Distance class algorithm

Schulten238 outlined the development of a multiple-time-scale approximation (distance class algorithm) for the evaluation of nonbonded interactions, as well as the fast multipole expansion (FME). efficiency of the FME was demonstrated when the method outperformed the direct evaluation of Coulomb forces for 5000 atoms by a large margin and showed, for systems of up to 24,000 atoms, a linear dependence on atom number. 

Various partitions, resulted from the different combinations of clustering parameters. The estimation of the number of classes and the selection of optimum clustering is based on separability criteria such as the one defined by the ratio of the minimum between clusters distance to the maximum of the average within-class distances. In that case the higher the criterion value the more separable the clustering. By plotting the criterion value vs. the number of classes and/or the algorithm parameters, the partitions which maximise the criterion value is identified and the number of classes is estimated. 

A better approach to validating the prediction is to compare the distance from the unknown sample to the predicted class relative to an expected distance for known members of that class. From the last example, the distance from Z to it.s nearest neighbor in class B is much larger than the distances between the samples within that class. This can be flagged by calculating a measure of expected interpoint distances for samples in each class. These distances are then compared to the distance of the unknown to the different classes to validate class membership. One algorithmic approach is discussed below illustrating the classification of unknown Z with respect to class B. 

Gordon and Herman have also proposed a variety of formulae which allow one to compute the distance between the original picture and the reconstructed matrix, and therefore to evaluate the efficiency of a reconstruction algorithm. The ART method, in conclusion, is simple, fast and versatile, which explains why it has become an ideal starting-point for research on a new class of reconstruction algorithms. 

First compute the fuzzy partition by using the GFNM algorithm and the usual distance. The diameters of the classes in this partition are computed. 

We may also adopt a different procedure in which the adaptive distances are computed at each iteration in the GFNM algorithm. At the first iteration the diameters are all equal 5, = 1, / = 1,2,..., n. The diameter of the fuzzy class A, obtained at iteration k induces an adaptive distance. The adaptive distances are used to compute the fuzzy classes at iteration k + For a relatively large data set the computation of diameters may involve the storage of a large distance matrix. We may avoid the computational difficulties in such cases by using a simpler adaptive distance. We have supposed the classes are approximately spherical (or ellipsoidal). The mean of a fuzzy class may thus be considered as an approximation of the geometric center of the class. On this basis we may define the radius r, of the fuzzy class A, as... 

Clustering problems can have numerous formulations depending on the choices for data structure, similarity/distance measure, and internal clustering criterion. This section first describes a very general formulation, then it details special cases that corresponds to two popular classes of clustering algorithms partitional and hierarchical. 

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