Asymmetric least squares

Figu re 2.2 (a) Raw spectra of a Raman emulsion layer image (b) Spectra after de-noising by principal component analysis (PCA) (c) Spectra after de-noising and baseline correction by asymmetric least squares. 

When the polynomial is fitted to signal points, which are manually selected, that can be attributed only to baseline (background), then baseline (background) removal is achieved and the drawbacks of detrending are overcome. A variant for baseline correction is to do adopt a weighted least squares automatic procediue (asymmetric least squares [63]). This is an automatic approach to determine which points most likely belong to baseline only, by... 

Iqdari and Velde (unpub. data, 1992, see Table 8.2) described experiments of Ce diffusion in apatite soaked in CeCl2 with asymmetric diffusion profiles. For one of their runs carried out at 1100°C for 15 days, and described as an example of a non-linear least-square fit in Section 5.2, it has been found that the relationship between the Ce concentration CCt and the distance X to the mineral surface is described by... 

The phase problem and the problem of arbitration. Fibrous structures are usually made up of linear polymers with helical conformations. Direct or experimental solution of the X-ray phase problem is not usually possible. However, the extensive symmetry of helical molecules means that the molecular asymmetric unit is commonly a relatively small chemical unit such as one nucleotide. It is therefore not difficult to fabricate a preliminary model (which incidently provides an approximate solution to the phase problem) and then to refine this model to provide a "best" solution. This process, however, provides no assurance that the solution is unique. Other stereochemically plausible models may have to be considered. Fortunately, the linked-atom least-squares approach provides a very good framework for objective arbitration independent refinements of competing models can provide the best models of each kind the final values of n or its components (eqn. xxiv) provide measures of the acceptability of various models these measures of relative acceptability can be compared using standard statistical tests (4) and the decision made whether or not a particular model is significantly superior to any other. 

Fig. 3. Conformation of the switch-2 cluster and neck linker/neck region in various members of the kinesin superfamily. The upper four panels (A, B, E, F) show crystal structures of N-type kinesins with their motor domain at the N-terminus and the neck at the C-terminus. (C), (D), (G), and (H) show C- and M-type kinesins with their neck N-terminal to the motor domain, except for PoKCBP (G) where the C-terminal neck mimic is shown instead of the N-terminal neck (which is not included in the crystal structure). Each structure is shown in two orientations that differ by a rotation of 90 degrees. Rat conventional kinesin (RnKHC [A]) has been chosen to define standard orientations with the neck helix a7 parallel/perpendicular to the drawing area. Orientations for the other structures have been determined by least-squares superposition of their P-loop regions with that of RnKHC (using 11 Ca-atoms of residues F83-T93 in RnKHC). (B), (C), and (D) show the structures of dimeric constructs with the second motor domain in pale colors. The Ned structure in (C) is 180-degree symmetric the symmetry axis is oblique to the drawing plane and coincides with the axis of the coiled-coil that is formed by the two neck helices. In the asymmetric structure of the Ned N600K mutant (D), the second motor domain (pale) is rotated by about 75 degrees around an axis perpendicular to the coiled-coil. The structures shown in (A), (B), (F), and (G) have their switch-2 cluster in permissive conformation, whereas the conformation of structures (C), (D), (E), and (H) is obstructive, as can be told by observing the slope of the extended switch-2 helix a4. Color code red, switch-2 cluster including the extended...

The 1378 cm"1 band is from the CH3 symmetric bending and the 1156 cm 1 band is a complex skeletal vibration involving the CH3 branch of propylene. The 722 cm"1 band represents the CH2 rock and the 1462 cm"1 band is a combination of the CH2 scissor and the asymmetric CH3 bend. In the photoacoustic spectra the 1378 and 1462 bands are strong while the 1154 and 722 cm"1 bands are weak. Least squares linear regression... 

In order to obtain a Larmor resonance line we have to vary the frequency of the microwave field and count the number of spin Hips per unit time. In order to avoid saturation effects the microwave field amplitude was kept low. The resonance curve obtained in the described manner is rather asymmetric. The lineshape can be described using the known spatial configuration of the magnetic field and a thermal distribution of the axial energy. A least squares fit to the data points as shown in Fig. 9 leads to a fractional uncertainty of about 10-6 and the g factor can be quoted with the same error [9]. 

According to a complete X-ray diffraction analysis, Se6 consists of ring molecules with the molecular symmetry of Dzd the crystal and molecular parameters are listed in Table II (17) and the crystal structure is shown in Fig. 2. Refinement by the least squares method resulted in the following atomic parameters of the single atom in the asymmetric unit x = 0.1602 0.00048, y = 0.20227 0.00047, z = 0.12045 0.00120 calculated density, 4.71 g/cm3. An earlier investigation of selenium vapor by electron diffraction led to an internuclear distance of 234 1 pm and an average bond angle of 102 0.5° for the chairlike cyclic Se6 molecule (23). 

Typke has introduced the rs-fit method [7] where Kraitchman s basic principles are retained. A system of equations is set up for all available isotopomers of a parent (not necessarily singly substituted) and is solved by least-squares methods for the Cartesian coordinates (referred to the PAS of the parent) of all atomic positions that have been substituted on at least one of the isotopomers The positions of unsubstituted atoms need not be known and cannot be determined. The method is presented here with two recent improvements true derivatives are used for the Jacobian matrix X, and the problem of the observations and theircovariances, which is rather elaborate, is fully worked out. The equations are always given for the general asymmetric rotor, noting that simplifications occur in more symmetric situations, e.g. for linear molecules, which could nonetheless be treated within the framework presented. 

