Nearest-neighbor

If this is not possible uneunbiguously, the decision is made randomly. In KNN regression, the value of the predicting function is calculated as the eirithmetic meeui of the k nearest neighbors of x  

In KNN classification, the class to be predicted is selected by majority vote eunong the k neeuest neighbors of x  

if class determination is not possible unambiguously, a reuidom decision is made. An alternative to a random decision is e.g. weighted methods that e.g. favor the neeuest neighbor in a tie decision. In the present work we will use KNN classification in the binary case only, see Section 7.6. In this case random decisions are avoided 

While we focussed on the methods of supervised statistical learning in this chapter, the next chapter builds on this with a specific focus on applying these to quantitative structure-property relationships (QSPRs). 

So far, all interactions have assumed that MP can be considered to be a dipole. This is a good approximation for repulsions, but not for attractive interactions at shorter range, which require a multipole approach to improve accuracy. Since all calculations were numerical, it was easy to regard the water molecules as having point charges. For simplicity, in initial simulations the center of each positive charge was 

The nearest neighbor distance between a first shell dipole and one in the second shell was initially assumed to be the diameter of a water molecule (2.76 A). Under these conditions, the distance between the central ion and the center of a second shell dipole is 3.30 A when the primary shell ion-dipole center distance 2.5 A. After various iterations described below, the view that adjacent water molecules had their geometric centers at 2.76 A was abandoned. 

b0 is about 6.2 A. Thus, the energy of cavity formation is about 4.5 x 10 12 ergs for a trikisoctahedron of 14 water molecules, or 280 kJ/14 per mole of water molecules. This corresponds almost exactly to the energy of one hydrogen bond. 

Initial Simulation Results for First and Second Trikisoctahedron 

Some EXAFS data on metal ion-oxygen distances are available for hexaaquo transition metal ions.94 These are generally about 2.1 A for 2+, and 2.0 A for 3+ ions, and correspond to those determined by X-ray diffraction in crystals.94 Some symmetrical breathing mode vibrational frequencies are also available, allowing force constants for a reduced 

---

One may consider a molecule in the surface region as being in a state intermediate between that in the vapor phase and that in the liquid. Skapski [11] has made the following simplified analysis. Considering only nearest-neighbor interactions, if n, and denote the number of nearest neighbors in the interior of the liquid and the surface region, respectively, then, per molecule... 

The next point of interest has to do with the question of how deep the surface region or region of appreciably unbalanced forces is. This depends primarily on the range of intermolecular forces and, except where ions are involved, the principal force between molecules is of the so-called van der Waals type (see Section VI-1). This type of force decreases with about the seventh power of the intermolecular distance and, consequently, it is only the first shell or two of nearest neighbors whose interaction with a given molecule is of importance. In other words, a molecule experiences essentially symmetrical forces once it is a few molecular diameters away from the surface, and the thickness of the surface region is of this order of magnitude (see Ref. 23, for example). (Certain aspects of this conclusion need modification and are discussed in Sections X-6C and XVII-5.)... 

This region has been divided into two subphases, L and S. The L phase differs from the L2 phase in the direction of tilt. Molecules tilt toward their nearest neighbors in L2 and toward next nearest neighbors in L (a smectic F phase). The S phase comprises the higher-ir and lower-T part of L2. This phase is characterized by smectic H or a tilted herringbone structure and there are two molecules (of different orientation) in the unit cell. Another phase having a different tilt direction, L, can appear between the L2 and L 2 phases. A new phase has been identified in the L 2 domain. It is probably a smectic L structure of different azimuthal tilt than L2 [185]. 

If the interaction between atoms that are not nearest neighbors is neglected, then the ratios B/A are each equal to the ratio of the number of nearest neighbors to a surface atom (across the dividing plane) to the number of nearest neighbors for an interior atom. The calculation then reduces to that given by Eq. Ill-15. 

The calculation of the surface energy of metals has been along two rather different lines. The first has been that of Skapski, outlined in Section III-IB. In its simplest form, the procedure involves simply prorating the surface energy to the energy of vaporization on the basis of the ratio of the number of nearest neighbors for a surface atom to that for an interior atom. The effect is to bypass the theoretical question of the exact calculation of the cohesional forces of a metal and, of course, to ignore the matter of surface distortion. 

Westwood and Hitch suggest, incidentally, that the cleavage experiment, not being fully reversible, may give only a bond-breaking or nearest-neighbor type of surface energy with little contribution from surface distortion. 

Make the following approximate calculations for the surface energy per square centimeter of solid krypton (nearest-neighbor distance 3.97 A), and compare your results with those of Table VII-1. (a) Make the calculations for (100), (110), and (111) planes, considering only nearest-neighbor interactions, (b) Make the calculation for (100) planes, considering all interactions within a radius defined by the sum... 

Taking into account only nearest-neighbor interactions, calculate the value for the line or edge tension k for solid argon at 0 K. The units of k should be in ergs per centimeter. 

Metals A and B form an alloy or solid solution. To take a hypothetical case, suppose that the structure is simple cubic, so that each interior atom has six nearest neighbors and each surface atom has five. A particular alloy has a bulk mole fraction XA = 0.50, the side of the unit cell is 4.0 A, and the energies of vaporization Ea and Eb are 30 and 35 kcal/mol for the respective pure metals. The A—A bond energy is aa and the B—B bond energy is bb assume that ab = j( aa + bb)- Calculate the surface energy as a function of surface composition. What should the surface composition be at 0 K In what direction should it change on heaf)pg, and why ... 

The various spectroscopic methods do have in common that they typically allow analysis of the surface composition. Some also allow an estimation of the chemical state of the system and even of the location of nearest neighbors. 

