Least-squares fitting, effective

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. ... 

There have also been attempts to describe the temporal aspects of perception from first principles, the model including the effects of adaptation and integration of perceived stimuli. The parameters in the specific analytical model derived were estimated using non-linear regression [14]. Another recent development is to describe each individual TI-curve,/j(r), i = 1, 2,..., n, as derived from a prototype curve, S t). Each individual Tl-curve can be obtained from the prototype curve by shrinking or stretching the (horizontal) time axis and the (vertical) intensity axis, i.e. fff) = a, 5(b, t). The least squares fit is found in an iterative procedure, alternately adapting the parameter sets (a, Zi, for 1=1,2,..., n and the shape of the prototype curve [15],... 

In turbulent flow, the edge effect due to the shape of the support rod is quite significant as shown in Fig. 6. The data obtained with a support rod of equal radius agree with the theoretical prediction of Eq. (52). The point of transition with this geometry occurs at Re = 40000. However, the use of a larger radius support rod arbitrarily introduces an outflowing radial stream at the equator. The radial stream reduces the stability of the boundary layer, and the transition from laminar to turbulent flow occurs earlier at Re = 15000. Thus, the turbulent mass transfer data with the larger radius support rod deviate considerably from the theoretical prediction of Eq. (52) a least square fit of the data results in a 0.092 Re0 67 dependence for... 

The algorithms developed in this chapter can model any situation, e.g. they can serve to demonstrate the effects of initial concentrations and rate constants in kinetics and of total concentration and equilibrium constants in equilibrium situations. Very importantly, these algorithms further form the core of non-linear least-squares fitting programs for the determination of rate or equilibrium constants, introduced and developed in Chapter 3, Model-Based Analyses. 

Historically, factorial designs were introduced by Sir R. A. Fisher to counter the then prevalent idea that if one were to discover the effect of a factor, all other factors must be held constant and only the factor of interest could be varied. Fisher showed that all factors of interest could be varied simultaneously, and the individual factor effects and their interactions could be estimated by proper mathematical treatment. The Yates algorithm and its variations are often used to obtain these estimates, but the use of least squares fitting of linear models gives essentially identical results. 

Four methods of performing this calculation have been proposed, the tracer property, linear programming, ordinary linear least squares fitting, and effective variance least squares fitting. 

The main advantage of the effective potential method consists in the relative simplicity of the calculations, conditioned by the comparatively small number of semi-empirical parameters, as well as the analytical form of the potential and wave functions such methods usually ensure fairly high accuracy of the calculated values of the energy levels and oscillator strengths. However, these methods, as a rule, can be successfully applied only for one- and two-valent atoms and ions. Therefore, the semi-empirical approach of least squares fitting is much more universal and powerful than model potential methods it combines naturally and easily the accounting for relativistic and correlation effects. 

The proton affinity results reflect changes in polarizability and inductive effects with the substituent group R. Comparison of proton affinities in the nitrile and primary amine series (Figure 10) reveals that the magnitude of these effects is linearly related in the two series but larger for the nitriles. A least-squares fit to the data is given by equation 26... 

If the analysis of a dynamic NMR spectrum is carried out by an iterative least-squares fitting method, the results are accompanied by estimates of the errors. These are proportional to the square root of the sum of the squares of the deviations of the theoretical spectrum from the experimental one, as well as to the sensitivity of the sum to changes in the value of the parameter considered within the region where the sum attains a minimum. These estimates constitute a measure of the effects, on the resulting parameter values, of random errors. They do not include any effects due to systematic errors such as those involved in the assumed values of certain parameters. Moreover, because of the nonlinearity of the least-squares fitting procedure employed, estimates of the errors have only an approximate statistical significance (Section IV.B.2 and reference 67). 

Primary kinetic hydrogen isotope effects in the reductions of triarylmethyl cations by formate in aqueous trifluoroacetic acid have been determined (Stewart and Toone, 1978). For 16 cations with pK + ranging from —2.25 (4-methoxy-4 -methyl) to — 7.63 (4,4 -dichloro), plots of log kH and log kD against pA"R+ show poor linear fits with the best straight lines (least squares) having slopes of 0.473 (r = 0.973) for the H-data and 0."491(r = 0.983) for the D-data. Both plots show downward curvature, with the effect being more emphatic for the H-data. Least squares fit to quadratics better... 

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