indicated by the dashed curves.

Figure 15. The average multifractal spectrum for middle cerebral blood flow time series is depicted by flh). (a) The spectrum is the average of 10 time series measurements from five healthy subjects (filled circles). The solid curve is the best least-squares fit of the parameters to the predicted spectrum using Eq. (157). (b) The spectrum is the average of 14 time series measurements of eight migraineurs (filled circles). The solid curve is the best least-squares fit to the predicted spectrum using Eq. (157). (Taken from [90].)...

In Table 3 the values of selected parameters of 2-chloropropane for the complete rt calculation and the p-Kr (rs) calculation from Table 2 are compared with closely related calculations performed after omission of the experimental data for the species with13 C substitution at the central carbon atom. The purpose of this calculation is to demonstrate how different the structure reported would have been if data for only seven isotopic species had been available. For the rs calculation with seven species the coordinates of the central carbon atom were determined by adjusting them to give a best least squares fit to the principal axis conditions. For the p-Kr calculation with seven species the moments of inertia of the parent species were included in the fit along with the six remaining sets of differences in moments of inertia. The conclusion we draw... 

The first program (obtained from Dr. Leo J. Lynch, Division of Textile Physics Wool Research Labs., 338 Blaxland Rd., Rydel Sydney, NSW Australia.) uses the model to predict oq and B at min as estimates for the second program. The second program (obtained from Dr. Henry A. Resing, Dept, of Chemistry, Code 6173, Naval Research Labs., Washington, DC 20390) uses numerical integration techniques to calculate the parameters for the best least squares fit [see Figure 2 solid lines for Ti and T2 curves... 

Davidson suggested that the wavefunction be projected onto a set of orbitals that have intuitive significance. These orbitals are a minimum set of atomic orbitals that provide the best least-squares fit of the first-order reduced-density matrix. Roby expanded on this idea by projecting onto the wavefunction of the isolated atom. One then uses the general Mulliken idea of counting the number of electrons in each of these projected orbitals that reside on a given atom to obtain the gross atomic population. 

An improved method was developed by Chirlian and Francl and called CHELP (CHarges from ELectrostatic Potentials). Their method, which uses a Lagrangian multiplier method for fitting the atomic charges, is fast and noniterative and avoids the initial guess required in the standard least-squares methods. In this approach, the best least-squares fit is obtained by minimizing y ... 

Figure 5.25. Change in magnetization. Mj(t)-Mj(oo), of the 6-H resonance offree trimethoprim as a function of the time, X,for which the bound resonance was irradiated. The curve is the best least-squares fit to the data, calculated by using the parameters in the table.

Fig. 5.8.2. Spectra of light scattered from the protoplasm of Nitella. The horizontal axis is frequency in Hz and the vertical axis is relative intensity. Spectrum (a) was taken at a scattering angle of 19.5 deg. Spectra (b) and (c) were taken at a scattering angle of 36.1 deg. Spectrum (c) was taken from the same point on the cell as spectrum (b), immediately after addition of parachloromercuribenzoate, a streaming inhibitor. (Each of these spectra was collected in about 30 sec. The points are the output of the spectrum analyzer, and the dark lines have been drawn merely to make the data more perspicuous and to emphasize the reproducible features of the data.) Part (d) is a plot of the magnitude of the Doppler shift from a fixed point on a Nitella cell as a function of the sine of the scattering angle 9. The line is the best least-square fit to the points. The predicted linear dependence is verified, and the deviations from the line provide an estimate of the experimental precision. (From Mustacich and Ware, 1974.)...

Iteration for Coexisting Densities. Orthobaric densities near the critical point generally cannot be obtained accurately from isochoric PpT data by extrapolation to the vapor-pressure curve because the isochore curvatures become extremely large near the critical point. The present, nonanalytic equation of state, however, can be used to estimate these densities by a simple, iterative procedure. Assume that nonlinear parameters in the equation of state have been estimated in preliminary work. For data along a given experimental isochore (density), it is necessary merely to find the coexistence temperature, Ta(p), by trial (iteration) for a best, least-squares fit of these data. 

Once a general equation of state involving Wq versus P and T such as (15.50) is available, it can be used to calculate a statistically smoothed solvus. This is the solvus that provides the best least-squares fit to the experimental data. With caution, it can also be used to extrapolate beyond the T and P range of the experimental data (this is safer if the excess properties such as Wh and Wv are derived directly from real data and not calculated from Wg or otherwise estimated). First, an equation for the total free energy of the system as a function of T, P, and concentration is derived using (15.50) for and the relation... 

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