Centroid mapping

Now let us examine the centroid map for node 29. The centroid map is shown in figure 5 for 4 unit cells. For reference the true map using image derived phases is shown in Figure 6. The agreement is excellent with a map correlation coefficient of 0.95. 

After analysis nodes 42(25), 50(18), 53(9), 98(56), 125(40), 189(26) are kept (the mean absolute phase errors are in parentheses). For reference the correct map using experimental phases is shown in Figure 6, and Figure 7 shows the best centroid map (for node 53). For reference the correct map using experimental phases is shown in Figure 6. The map correlation coefficient is 0.94. 

Figure 7. The centroid map for node 53 with a mean phase error of 90 and a map correlation coefficient of 0.94.

The location of the hydrogen atoms in hydrogen bonded systems is often difficult to ascertain. When X-ray diffraction is used there is an experimental limitation to face, as it is usually difficult to locate the very light H-atom in Fourier maps and, even when this is possible, the technique can provide information on electron density centroids rather than on the position of the light nucleus. Neutron diffraction is required for an unambiguous location of the H-atom. In ionic hydrogen bonds the situation may occur where a knowledge of the proton position in a donor-acceptor system is necessary to know whether proton transfer, i.e. protonation of a suitable base, has occurred or not. 

The practical differences between centroid and maximum entropy maps can be seen with reference to membrane data in [7],... 

Fig. 7. Simplified nonlinear mapping of Italian red wines (category centroids as basepoints). 8 variables...

Fig. 5. Difference-Fourier maps corresponding to the six non-crystallographic mirror planes of the Re4 tetrahedron. In each section, the two dots on the left represent Re atoms, the dot in the center represents the centroid of the tetrahedron, and the dot on the right corresponds to the mid-point of a Re-Re bond...

Fig. 6. A composite difference-Fourier section for H4Re4(CO)i2, calculated by superimposing the maps in Fig. 5. Although the six individual sections contained many spurious peaks, in the composite shown here they are effectively averaged out Point Y corresponds to the mid-point of a Re-Re bond (which is perpendicular to the plane of the paper), and point X represents the centroid of the tetrahedron. The H peaks shown here correspond to face-bridging positions, with a measured Re-H distance of 1.75—1.79 A. The alternative edge-bridging position (marked E) and the terminal positions (marked T), which also lie on the mirror planes of the tetrahedron, show no electron density (Ref. 12)...

Refinement of the framework coordinates was difficult because of the acentricity and pseudosymmetry. A stable least-squares solution was obtained for the framework atoms in which the T-0 distances indicate alternation of A1 and P atoms. However, the 0 atoms showed large displacements from the centroid, particularly 0(2) for which a difference-Fourier map indicated three spearate peaks (Figure 3). Because there is no optical or X-ray evidence for symmetry lower than hexagonal, it is assumed that there are microdomains with tilted tetrahedra, as proposed for high-cristobalite(6) and high-tridymite( 7). For convenience, the displacements of the oxygen atoms are approximated by ellipsoids. 

The iterative adjustment of weight vectors is similar to the iterative refinement of k-means clustering to derive cluster centroids. The main difference is that adjustment affects neighboring weight vectors at the same time. Kohonen mapping requires O(Nmn) time and 0(N) space, where m is the number of cycles and n the number of neurons. 

Figure 28 (a) An electron density map calculated from the most likely phases (also known as centroid phases) of selenium SAD-phasing of conserved dopamine neurotrophic factor (CDNF) (V.-M. Leppanen, unpublished), (b) The final map with the refined model. 

Unlike other multiclass classifiers VVRKFA [23, 24], utilizes the regression or function approximation approach to map the feature vectors to its label vectors in a space whose dimension is equal to the number of categories present in the classification problems. The classification is performed in this label space by comparing the Maha-lanobis distance of a test pattern from the respective class centroids. The idea of VVRKFA classifier is graphically illustrated in Fig. 6 for a three class classification problem. The performance of VVRKFA classifier is comparable to that of SVM classifier and it takes less time to train a large data set [23],... 

Figure 17.7 Hydrogen exchange kinetics of free ACTR and CBP and their complex. Representative spectra (a) and uptake curves (b) from peptides in CBP and ACTR that become a-helical in the complex. Secondary structure elements shown In diagram indicate a-helical (boxes), loop (lines), and unstructured (dotted lines) that ACTR adopts in complex with CBP [29]. The vertical dashed lines in the spectra denote the centroids of the undeuterated and fully deuterated states. The dashed lines in the uptake curves denote data fitting using Equation 17.20. (c) The kinetic analysis provides estimates of peptide-averaged protection, depicted as different colored bars mapped onto the sequence and secondary structure found in the complex. Adapted from Ref [81] with permission, 2011 American Chemical Society. (See insert for color representation of the figure.)...

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