Distance measures nearest neighbour

A mathematically very simple classification procedure is the nearest neighbour method. In this method one computes the distance between an unknown object u and each of the objects of the training set. Usually one employs the Euclidean distance D (see Section 30.2.2.1) but for strongly correlated variables, one should prefer correlation based measures (Section 30.2.2.2). If the training set consists of n objects, then n distances are calculated and the lowest of these is selected. If this is where u represents the unknown and I an object from learning class L, then one classifies u in group L. A three-dimensional example is given in Fig. 33.11. Object u is closest to an object of the class L and is therefore considered to be a member of that class. 

In summary, ionisation potentials, dissociation and cohesive energies for mercury clusters have been determined. The mass spectrum of negatively charged Hg clusters is reported. The influence of the transition from van der Waals (n < 13), to covalent (30 < n < 70) to metallic bonding (n > 100) is discussed. A cluster is defined to be metallic , if the ionisation potential behaves like that calculated for a metal sphere. The difference between the measured ionisation potential and that expected for a metallic cluster vanishes rather suddenly around n 100 Hg atoms per cluster. Two possible interpretations are discussed, a rapid decrease of the nearest-neighbour distance and/or the analogue of a Mott transition in a finite system. Electronic correlation effects are strong they make the experimentally observed transitions van der Waals/covalent and covalent/metallic more pronounced than calculated in an independent electron theory. 

The contributions of the modifier components to the RDF of the simulated NaRbSi205 glass are shown in Fig. 12.12. The cation-cation correlations responsible for the small oxygen bond angles discussed above can be clearly distinguished. Na-Na distances peak initially at 3.1 A and Rb-Rb distances close to 3.9 A. This separation relates mainly to the different nearest neighbour distances 2.4 A for Na-O compared to 3.0 A for Rb-O. In the total RDF measured by diffraction techniques these distances fall either side of the strong O-Si-O peak at 2.65 A and can only be accurately measured by EXAFS. In... 

Each pattern is characterized in the usual way (Chapter 1.2) by a set of d components (features, measurements) and can be considered as a point in a d-dimensional space. An additional component as described in Chapter 1.3 is not necessary for the KNN-method. Classification of an unknown pattern x is made by examination some pattern points with known class membership which are closest to x- In order to find the nearest neighbours of the unknown it is necessary to compute the distances between X and all other pattern points of the available data set. The number of neighbours which are considered for classification is usually denoted by K. If only one neighbour is used for the classification (K=1, 1NN-method") the class membership of the first (nearest) neighbour gives the class membership of the unknown. If more than one neighbour is used a voting scheme or some other procedure is applied to determine the class of the unknown. 

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