Cluster index

Figure 5.11 Subgraphs of different order and type shown for 2-methylbutane. The cluster index of fourth order is illustrated with 2,2-dimethylpropane.

There were eight clusters generated by variable clustering the 89 TIs used for the aromatic amine data set. The total variation explained by these clusters was 88.6% of thejotal. In Table 9, we present the indexes selected from each of the clusters. Indexes O, IC, IC2, SIC3, CIC3, and xc were retained because of their poor cluster membership (r<0,7). 

A bivariant linear eq uation, contain ing factors such as phase percentages, crystal sizes, morphology and cluster indexes, for predicting the 28-day mortar strength was presented by Sinha, Rao, and Akhouri (1991). The standard deviation was said to be 17.9 kg/ cm agreeing closely with the actual strength. 

Figure 12.17 Carbon K-edge X-ray absorption spectra of (a) a calcined H-ZSM-5 (ZSM-5-C) sample and (c) steamed H-ZSM-5 (ZSM-5-S) sample, obtained from a cluster analysis performed after 150 min time-on-stream. Cluster index for (b) ZSM-5-C and...

There is other useful information that can be determined from an n-ad distribution, for example how well a secondary comonomer is dispersed among the primary repeat units. Two approaches are commonly used to indicate the degree of clustering of ethylene units in an ethylene-1-olefin copolymer [4]. One involves a direct empirical measurement of clustering, the monomer dispersity (MD), and the other utilizes the Bernouillian distribution as a reference point for establishing the degree of clustering and is denoted the cluster index (Cl). These numbers are... 

Ke5Words cluster index, comonomer distribution, monomer dispersity, NMR, H NMR, NMR, tacticity, sequence length, copol3uner. 

Keywords— Electromyography (EMG) signal, feature extraction, feature selection, cluster index, prosthesis. 

Keywords Information theory Mutual information Information integration Dynamical cluster index Dynamical system Boolean networks... 

The DCI method has been introduced in previous work and the interested reader can find details in [1,12,13]. For the sake of completeness, we provide a brief summary of the main notions and the method itself. Let us consider a system modelled with a set U oin variables ranging in finite and discrete value domains. The cluster index of a subset S of variables in C/, 5 c I/, as defined by Tononi et al. [10], estimates the ratio between the amount of information integration among the variables in S and the amount of integration between S and U. These quantities are based on the Shannon entropy of both the single elements and sets of elements in U. The entropy of an element Xi is defined as ... 

In this work, we deal with observational data, therefore probabilities are estimated by means of relative frequencies. The cluster index C(S ) of a set S of k elements is defined as the ratio between the integration I S) of S and the mutual information between S and the rest of the system U — S. The integration of S is defined as ... 

Since C is defined as a ratio, it is undefined in all those cases where M S-,U— S) vanishes. In this case, the subset S is statistically independent from the rest of the system and it has to be analyzed separately. As C S) scales with the size of S, cluster index values of systems of different size need to be normalized. To this aim, a reference system is defined, i.e., the homogeneous system Uh, that keep some statistical properties of the original system but does not contain clusters. There are several ways to generate Uh, which will be discussed in Sect. 4. In general, for each subsystem size of Uh the average integration Ih and the average mutual information Mh are computed. The cluster index value of S is normalized by means of the appropriate normalization constant ... 

Furthermore, to assess the significance of the differences observed in the cluster index values, a statistical index Tc is computed ... 

We now describe a relatively simple MD model of a low-index crystal surface, which was conceived for the purpose of studying the rate of mass transport (8). The effect of temperature on surface transport involves several competing processes. A rough surface structure complicates the trajectories somewhat, and the diffusion of clusters of atoms must be considered. In order to simplify the model as much as possible, but retain the essential dynamics of the mobile atoms, we will consider a model in which the atoms move on a "substrate" represented by an analytic potential energy function that is adjusted to match that of a surface of a (100) face-centered cubic crystal composed of atoms interacting with a Lennard-Jones... 

In this Section we want to present one of the fingerprints of noble-metal cluster formation, that is the development of a well-defined absorption band in the visible or near UV spectrum which is called the surface plasma resonance (SPR) absorption. SPR is typical of s-type metals like noble and alkali metals and it is due to a collective excitation of the delocalized conduction electrons confined within the cluster volume [15]. The theory developed by G. Mie in 1908 [22], for spherical non-interacting nanoparticles of radius R embedded in a non-absorbing medium with dielectric constant s i (i.e. with a refractive index n = Sm ) gives the extinction cross-section a(o),R) in the dipolar approximation as ... 

Figure 6. Absorption spectra of spherical non-interacting nanoclusters embedded in no absorbing matrices (a) effect of the size for Ag nanoclusters in silica (b) effect of the matrix for R = 2.5 nm Au clusters (the refractive index n = and the position of the plasma resonance are reported for each considered matrix) (c) effect of the cluster composition for i = 5 nm noble-metal clusters (Ag, Au, Cu) in silica. (Reprinted from Ref [1], 2005, with permission from Italian Physical Society.)...

J. Jurs and R.G. Lawson, New index for clustering tendency and its application to chemical problems. Chemom. Intell. Lab. Syst., 10 (1991) 81-83. 

Many other subjects are important to achieve successful pattern recognition. To name only two, it should be investigated to what extent outliers are present, because these can have a profound influence on the quality of a model and to what extent clusters occur in a class (e.g. using the index of clustering tendency of Section 30.4.1). When clusters occur, we must wonder whether we should not consider two (or more) classes instead of a single class. These problems also affect multivariate calibration (Chapter 36) and we have discussed them to a somewhat greater extent in that chapter. 

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. 

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