Histogram matching

Electron density statistics. At high resolution we know the shape of the electron density of an atom, in which case we only need to know its exact location to reconstruct the electron density in its immediate vicinity. At lower resolution we can impose an expected shape on the uni- or multivariate distributions of electron density within the protein region in a procedure that is known as histogram matching. 

Figure 10.6 Histogram matching. In (a) are shown a histogram from a first map in phase refinement (dashed line) and a theoretical protein histogram (solid line). In (b), the protein histogram (dashed line) and a sharp solvent histogram (solid line) are shown.

Figure 3. Product energy distributions for the Cl" CHaBr - CICH3 + Br reaction histogram, trajectory result 6 dashed line, experiment 29 and solid line, prediction of OTS/PST. The trajectory results are scaled to match the experimental exothermicity.

Table HI compiles MC results obtained over the years for the critical temperature and critical density of the RPM. Table in includes also results from the cluster calculations of Pitzer and Schreiber [141]. In a critical assessment of earlier work [40, 141, 179-181, 246], Fisher deduced in 1994 that T = 0.052-0.056 and p = 0.023-0.035 represent the best values [15]. Since then, however, the situation has substantially changed. Caillol et al. [53,247] performed simulations of ions on the surface of a four-dimensional hypersphere and applied finite-size corrections. Valleau [248] used his thermodynamic-scaling MC for systems with varying particle numbers to extract the infinite-size critical parameters. Orkoulas and Panagiotopoulos [52] performed grand canonical simulations in conjunction with a histogram technique. All studies indicate an insufficient treatment of finite-size effects in earlier work. While their results do not agree perfectly, they are sufficiently close to estimate T = 0.048-0.05 and p = 0.07-0.08, as already quoted in Eq. (6). Critical points of some real Coulombic systems match quite well to these figures [5]. The coexistence curve derived by Orkoulas and Panagiotopoulos [52] is displayed in Fig. 9.

Figure 2. O O hydrogen bond length histograms indicate a difference in the intermolecular potential for various chemical species. Closely matched (symmetric) hydrogen bonds clearly adopt shorter O - O distances. Both X-ray and neutron diffraction data are from the CSD. From top to bottom all O—H O containing species carboxyl-carboxyl, ApKa > 15 carboxyl-carboxylate, ApXa 0 water-water, ApXa 16 hydroxium-water, ApXa 0 carboxyl-water, ApXa 5 water-carboxylate, ApXa 10. The pK values used here are based on typical aqueous values.

FIGURE 8 Histogram of the peak wavelength (shaded bars) and the stimulated emission spectrum by optical pumping (solid line). The histogram has a periodic modulation with a period of 1 nm. The stimulated emission line is a good match with the modulation. 

A histogram Hk E) monitors how often each state is visited in the window [ / E. Care must be exercised at the boundaries of a window to fulfill detailed balance [36] If a move attempts to leave the window, it is rejected and iP(windowedge) is incremented by unity. Another question which may arise from the discussion of the boundary is the optimum amount of overlap to minimize the uncertainty of the overall ratio. Here we choose the minimal overlap of one state at the interval boundaries, i.e., E = E -. This is simple to implement and sufficient to match the probability distributions at their boundaries. A larger overlap may reduce the uncertainty but requires a higher computational effort. 

---