Least squares method sample calculation

The ferrite samples were analysed by X-ray diffractometer (XRD) with CuKa radiation (Rigaku D/max-RB), X=0.154nm, the scan range (20) is 20-80°. The lattice constant was calculated by extrapolating the values of ao vs the Nelson-Riley function, cos"0/sin01 cos O/O. to zero using the least squares method. 

Table 19.2 files the integrated fractions of lines I, II and III of the CH triads obtained for the crystalline components in different PVA samples by the lineshape analyses described above. Here, the observed integrated fractions were corrected by considering the difference in Tic for the respective CH lines. In this table, the calculated fractions of lines I, II and III, which were obtained to fit the observed fractions by the least-square method, are also shown together with and Fa values. The calculated fractions agree... 

The silanol numbers obtained for the 100 samples are plotted versus specific surface area in Figure 16 (66). The average silanol number of the 231 independent measurements is 4.9 OH groups per square nanometer. Calculations by the least-squares method gave 4.6 OH groups per square nanometer as an average silanol number. 

The quantity V,(C, - Co) was then plotted versus time t, a line fit to the data by least-square methods, and the amount of material entering the water per unit time determined as the slope. This number was divided by the surface area sampled by the flux box to calculate the flux of material out of the sediment per unit area per time. The error is taken as the standard deviation of the slope and is about 10%. The actual uncertainty must be higher, as will become clear later (Nixon et al., 1980). The last sample points (—30-50 hr) in both the summer and fall sets of flux cores were not used in the calculations. Inclusion of these points often, but not always, resulted in calculation of a higher (or Mn, see Part II) flux. (In some cases a lower flux is calculated.) This suggested that lower O2 levels were beginning to cause changes in pore-water profiles and perhaps reaction distribution in the sediment. For consistency in calculation none of these later sample points were used. These points were arbitrarily included or discarded in previous calculations (Aller, 1977) the present... 

A//sub may be calculated in kJ. Rates of mass-loss of powdered 50-100-mg samples, contained in a platinum boat, were recorded at a series of five or six temperatures over a 20-3CP range. By choosing the temperature to give low rates of mass-loss and low (<2%) overall Joss, good straight-line plots were obtained from which the slopes, as calculated by the least-squares method, were reproducible to about 5%. Enthalpies of sublimation obtained by this method are shown in Table 4.9. There was good agreement between the... 

The amplitudes of the histogram of the distribution function are calculated by a non-negative least square method. This procedure is known as the exponential sampling method and is applicable to both monomodal and bimodal distributions. However, in view of the limitations of the Laplace inversion, it is difficult to resolve bimodal distributions with a ratio between the two particle species below 2. 

The values of the silanol number, aon, of 100 silica samples, with a completely hydroxylated surface, were established [3-5]. The average silanol number (arithmetical mean) was found to be aoH,av = 4.9 OH/nm. Calculations by the least-squares method yielded aon,av = 4.6 OH/nm. These values are in agreement with those reported by De Boer and Vleeskens [11] as well as with results reported by other researchers. To sum up, the magnitude of the silanol number, which is independent of the origin and structural characteristics of amorphous silicas is considered to be a physicochemical constant. The results fully confirmed the idea predicted earlier by Kiselev and co-workers [13,14] on the constancy of the silanol number for a completely hydroxylated silica surface. This constant now has a numerical value cioH,av = 4.6 0.5 OH/nm [3-5] and is known in literature as the Kiselev-Zhuravlev constant. 

Lattice parameter of solid powder sample is not only affected by its element constitution, but also the outside conditions such as temperature, and pressure. At the same time, the change of crystal lattice parameter of series samples is often very small, which can only be found in a more precise determination. The key issue is to accurately measure the 29 value of diffraction line, and calculate the lattice parameter by 29 value. In order to reduce systematic errors in experiments, one should strictly control the temperature and facture of flat plate sample, and use single-color, step-scan method of data collection. At the same time, when measm-ing the crystal lattice parameter of the cubic crystal system of Fes04, systematic errors can be eliminated by extrapolation, Cohen least square methods. 

When calculating the lattice parameter, the diffraction peak data Kai and Ka2 of the sample should be separated first, and then peak values can be calculated by parabola function fitting method. As the accuracy of the determination of lattice parameters is determined by the precision of the angle measured, so in order to enhance the accuracy when reading value of angles and to avoid errors caused by human factors, as shown in Fig. 7.26, in the vicinity of peak 20m, 10 to 15 data of diffraction intensity are collected, with parabola equation fitting near the peak intensity distribution curve by the least square method to obtain the best 20 peak value. With known lattice parameters of the silica powder and sample together, the test results are obtained under the same condition to establish curve correction. 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

The optimal number of components from the prediction point of view can be determined by cross-validation (10). This method compares the predictive power of several models and chooses the optimal one. In our case, the models differ in the number of components. The predictive power is calculated by a leave-one-out technique, so that each sample gets predicted once from a model in the calculation of which it did not participate. This technique can also be used to determine the number of underlying factors in the predictor matrix, although if the factors are highly correlated, their number will be underestimated. In contrast to the least squares solution, PLS can estimate the regression coefficients also for underdetermined systems. In this case, it introduces some bias in trade for the (infinite) variance of the least squares solution. 

Selected entries from Methods in Enzymology [vol, page(s)] Association constant determination, 259, 444-445 buoyant mass determination, 259, 432-433, 438, 441, 443, 444 cell handling, 259, 436-437 centerpiece selection, 259, 433-434, 436 centrifuge operation, 259, 437-438 concentration distribution, 259, 431 equilibration time, estimation, 259, 438-439 molecular weight calculation, 259, 431-432, 444 nonlinear least-squares analysis of primary data, 259, 449-451 oligomerization state of proteins [determination, 259, 439-441, 443 heterogeneous association, 259, 447-448 reversibility of association, 259, 445-447] optical systems, 259, 434-435 protein denaturants, 259, 439-440 retroviral protease, analysis, 241, 123-124 sample preparation, 259, 435-436 second virial coefficient [determination, 259, 443, 448-449 nonideality contribution, 259, 448-449] sensitivity, 259, 427 stoichiometry of reaction, determination, 259, 444-445 terms and symbols, 259, 429-431 thermodynamic parameter determination, 259, 427, 443-444, 449-451. 

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