K clusters

Figure Bl.19.13. (a) Tliree STM images of a Pt(l 11) surface covered witli hydrocarbon species generated by exposure to propene. Images taken in constant-height mode. (A) after adsorption at room temperature. The propylidyne (=C-CH2-CH2) species that fomied was too mobile on the surface to be visible. The surface looks similar to that of the clean surface. Terraces ( 10 mn wide) and monatomic steps are the only visible features. (B) After heating the adsorbed propylidyne to 550 K, clusters fonn by polymerization of the C H...

This index is employed by both the k-means (MacQueen, 1967) and the isodata algorithms (Ball and Hall, 1965), which partition a set of data into k clusters. With the A -means algorithm, the number of clusters are prespecified, while the isodata algorithm uses various heuristics to identify an unconstrained number of clusters. 

Select a number k of desired clusters and initialize k cluster centroids c , for example, by randomly selecting k different objects. 

FIGURE 6.13 Fuzzy clustering uses membership coefficients to assign each object with varying probabilities to all k clusters. 

The square-error across all K clusters in a partition is the sum of the square-errors for each of the K clusters. (Note also that the standard deviation would be the square root of the square-error.)... 

Although stability can be guaranteed after a finite number of steps, this number can be reduced if step (3) is modified and the centroids are recalculated after each assignation of the observations. As a result of applying this technique, as well as a description of k clusters, computer programs usually provide the mean values of the variables in each of these, and the comparison of these mean values. 

Kmeans Clustering. Type of partitioning cluster analysis in which an object, such as a chemical structure, is placed into one of K clusters, based on how similar the structure is to the average value (or centroid) of each cluster. The average of the cluster may be an actual structure itself, in which case the technique is referred to as K-medoids clustering. 

Fig. 11 Cluster plot of the first and second principal components derived by PCA from spectral data. The (5-receptor cluster covers DTLET, DSLET, DADLE, aLLeu -enkephalin, and DPDPE. The p-receptor cluster includes DAGO, MeC-enkephalin, and P-endorphin. The alternate (5-receptor cluster is composed of Leu -enkephalin and Leu -enkephalin amide. The K-cluster of the d5morphins contains B(l-13), A(l-13) A(l-9) A(l-ll) and A(l-13) amide. No receptor preference was reported for PLO 17 or ICI 174 864. (From Refs. ...

Figure 15 Dissociation rate of CO on Rh (grown at 90 and 300 K) and Ir (grown at 300 K) clusters supported on ultrathin alumina thin films, measured by XPS. The level of dissociation on extended Rh (111) and (210) surfaces is indicated. (From M. Frank and M. Baumer [183].)...

Step 2. At the h -th iterative step distribute the n samples among k clusters, using the... 

One of the most popular and widely used clustering techniques is the application of the. -Means algorithm. It is available with all popular cluster analysis software packages and can be applied to relatively large sets of data. The objective of the method is to partition the m objects, characterized by n variables, into K clusters so that the square of ffie within-cluster sum of distances is minimized. Being an optimization-based technique, the number of possible solutions cannot be predicted and the best possible partitioning of the objects may not be achieved. In practice, the method finds a local optimum, defined as being a classification in which no movement of an observation from one cluster to another will reduce the within-cluster sum of squares. 

Step 1 Given K clusters and their initial contents, calculate the cluster means Blj and the initial partition error, e. 

As with A-means clustering, the fuzzy fc-means technique is iterative and seeks to minimize the within-cluster sum of squares. Our data matrix is defined by the elements Xy and we seek K clusters, not by hard partitioning of the variable space, but by fuzzy partitions, each of which has a cluster centre or prototype value, Bkj, i [Pg.118]

Other distance measures include the Canberra metric and the Czekanowski coefficient [126]. Clustering can be hierarchical such as grouping of species and subspecies in biology or nonhierarchical such as grouping of items. For fault diagnosis nonhierarchical clustering is used to group data to k clusters... 

Fig. 12.6. Binding energies for K clusters, showing the discontinuities observed at the magic numbers (after W.D. Knight et al. [684]).

Fig. 12.15. Example of a giant dipole resonance in a metal cluster with a closed shell, in this case a singly ionised K cluster with eight delocalised electrons (after C. Brechignac and J.-P. Connerade [714]).

K-containing species K+ clusters, 544-556 K clusters, 557 neutrals, 541-544 Ketones RC(0)R ... 

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