After the positions of the atoms in an asymmetric unit have been located in electron density maps, the best values for the x,y,z coordinates are obtained by means of a least-squares refinement in which the squared values of the differences between the values of the experimentally observed Fhk, and the calculated Ehkh based on the determined coordinates of each atom, are minimized. A measure of the minimization and agreement is an R factor. 

Another factor to be taken into account is the degree of over determination, or the ratio between the number of observations and the number of variable parameters in the least-squares problem. The number of observations depends on many factors, such as the X-ray wavelength, crystal quality and size, X-ray flux, temperature and experimental details like counting time, crystal alignment and detector characteristics. The number of parameters is likewise not fixed by the size of the asymmetric unit only and can be manipulated in many ways, like adding parameters to describe complicated modes of atomic displacements from their equilibrium positions. Estimated standard deviations on derived bond parameters are obtained from the least-squares covariance matrix as a measure of internal consistency. These quantities do not relate to the absolute values of bond lengths or angles since no physical factors feature in their derivation. 

Figure 2 shows the reconstructed spectrum. The symbols correspond to the intensities of the fluorescence at the measured wavelength at a given time r. The lines are the least-square-fits to the points using a log-normal function. The log-normal function describes an asymmetric line shape and is often used to fit broad featureless absorption or fluorescence spectra. 

A three-dimensional Patt son synthesis provided for a ready location of the two rhodium atoms of the asymmetric unit. Three cycles of full-matrix least-squares reflnement, in which isotropic thermal parameters were used for the Rh atoms, resulted in a conventional R factor of 0.25 for 1504 reflections for which / > a(7). 

A Fourier synthesis revealed ten major peaks, in addition to the two Rh atom peaks in the asymmetric unit. These ten peaks were assigned as F atoms and refinement, employing isotropic thermal parameters, yielded a final conventional R =0.067. Three cycles of least-squares refinement, in which anisotropic thermal parameters were permitted for the rhodium atoms, resulted in an R factor of 0.049. 

There is peak overlap. In this case, it is necessary to decompose the peak by modelling using software. A least squares error minimisation procedure can be employed to adjust the positions, intensities and full width half maxima of the components and provides an indication of the quality of the model in relation to the actual shape of the peak. Various peak shapes are available for modelling purposes in particular the use of experimentally obtained shapes becomes extremely useful in the case of asymmetric peaks of transition metals (Fig. 5.7). 

Konnert, J. H. A restrained-parameter structure-factor least-squares refinement procedure for large asymmetric units. Acta Cryst. A32, 614-617 (1976). Sussman, J. L., and Podjarny, A. D. The use of a constrained-restrained least-squares procedure for the low-resolution refinement of a macromolecule, yeast tRNAfActa Cryst. B39, 495-505 (1983). 

Besides the classical Discriminant Analysis (DA) and the k-Nearest Neighbor (k-NN), other classification methods widely used in QSAR/QSPR studies are SIMCA, Linear Vector Quantization (LVQ), Partial Least Squares-Discriminant Analysis (PLS-DA), Classification and Regression Trees (CART), and Cluster Significance Analysis (CSA), specifically proposed for asymmetric classification in QSAR. 

The amino acid isoleucine was investigated at a time when63 the relative configuration at the two asymmetric carbon atoms was not known. Phase determination was facilitated by a parallel study of the isotypic hydrobromide. The u-axis projection of the first trial structure is shown in Figure 2a. The visual data used to calculate this projection refined to an R of 23 percent. A difference map calculated at this point revealed that one of the methyl groups must be displaced, and least squares re-... 

According to Craven et al, there are two types of puckered conformations of the barbituric acid ring.149 In the symmetric puckered conformation, the C-2 and C-5 atoms lie on the C2 line (Fig. 7). This conformation is observed for a few barbiturates.20 149,158 The asymmetric puckered conformation has the C-5 atom displaced from the best least squares plane of the other ring atoms (Fig. 7). This usually slight deviation of the C-5 atom varies from one barbiturate to another.149,152 155,156,159 162,173-175 For example, dihedral angles between the C-4-C-5-C-6 plane and the best least squares plane for 5-(l -cycloheptenyl)-5-ethyl-, 5,5-diallyl-, and 5-cyclohexenyl-l,5-dimethylbarbituric acid are 3,10.4, and 29°, respectively.157 Deviations from the planarity of the barbituric acid ring were also predicted by MINDO/3 calculations.175... 

In the course of a typical rotational Zeeman effect investigation of an asymmetric top molecule 40 to 100 Zeeman satellites of different rotational transitions are recorded with both AM=0 and AM = 1 selection rules. According to Eq. (III. 13), this corresponds to a set of 40 to 100 linear equations from which the g-values and susceptibility anisotropies are calculated by a least squares procedure. As an illustration. Fig. III.12 shows recordings of the 2i2 - -22i rotational transition of ethylene oxide in exterior magnetic fields close to 25 kG. The +2.259... 

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