Least squares 506 INDEX

Multichannel time-resolved spectral data are best analysed in a global fashion using nonlinear least squares algoritlims, e.g., a simplex search, to fit multiple first order processes to all wavelengtli data simultaneously. The goal in tliis case is to find tire time-dependent spectral contributions of all reactant, intennediate and final product species present. In matrix fonn tliis is A(X, t) = BC, where A is tire data matrix, rows indexed by wavelengtli and columns by time, B contains spectra as columns and C contains time-dependent concentrations of all species arranged in rows. 

The difference electron density map following the last cycle of least squares refinement did not show evidence for a simple disorder model to explain the anomalously high B for the hydroxyl oxygen. Attempts to refine residual peaks with partial oxygen occupancies did not significantly improve the agreement index. 

Figure 1 shows the powder X-ray diffraction (XRD) pattern of the as-prepared Li(Nio.4Coo.2Mno.4)02 material. All of the peaks could be indexed based on the a-NaFeC>2 structure (R 3 m). The lattice parameters in hexagonal setting obtained by the least square method were a=2.868A and c=14.25A. Since no second-phase diffraction peaks were observed from the surface-coated materials and it is unlikely that the A1 ions were incorporated into the lattice at the low heat-treatment temperature (300°C), it is considered that the particle surface was coated with amorphous aluminum oxide. 

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. 

Undoubtedly the most popular multivariable controller is the multivariable extension of dynamic matrix control. We developed DMC for a SISO loop in Chap. 8. The procedure was a fairly direct least-squares computational one that solved for the future values of the manipulated variable such that some performance index was rninirnized. 

Therefore, uniaxially oriented samples should be prepared for this purpose, which give so-called fiber pattern in X-ray diffraction. The diffraction intensities from the PPX specimen of P-form, which had been elongated 6 times at 285°C, were measured by an ordinary photographic method. The reflections were indexed on the basis of the lattice constants a=ft=2.052nm, c(chain axis)=0.655nm, a=P=90°, and y=120°. Inseparable reflections were used in the lump in the computation by the least square method. 

I estimated a version of equation (7.1) in which i denotes vaccine i (i = 1, 2,..., 13), and the continuous variable f was replaced by a set of year dummies. The model was estimated via weighted least squares, where the weight was equal to the market value (price times quantity) of that vaccine in thatyear. The coefficients on these year variables maybe considered values of a Center for Disease Control vaccine price index. Nominal FSS and Centers for Disease Control vaccine price indexes are compared in Figure 7.2... 

Fig. 10. Concentration dependence of the refractive index difference (An) between the polymer solution and solvent for a 13% PET-PCL copolymer. The lines represent the least-square fits to the measured data...

This method makes use of a test battery to derive a toxicity index that can be employed to classify effluents as a function of their overall toxicity. A formula is given as an example and a procedure to calculate the index using expert judgements and a PLS (Partial Least Square) regression procedure is described using data on 30 effluents. 

Although the direct location of H atoms from X-ray data is often possible, success is notoriously unpredictable. In some cases H atom positions will be readily apparent from difference-Fourier maps, while in other cases one has to apply the tricks mentioned in this section to ferret them out. A good R index does not necessarily guarantee that the search for H atoms will be successful conversely, structures with less impressive R indices will sometimes produce acceptable H positions6). Then there is the problem that H coordinates do not always converge during least-squares refinement. Even when they do, there is no guarantee that the refined positional parameters will be any better than the raw peak positions obtained from... 

In order to compare our results with calculations based on an expansion in Za, we approximate our data for the function G ai by a least-squares fit with five parameters a50, O63, Cf62, a i, and ago (the first index of the coefficients aij indicates the power of Za and the second one corresponds to the power of In (Za)-2). The fit yields... 

Powder X-ray diffiactograms of the alloys are taken using Fe Ka radiation. The X-ray powder patterns are indexed on the basis of hexagonal C14 Laves phase structure. The lattice constants are evaluated by least square refinement. 

The matrix of the regression parameters is estimated by using the least squares approach A = (X7 X) (XrY), where the index T means the transposed... 

X-ray powder diffraction investigations were made in cooperation with P. Norby (Oslo University) and I.G. and E. Krogh Andersen (Odense University). For phase identification and lattice costants determination a Guinier-Hagg camera has been used (CuKa =1.5451 (10) quartz internal standard a=4.91309 A, c=5.40426 A (11). The diagrams were indexed and lattice constants refined by least square... 

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