Diagonalization-least squares

The structure was refined by block-diagonal least squares in which carbon and oxygen atoms were modeled with isotropic and then anisotropic thermal parameters. Although many of the hydrogen atom positions were available from difference electron density maps, they were all placed in ideal locations. Final refinement with all hydrogen atoms fixed converged at crystallographic residuals of R=0.061 and R =0.075. 

The structure was refined with block diagonal least squares. In cases of pseudo-symmetry, least squares refinement is usually troublesome due to the high correlations between atoms related by false symmetry operations. Because of the poor quality of the data, only those reflections not suffering from the effects of decomposition were used in the refinement. With all non-hydrogen atoms refined with isotropic thermal parameters and hydrogen atoms included at fixed positions, the final R and R values were 0.142 and 0.190, respectively. Refinement with anisotropic thermal parameters resulted in slightly more attractive R values, but the much lower data to parameter ratio did not justify it. 

Crystals of the compound of empirical formula FiiPtXe are orthorhombic with unit-cell dimensions a = 8-16, h = 16-81. c = 5-73 K, V = 785-4 A . The unit cell volume is consistent with Z = 4, since with 44 fluorine atoms in the unit cell the volume per fluorine atom has its usual value of 18 A. Successful refinement of the structure is proceeding in space group Pmnb (No. 62). Three-dimensional intensity data were collected with Mo-radiation on a G.E. spectrogoniometer equipped with a scintillation counter. For the subsequent structure analysis 565 observed reflexions were used. The platinum and xenon positions were determined from a three-dimensional Patterson map, and the fluorine atom positions from subsequent electron-density maps. Block diagonal least-squares refinement has led to an f -value of 0-15. Further refinements which take account of imaginary terms in the anomalous dispersion corrections are in progress. 

The positional and thermal parameters, and a scale factor, were refined by block-diagonal least-squares methods. After six cycles with isotropic thermal parameters R was 0-136. Five further cycles with anisotropic parameters for the gold atom and isotropic parameters for the fluorines reduced R to 0-099, and t ee cycles with all atoms anisotropic completed the refinement, the final R being 0-090. Measured and calculated structure factors are listed in Table 1. 

Principal programs used were as follows REDUCE and UNIQUE, data reduction programs by M. E. Lewonowicz, Cornell University, 1978 BLS78A, an anisotropic block-diagonal least-squares refinement written by K. Hirotsu and E. Arnold, Cornell University, 1980 and PLIPLOT, by G. VanDuyne, Cornell Universitv, 1984. 

For simple spectra, the various interaction parameters are evaluated by a direct comparison of the theoretical formulae for the energy levels with their experimental values. Since in most cases the number of parameters is much smaUer than the number of levels, the energy level calculation is based on an iterative diagonalization-least-squares process. This process comprises four peu ts ... 

The structure was solved by an application of the symbolic addition method and refined by the block-diagonal least-squares method. Anisotropic thermal vibrations were assumed for the nonhydrogen atoms. All the hydrogen atoms were clearly found from a difference Fourier map and their positional and isotropic thermal parameters were refined. The final conventional i index was 0.037. 

Damp the diagonal of the least squares matrix. This stops the calculation from locating false minima by moving parameters too far in a given cycle of refinement. 

Fig. 3.13. Partial least-squares (PLS) calibration of the API data set (5 s accumulation time). Spectra were baseline corrected, normalised to unit length and mean centred. The data set was randomly split into a calibration set (two-thirds) and a prediction set (one-third) obvious outliers from the PCA analysis were excluded from the analysis. The graph shows predicted versus measured API concentration of the prediction set. The straight line represents the 45° diagonal (this figure was published in [65], Copyright Elsevier (2008))...

The regression analysis of multicollinear data is described in several papers e.g. [MANDEL, 1985 HWANG and WINEFORDNER, 1988], HWANG and WINEFORD-NER [1988] also discuss the principle of ridge regression, which is, essentially, the addition of a small contribution to the diagonal of correlation matrix. The method of partial least squares (PLS) described in Section 5.7.2 is one approach to solving this problem. 

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