Cluster method INDEX

Where lakes are geographically clustered, the index-lake method can be useful for estimating the groundwater components of additional lake budgets after calculating the budget for the index lake. 

Table VII. Use Pair Clusters in Merck Index Date Base, Derived by Alphanumeric Clustering Method III.

The upper bound of the adjusted Rand index is unity and it takes on the value of zero when the similarity between the two clustering methods matches the expectation of the hypergeometric distribution. A higher value of the adjusted Rand index indicates a greater similarity of membership between the clusters of the two methods being compared. 

Hohm et al. have calculated the static dipole polarizability of P4 clusters using ab initio finite-field MP and coupled-cluster methods. The results have been compared with frequency-dependent measurements obtained fi-om the gas phase refi active index. 

A calculation of the similarity index of the new cluster versus all the other objects is performed. The similarity index calculation criterion differs according to the chosen clustering method, but the operation has the common result of substituting in the similarity matrix the rows and colunms related to the two objects which have just been linked with a new row and column that report the similarity index of the new cluster with all the remaining objects. 

Of the four different methods of cluster analysis applied, the method of Ward described in the Clustan User Manual (10), worked best when compared to the single-, complete-, or average-linkage methods. Using Ward s method, two clusters, Gn and Gm, are fused when by pooling the variance within two existing clusters the variance of the so formed clusters increases minimally. The variance or the sum of squares within the classes will be chosen as the index h of a partition. 

Figure 1. Pyridazinone structures belonging to clusters identified by the method of Ward (partition index of 1.97).

In 2008 Ponomarenko et al. (36) developed the ElliPro Web server with approximation of the protein shape as ellipsoid. They implemented Thornton s method (15), which was originally developed for continuous epitopes and, together with protrusion index and neighboring residue clustering, allows the prediction of antibody epitopes in a given protein sequence or structure. 

Chemometric methods such as analysis of correlation coefficients, cluster analysis or neural network analysis are used, for example, in the classification of fragments of glass on the basis of their elemental composition or refractive index. Such methods allow the test material to be classified into the appropriate group of products on the basis of the measured parameter. 

The work functions for low-index surfaces of the 4d transition metals have been calculated by a full-potential linear-muffin-tin-orbital method using a slab geometry (a periodic arrangement of 7-layer metal slabs and 10-layer vacuum slabs) (29), and the results (Fig. 8) agree well with experimental results. This is a considerable improvement with respect to extended Hiickel calculations for slab [30 (see footnote 21) ] or cluster [i0 (see Chapter 3)] geometries, which usually yield values closer to the atomic ionization energies. However, the shape of the DOS curves and the relative position of the Fermi level as found by the extended Hiickel calculations are reasonably similar to those obtained by the more sophisticated methods. Therefore, it seems that the major error is in the determination of the dipole layer potential. (Further analysis of this topic would lead us beyond the scope of this chapter.)... 

The distance between points in the graphs represents the dissimilarity of the methods within the statistical error of the construction. Therefore, clusters of methods indicate similar performance for the specific index, whereas points far apart indicate that the methods diverge in predictive terms. 

In another work, Poteau et al.61 studied the temperature-dependent properties of the Na clusters using a tight-binding model together with a Monte Carlo thermodynamic method (cf. Section 2.2). By studying the Lindemann index [Eq. (60)] as a function of temperature they could identify phase-transition temperatures, although, as also seen by analysing the heat capacity and the moments of inertia as functions of temperature, these transitions are not sharp. 

The actual implementation of the generalized GCA involves a variety of issues. The point reflection with respect to the pivot requires careful consideration of the periodic boundary conditions. Furthermore, as mentioned above, particles that have been translated via a point reflection must not be translated again within the same cluster move, and particles that interact with a given cluster particle both before and after the translation of that cluster particle must be considered only once, on the basis of the difference in pair potential. One way to account for all interacting pairs in an efficient manner is the use of the cell index method [25]. For mixtures with large size asymmetries (the situation where the generalized GCA excels), it is natural to set up different cell structures, with cell lengths based upon the cutoffs of the various particle... 

The adjusted Rand index (86) measures the extent of agreement between two different cluster structures obtained for the same set of data points. This is a useful measure when comparing two methods producing a different number of clusters. The Rand index is simply the proportion of agreement between the two methods and is defined as... 

N is the total number of genes in the data set. The index ranges from 0 (when the two cluster results are completely different) to 1 (when the two methods agree completely). The expectation of the Rand index for two random partitions is not a constant and it is preferable to use the adjusted Rand index, which corrects the Rand index for the case of random partitions of the data. The adjusted Rand index is given by... 

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