Nearest-neighbour

The discriminant analysis techniques discussed above rely for their effective use on a priori knowledge of the underlying parent distribution function of the variates. In analytical chemistry, the assumption of multivariate normal distribution may not be valid. A wide variety of techniques for pattern recognition not requiring any assumption regarding the distribution of the data have been proposed and employed in analytical spectroscopy. These methods are referred to as non-parametric methods. Most of these schemes are based on attempts to estimate P(x g and include histogram techniques, kernel estimates and expansion methods. One of the most common techniques is that of K-nearest neighbours. 

The basic idea underlying nearest-neighbour methods is conceptually very simple, and in practice it is mathematically simple to implement. The general method is based on applying the so-called AT-nearest neighbour classification rule, usually referred to as AT-NN. The distance between the pattern vector of 

For objects 1 and 2 characterized by multivariate pattern vectors Xi and X2 defined by 

For a total training set of N objects comprised of /i samples known to belong to each group i, the procedure adopted is to determine the /Tth nearest neighbour to the unclassified object defined by its pattern vector x, ignoring group membership. From this, the conditional probability of the pattern vector 

Since the volume term is constant to both sides of the equation, the rule simplifies to, 

Application of Equation 5.39 to the AT-NN rule serves to define a sphere, or circle for bivariate data, about the unclassified sample point in space, of radius rx which is the distance to the Afth nearest neighbour, containing K nearest 

If the number of objects in each training set, is proportional to the unconditional probability of occurrence of the groups, P(G.i) then Equation 5.42 simplifies further to  

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Our discussion of solids and alloys is mainly confined to the Ising model and to systems that are isomorphic to it. This model considers a periodic lattice of N sites of any given symmetry in which a spin variable. S j = 1 is associated with each site and interactions between sites are confined only to those between nearest neighbours. The total potential energy of interaction... 

An alternative fomuilation of the nearest-neighbour Ising model is to consider the number of up f T land down [i] spins, the numbers of nearest-neighbour pairs of spins IT 11- U fl- IT Hand their distribution over the lattice sites. Not all of the spin densities are independent since... 

Thus, only two of the five quantities Itl lJ-l-Utl-UlMfi lare independent. We choose the number of down spins [i] and nearest-neighbour pairs of down spins [ii] as the independent variables. Adding and subtracting the above two equations. 

Making use of the relations between the spin densities, the energy of a given spm configuration can be written in tenns of the numbers of down spins [4] and nearest-neighbour down spins [44] ... 

The Ising model is isomorphic with the lattice gas and with the nearest-neighbour model for a binary alloy, enabling the solution for one to be transcribed into solutions for the others. The tlnee problems are thus essentially one and the same problem, which emphasizes the importance of the Ising model in developing our understanding not only of ferromagnets but other systems as well. 

No more than one particle may occupy a cell, and only nearest-neighbour cells that are both occupied mteract with energy -c. Otherwise the energy of interactions between cells is zero. The total energy for a given set of occupation numbers ] = (n, of the cells is then... 

The relationship between tlie lattice gas and the Ising model is also transparent in the alternative fomuilation of the problem, in temis of the number of down spins [i] and pairs of nearest-neighbour down spins [ii]. For a given degree of site occupation [i]. 

A binary alloy of two components A and B with nearest-neighbour interactions respectively, is also isomorphic with the Ising model. This is easily seen on associating spin up with atom A and spin down with atom B. There are no vacant sites, and the occupation numbers of the site are defined by... 

This is the quasi-chemical approximation introduced by Fowler and Guggenlieim [98] which treats the nearest-neighbour pairs of sites, and not the sites themselves, as independent. It is exact in one dimension. The critical temperature in this approximation is... 

In the simplest model the eoeflfieient K depends only on the differenees of the attraetive energies -e of the nearest-neighbour pairs (these energies are negative relative to those of the isolated atoms, but here their magnitudes e are expressed as positive numbers)... 

The parameter J.j is a measure of the energy of interaction between sites and j while h is an external potential or field common to the whole system. The tenn ll, 4s a generalized work temi (i.e. -pV, p N, VB M, etc), so is a kind of generalized enthalpy. If the interactions J are zero for all but nearest-neighbour sites, there is a single nonzero value for J, and then... 

Figure Bl.23.14. Schematic illustration of the Pt 111 ] -(1 x 1) surface. Arrows are drawn to indicate the nearest-neighbour first-first-, second-first-, and third-first-layer interatomic vectors.

Figure B3.3.7. The cell structure. The potential cutoff range is indicated. In searching for neighbours of an atom, it is only necessary to examine the atom s own cell, and its nearest-neighbour cells.

To introduce protein-like character tire interactions between beads (Arose separated by at least tliree bonds) that are nearest neighbours on a lattice are assumed to depend on tire nature of tire beads. The energy of a confomration. 

The simplest example is that of tire shallow P donor in Si. Four of its five valence electrons participate in tire covalent bonding to its four Si nearest neighbours at tire substitutional site. The energy of tire fiftli electron which, at 0 K, is in an energy level just below tire minimum of tire CB, is approximated by rrt /2wCplus tire screened Coulomb attraction to tire ion, e /sr, where is tire dielectric constant or the frequency-dependent dielectric function. The Sclirodinger equation for tliis electron reduces to tliat of tlie hydrogen atom, but m replaces tlie electronic mass and screens the Coulomb attraction. 

Figure C3.2.9. Both nearest neighbour and nonnearest neighbour coupling interactions mediate superexchange between tire temrinal pi-electron groups of rigid dienes witlr saturated bridging units. From [31],...

This then is the limiting radius ratio for six nearest neighbours— when the anion is said to have a co-ordination number of 6. Similar calculations give the following limiting values ... 

For eight nearest neighbours (a co-ordination number of 8) the radius ratio r /r must not be less than 0.73. 

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