Best fitting procedure

P-outine chemical analysis. This implies analysis of many samples, and use of calibration curves is an economic necessity. In general, the two-standard method, with standards bracketing each sample analyzed, is economical for the analysis of up to about 10 samples. Conventional least squares curve of best fit procedures are economical for analysis of 10 to 500 samples. The procedures described here are cost effective for the analysis of 500 samples or more. 

The least-squares curve-of-best-fit procedure implicitly assumes the same variance (standard deviation) at all concentrations. This assumption is rarely correct. Figure 3a shows hypothetical replicate standard analysis data with constant variance. This pattern is almost never seen in routine chemical analyses. Figure 3b shows a much more realistic pattern in which the variance increases with concentration. 

These factors are used in the equations given in Table I. The computation requires only that the variance ratios be accurately known. The absolute precision of the method may change from day to day without affecting the validity of either the least-squares curve-of-best fit procedure or the confidence band calculations. (It is not practical to regularly monitor local variances, and errors may develop in variance ratios. Eowever, the error due to incorrect ratios will almost always be much less than the error due to assuming constant variance. Even guessed values of, say, S a concentration are likely to yield more precise data.)... 

The NMRD profiles of water solution of Ti(H20)g" have been shown in Section I.C.7 and have been already discussed. We only add here that the best fit procedures provide a constant of contact interaction of 4.5 MHz (61), and a distance of the twelve water protons from the metal ion of 2.62 A. If a 10% outer-sphere contribution is subtracted from the data, the distance increases to 2.67 A, which is a reasonably good value. The increase at high fields in the i 2 values cannot in this case be ascribed to the non-dispersive term present in the contact relaxation equation, as in other cases, because longitudinal measurements do not indicate field dependence in the electron relaxation time. Therefore they were related to chemical exchange contributions (see Eq. (3) of Chapter 2) and indicate values for tm equal to 4.2 X 10 s and 1.2 X 10 s at 293 and 308 K, respectively. 

The number of inner sphere water molecules has an obvious effect on the relaxivity. Doubling the value of q will imply a doubling of and then a substantial increase in the relaxation efficacy of the Gd(III) complex, as shown in Pig. 7. Knowing q is then crucial for a proper analysis of the NMRD profile, but this parameter cannot be extracted from a best fit procedure as, at best, only the term q/r can be obtained. Prom a qualitative point of view, for relatively small Gd(III) complexes and in the absence of... 

The water proton NMRD profile of Cu(II) aqua ion at 298 K [108] (Fig. 5.36) is in excellent accordance with what expected from the dipole-dipole relaxation theory, as described by the Solomon equation (Eq. (3.16)). The best fitting procedure applied to a configuration of 12 water protons bound to the metal ion provides a distance between water protons and the paramagnetic center equal to 2.7 A, and a correlation time equal to 2.6 x 10 11 s, which defines the position of the cos dispersion. The correlation time is determined by rotation as expected from the Stokes-Einstein equation (Eq. (3.8)). The electron relaxation time is in fact expected to be one order of magnitude longer (see Table 5.6). This also ensures... 

The H NMRD profiles of Mn(OH2)g+ in water solution show two dispersions (Fig. 5.43). The first (at ca. 0.05 MHz, at 298 K) is attributed to the contact relaxation and the second (at ca. 7 MHz, at 298 K) to the dipolar relaxation. From the best fit procedure, the electron relaxation time, given by rso = 3.5 x 10 9 and r = 5.3 x 10 12 s, is consistent with the position of the first dispersion, the rotational correlation time xr = 3.2 x 10 11 s is consistent with the position of the second dispersion and is in accordance with the value expected for hexaaquametal(II) complexes, the water proton-metal center distance is 2.7 A and the constant of contact interaction is 0.65 MHz (see Table 5.6). The impressive increase of / 2 at high fields is due to the field dependence of the electron relaxation time and to the presence of a non-dispersive zs term in the equation for contact relaxation (see Section 3.7.2). If it were not for the finite residence time, xm, of the water molecules in the coordination sphere, the increase in Ri could continue as long as the electron relaxation time increases. 

Equation (19) contains four unknown parameters, K, Ng, k, and Cl,o, whose determination may be carried out by a best fitting procedure. 

The slopes and linear coefficients of lines allow to determine the Ng and values corresponding to these runs these values are almost the same as those determined by means of Equation (19) (error percentage, +4%). Eigure 8 also reports the adsorption results obtained in the absence of irradiation, that is, in the dark. A least-squares best fitting procedure allows to determine the values (R > 0.99) of the Langmuir equilibrium constant and the maximum adsorption capacity in the absence of irradiation, that is, fCL = 350 and Ns = 4.57 x 10 mol/g of catalyst. 

Equation (A24) contains five unknown parameters, Kj, Ng, k, /, and Cl.o/ whose determination may be carried out by a best fitting procedure. As done in the cases previously described, integration of Equation (A24) must be performed with the condition that at f = 0 the substrate concentration in the liquid phase is that in equilibrium with the initial photoadsorbed amoimt, Cl.o this initial concentration is unknown, but it may be determined by the regression analysis carried out with the experimental data obtained after the start of irradiation. 

