Least-squares fit

Constants A, and c are estimated by the least squares fitting procedure. 

Door R and Gangler D 1995 Multiple least-squares fitting for quantitative eleotron energy-loss... 

Figure B2.4.6. Results of an offset-saturation expermient for measuring the spin-spin relaxation time, T. In this experiment, the signal is irradiated at some offset from resonance until a steady state is achieved. The partially saturated z magnetization is then measured with a kH pulse. This figure shows a plot of the z magnetization as a fiinction of the offset of the saturating field from resonance. Circles represent measured data the line is a non-linear least-squares fit. The signal is nonnal when the saturation is far away, and dips to a minimum on resonance. The width of this dip gives T, independent of magnetic field inliomogeneity.

Figure B2.4.8. Relaxation of two of tlie exchanging methyl groups in the TEMPO derivative in figure B2.4.7. The dotted lines show the relaxation of the two methyl signals after a non-selective inversion pulse (a typical experunent). The heavy solid line shows the recovery after the selective inversion of one of the methyl signals. The inverted signal (circles) recovers more quickly, under the combined influence of relaxation and exchange with the non-inverted peak. The signal that was not inverted (squares) shows a characteristic transient. The lines represent a non-linear least-squares fit to the data.

STO-3G bases [4T] were employed some years ago, but have recently become less popular. These bases are constructed by least-squares fitting GTOs to STOs which have been optimized for various electronic states of the atom. Wlien tlnee GTOs are employed to fit each STO, a STO-3G basis is fonned. 

Once the least-squares fits to Slater functions with orbital exponents e = 1.0 are available, fits to Slater function s with oth er orbital expon cn ts can be obtained by siin ply m ii Itiplyin g th e cc s in th e above three equations by It remains to be determined what Slater orbital exponents to use in electronic structure calculation s. The two possibilities may be to use the "best atom" exponents (e = 1. f) for II. for exam pie) or to opiim i/e exponents in each calculation. The "best atom expon en ts m igh t be a rather poor ch oicc for mo lecular en viron men ts, and optirn i/.at ion of non linear exponents is not practical for large molecules, where the dimension of the space to be searched is very large.. 4 com prom isc is to use a set of standard exponents where the average values of expon en ts are optirn i/ed for a set of sin all rn olecules, fh e recom -mended STO-3G exponents are... 

The coefficients and the exponents are found by least-squares fitting, in which the overlap between the Slater type function and the Gaussian expansion is maximised. Thus, for the Is Slater type orbital we seek to maximise the following integral ... 

Expand the three detemiinants D, Dt, and for the least squares fit to a linear function not passing through the origin so as to obtain explicit algebraic expressions for b and m, the y-intercept and the slope of the best straight line representing the experimental data. 

Potentiometric titration curves are used to determine the molecular weight and fQ or for weak acid or weak base analytes. The analysis is accomplished using a nonlinear least squares fit to the potentiometric curve. The appropriate master equation can be provided, or its derivation can be left as a challenge. 

It is usually advisable to plot the observed pairs of y versus r, to support the linearity assumption and to detect potential outhers. Suspected outliers can be omitted from the least-squares Tit and then subsequently tested on the basis of the least-squares fit. 

Figure 4 Sample spatial restraint m Modeller. A restraint on a given C -C , distance, d, is expressed as a conditional probability density function that depends on two other equivalent distances (d = 17.0 and d" = 23.5) p(dld, d"). The restraint (continuous line) is obtained by least-squares fitting a sum of two Gaussian functions to the histogram, which in turn is derived from many triple alignments of protein structures. In practice, more complicated restraints are used that depend on additional information such as similarity between the proteins, solvent accessibility, and distance from a gap m the alignment.

Data Source. The correlation constants were determined from a least-squares fit of data from the literature.-- - In most cases, average deviations between calculated and reported data were less than 0.6kjoules/g-mol. 

Figure 7 shows Eq for GaAs and Ga 82 0.18 function of temperature T to about 900 K. Additional measurements on samples having differing A1 contents would generate a family of curves. The solid line is a least-squares fit to a semi-empirical relation that describes the temperature variation of semiconductor energy gaps ... 

Figure 7 Temperature dependence of of GaAs (circles) and Gag AIg sAs (squares). The solid lines are least-squares fits to Equation (2).

The absolute precision of ERS therefore depends on that of da/dfl (Ej, (t>). Unfortunately, some disagreement prevails among measurements of the recoil cross section. One recent determination is shown in Figure 4a for (t> = 30° and 25°. The convergence of these data with the Rutherford cross section near 1 MeV lends support to their validity. The solid lines are least squares fits to the polynomial form used by Tirira et al.. For (t> = 30°, the expression reads ... 

Fig. 2.2. Compilation by Seah and Dench [2.3] of measurements of inelastic mean free path as a function of electron kinetic energy. The solid line is a least-squares fit.

The structure refinement program for disordered carbons, which was recently developed by Shi et al [14,15] is ideally suited to studies of the powder diffraction patterns of graphitic carbons. By performing a least squares fit between the measured diffraction pattern and a theoretical calculation, parameters of the model structure are optimized. For graphitic carbon, the structure is well described by the two-layer model which was carefully described in section 2.1.3. 

When evaluating gas concentrations in practical applications, a reterence spectrum is least squares fitted to the received absorption spectrum. This im proves the system accuracy, since the spectral fingerprint over the whole scanning range contributes to the result.- ... 

To extract the agglomeration kernels from PSD data, the inverse problem mentioned above has to be solved. The population balance is therefore solved for different values of the agglomeration kernel, the results are compared with the experimental distributions and the sums of non-linear least squares are calculated. The calculated distribution with the minimum sum of least squares fits the experimental distribution best. 

The size-dependent agglomeration kernels suggested by both Smoluchowski and Thompson fit the experimental data very well. For the case of a size-independent agglomeration kernel and the estimation without disruption (only nucleation, growth and agglomeration), the least square fits substantially deviate from the experimental data (not shown). For this reason, further investigations are carried out with the theoretically based size-dependent kernel suggested by Smoluchowski, which fitted the data best ... 

In order to determine the mesomixing time, a least square fit of the 300 ml eontinuous ealeium oxalate (CaOx) preeipitation results for the number mean size and nueleation rate was performed. From these ealeulations, the faetor A in equation 8.15 was obtained as 17.7. Using the kinetie parameters determined from the laboratory-seale eontinuous experiments (Zauner, 1999), the large-seale experiments were simulated with the SFM and eompared with the experimental findings. 

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. ... 

The high-level correction (item 7 above) is determined by a least squares fit for 55 molecules rather than just hydrogen atom and hydrogen molecule. 

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