A LEED pattern is obtained for the (111) surface of an element that crystallizes in the face-centered close-packed system. Show what the pattern should look like in symmetry appearance. Consider only first-order nearest-neighbor diffractions. 

Some studies have been made of W/O emulsions the droplets are now aqueous and positively charged [40,41 ]. Albers and Overbeek [40] carried out calculations of the interaction potential not just between two particles or droplets but between one and all nearest neighbors, thus obtaining the variation with particle density or . In their third paper, these authors also estimated the magnitude of the van der Waals long-range attraction from the shear gradient sufficient to detach flocculated droplets (see also Ref. 42). 

Such attractive forces are relatively weak in comparison to chemisorption energies, and it appears that in chemisorption, repulsion effects may be more important. These can be of two kinds. First, there may be a short-range repulsion affecting nearest-neighbor molecules only, as if the spacing between sites is uncomfortably small for the adsorbate species. A repulsion between the electron clouds of adjacent adsorbed molecules would then give rise to a short-range repulsion, usually represented by an exponential term of the type employed... 

As for the trough states, a statistical analysis has been earned out for the calculated cone states [12]. The nearest neighbor spacings are calculated by... 

Figure 3-20. Distribution of the dataset of 120 reactions in the Kohonen netv/ork, a) The neurons were patterned on the basis of intellectually assigned reaction types b) in addition, empty neurons were patterned on the basis of their k nearest neighbors.

Woodruff and co-workers introduced the expert system PAIRS [67], a program that is able to analyze IR spectra in the same manner as a spectroscopist would. Chalmers and co-workers [68] used an approach for automated interpretation of Fourier Transform Raman spectra of complex polymers. Andreev and Argirov developed the expert system EXPIRS [69] for the interpretation of IR spectra. EXPIRS provides a hierarchical organization of the characteristic groups that are recognized by peak detection in discrete ames. Penchev et al. [70] recently introduced a computer system that performs searches in spectral libraries and systematic analysis of mixture spectra. It is able to classify IR spectra with the aid of linear discriminant analysis, artificial neural networks, and the method of fe-nearest neighbors. 

The distance of a compound from a library can be given by the distance of this compound from a) its nearest neighbor in the library, b) its fe nearest neighbors in the library, and c) the centroid of the Ubraiy. 

One restriction imposed by Huckel theory that is rather easy to release is that of zero overlap for nearest-neighbor interactions. One can retain a — as the diagonal elements in the secular matrix and replace p by p — EjS as nearest-neighbor elements where S is the overlap integral. Now,... 

Place the equatorial bonds so as to approximate a tetrahedral arrangement of the bonds to each carbon The equatorial bond of each carbon should be parallel to the ring bonds of its two nearest neighbor carbons... 

The primary structure of a peptide is its ammo acid sequence We also speak of the secondary structure of a peptide that is the conformational relationship of nearest neighbor ammo acids with respect to each other On the basis of X ray crystallographic studies and careful examination of molecular models Linus Pauling and Robert B Corey of the California Institute of Technology showed that certain peptide conformations were more stable than others Two arrangements the a helix and the (5 sheet, stand out as... 

Secondary structure (Section 27 19) The conformation with respect to nearest neighbor ammo acids m a peptide or pro tern The a helix and the pleated 3 sheet are examples of protein secondary structures... 

You can choose to calculate all nonbonded interactions or to truncate (cut off) the nonbonded interaction calculations using a switched or shifted function. Computing time for molecular mechanics calculations is largely a function of the number of nonbonded interactions, so truncating nonbonded interactions reduces computing time. You must also truncate nonbonded interactions for periodic boundary conditions to prevent interaction problems between nearest neighbor images. 

The features that occur near 5 = 1.0 ppm are associated with the protons of the a-methyl group. The location of this peak depends on the configurations of the nearest neighbors. 

Enthalpies of mixing have their origin in the forces that operate between individual molecules. Intermolecular forces drop off rapidly with increasing distance of separation between molecules. This means that only nearest neighbors need be considered in the model. 

Each lattice site is defined to have z nearest neighbors, and 0i and 02 > respectively, can be used to describe the fraction of sites which are occupied by solvent molecules and polymer segments. The following inventory of interactions can now be made for the mixture ... 

Especially for large values of Aw, there could be an additional entropy effect beyond that calculated in the last section which arises from the interaction of nearest neighbors. That is, reaction (8.A) might be characterized by both a AHp jj. and a ASp jj.. In this case Aw might be viewed as the pairwise contribution to a free energy ACpaj with... 

Utilize periodic boundary conditions, which permit reduction of the number of nonbonded interactions at greater distances by involving only the "nearest neighbor" atoms from copies of the system which are in different but adjacent cells. 

Figure 19 compares experimental data with calculated curves (80). In the random Co—Cr alloy the Cr atoms are not distributed ia the most suitable way for reduciag the Af of the alloy. Therefore the maximum local content of Cr for this distribution is much higher than ia the case where Cr—Cr bonds are not present. The curve for no Cr—Cr bonds present shows that the Af becomes zero at 25 at. % Cr, based on the fact that for bulk material the measured Af for this composition is zero. Consequently 4 Cr nearest neighbor ia an hep lattice makes the final Af zero. 

The short-range order in a material is important in determining optoelectronic properties. For instance, x-ray and electron diffraction experiments performed on amorphous siHcon (i -Si) and germanium (a-Ge) have revealed that the nearest neighbor environments are approximately the same as those found in their crystalline counterparts (6) photoemission experiments performed on i -Si show that the DOS in valence and conduction bands are virtually identical to the corresponding crystal with the exception that the singularities (associated with periodicity) present in the latter are smeared out in the former. 

---