It is worth mentioning straightaway that the e values found by best fit procedures to spectral and magnetic data do indeed tend to support this qualitative view. It has been found that for ammonia and tertiary amines one can consistently associate vanishing e values for these ligands in various complexes. In M(pyridine)4(NCS)2 M=Co(II), Fe(II), the e parameters defined in the planes of the pyridine groups are zero and take small (positive) values (ca. 100 cm ) perpendicular to these groups. By contrast, the... 

From Eq. (2) the thickness of the transition layer, electron density and area per molecule of coating material can be estimated using the best-fit procedure. 

Parameters obtained by best-fit procedure between observed and calculated decay curves of fluorescence of Trp-29 in erabutoxin b. 

In order to check the validity of the model described in Section 2, integration of Eqs. (6, 7) was carried out, considering the experimental conditions of the three dynamic runs in the boundary conditions (10). As already indicated, once the boundary conditions and the equilibrium isotherms are assigned, the only parameters of the model are the mass transfer coefficients kg and A simple best fit procedure led to a very satisfactory agreement between model and experimental results. It is important to observe that all three breakthrough curves were obtained using the same value for the external mass transfer coefficient (namely, e=5.M0 m/s) this is in agreement with the physical nature of kg, which only depends on fluid dynamic conditions. On the other hand, the internal mass transfer coefficient ki varied with the total normality of the solution used. In particular, for A =0.89 eqW it was, =1.4-10 eq/m for A7=22.6eq/m it was eq/m for A =44.4eq/m it was... 

Some examples of the fitting of equations (13)- 15) to the experimental data of Figs. 1-3 are demonstrated in Figs. 4-6. The points are experimental data, and the curves are calculated by equations (13)-(15) using a computer best fit procedure. Satisfactorily good agreement between the experimental kinetic data and the calculations have been obtained for different compositions of NO-CO mixtures. 

At pH 7 (N-16-alkyl-N,N-dimethylammonio)-acetic acid bromide is transformed into a betaine (BHBCia). To determine optimum parameters of Eqs. (2.84)-(2.88), a best fit procedure was used, the details of which are discussed in Chapter 3. In brief, for each set of coi, C02 and a, the values of b, in each of the m experimental points flj = Ili(Ci), i = 1,2,. .., m are caleulated and then the mean value of b averaged weighted with the relative surface pressure interval... 

To And the variations in film density and elastic constants, the measured responses (Table 4.1) would be substituted in Equation 4.39. This yields a set of four simultaneous equations with the unknowns being Ap/p, AC /C, and AC44/C44. Solving the above equations using best-fit procedure, the results obtained within error limits of 20% are... 

Another way to estimate the number of free parameters is to start from the observation that deviations from the original theoretical potential by the best-fit potential occur for r < 1.5 fm (cf. Fig. 5). This range is equivalent to about 6m and more. Thus, the Yukawas with six and more pion masses are affected by the best-fit procedure. Since there are 12 Yukawas, there are seven larger or equal 6 this applies to the 14 components of the potential separately, yielding 7 x 14 = 98... 

Figs. 17.a and b and 18.a and b show a comparison of the density dependence upon cooling rate for the iPP grades studied, whereas Table 2 reports the crystallization kinetics model parameters calculated by a best fitting procedure not only on the basis of the final monoclinic and mesomorphic content of the quenched samples, taken from the deconvolution of the WAXD patterns, but also accoimting for results which provide the time and the temperature at the maxima of the crystallization rate (isothermal tests and DSC measurements) respectively. For this purpose a mulfiobjecfive optimization code was adopted. 

Urayama et al. [119-121] tested the diffused constraint model using both uniaxial compression and equibiaxial elongation data for end-linked PDMS networks in which trapped entanglements were dominant in number relative to chemical crosslinks. The parameter k was used as an empirical fitting parameter, and the best-fit procedure yielded k = 2.9. The structural parameters (v, jj., /)... 

Fig. 19.1. Temperature dependencies of the primary nucleation rate (I) (A) and the linear crystal growth rate (G) (Q) for poly(ethylene succinate) (PEISU) [14] with a molecular weight (M) of 8,770. The solid and broken lines are results from the best fitting procedure for G based on Eq. (19.2) and for I based on Exj. (19.11) by the Arrhenius and the WLF expressions of the molecular transport term, respectively...

The ratio of Tg and T°j is well known to be 2/3 for the most of polymers [44 6]. That is, To comes near to (l/2)T°j. This result is much used for the best fitting procedure for crystal growth data. Anyhow, the activation energy for molecular transport term can be expressed by either or Arrhenius... 

Auslander proposed an equation for best-fitting procedures of viscosity data of the type ... 

The shear stresses over the flow boundaries can be rigorously derived as an integral part of the solution of the flow field only in laminar flows. The need for closure laws arise already in single-phase, steady turbulent flows. The closure problem is resolved by resorting to semi-empirical models, which relate the characteristics of the turbulent flow field to the local mean velocity profile. These models are confronted with experiments, and the model parameters are determined from best fit procedure. For instance, the parameters of the well-known Blasius relations for the wall shear stresses in turbulent flows through conduits are obtained from correlating experimental data of pressure drop. Once established, these closure laws permit formal solution to the problem to be found without any additional information. 

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