Dissimilarity matrix

Generate an NxN dissimilarity matrix in which MilJ) contains the dissimilarity between molecules I and J. 

There are a number of models for polarization of heterogeneous systems, many of which are reviewed by van Beek (23). Brown has derived an exact, though unwieldly, series solution using point probability functions (24). For comparison to spectra for the thermoplastic elastomers of interest here, the most useful model seems to be the one derived by Sillars (25) and, in a slightly different form, by Fricke (26). The model assumes a distribution of geometrically similar ellipsoids with major radii, r-p and rj which are randomly oriented and randomly distributed in a dissimilar matrix phase. Only non-specific interactions between neighboring ellipsoids are included in the model. This model includes no contribution from the polarization of mobile charge carriers trapped on the interfacial surfaces. 

The matrix W(k) assesses the difference of Hasse diagrams induced by the two subsets of attributes with respect to a key element k. This matrix, which is at the heart of the analysis, is called the "dissimilarity-matrix", because the larger the matrix-entries are, the greater is the difference between the successor sets for the element k and hence between the Hasse diagrams (see for more details, below). We define the entry W(k, B, C) of matrix W to be ... 

The theoretical background of this dissimilarity matrix, which describes the influence of the attributes on the Hasse diagram, is given by Bruggemann and Carlsen, p. 61. 

Multidimensional scaling [70] is a method for obtaining the best low-dimensional representation of a high-dimensional data set. Normally, a two- or three-dimensional representation is required, since it can then be plotted and inspected visually for clusters. In the classical scaling technique, the low-dimensional representation is obtained by extracting the eigenvectors of the (Af xAf) dissimilarity matrix. However, it can be shown that this operation is equivalent to a PC A of the (ApXAp) covariance matrix, provided that the distances in the dissimilarity matrix are Euclidian or near-Euclidian [53]. Since the covariance matrix is invariably of smaller order than the dissimilarity matrix, PCA is to be preferred on computational grounds. The only exception is if the dissimilarity matrix is available but the covariance matrix is not, a circumstance that rarely arises in structural chemistry work. 

Table 2. The Similarity/Dissimilarity Matrix for Octane Isomers Based on Path Characterization...

In Table 4 we show the similarity/dissimilarity matrix based on orthogonal descriptors of Table 3. Observe how the degeneracy of the similarity/dissimilarity table for path numbers has almost completely disappeared. The most similar pairs of isomers are listed in Table 5 and are illustrated in Figure 13. 

Table 15. Similarity/Dissimilarity Matrix among the Nine Configurations of a Chain ...

In order to identify a possible segmentation, we first calculated cross-subject RV coefficients which were pooled into a square matrix of RVs. This matrix is then turned into a dissimilarity matrix of [1-RV] and submitted to hierarchical cluster analysis (Fig. 6.10). 

Figure 7.1 Transformation of the raw data (a) given by a fictitious subject into a similarity matrix (b), dissimilarity matrix (c), and a matrix of indicator variables (d).

The input of the MDS technique (Kruskal, 1964 Torgerson, 1958) is a dissimilarity matrix between the products. The aim is to seek a low dimensional space to... 

The aim of cluster analysis is to depict the dissimilarity among a set of products as a hierarchical tree (also called dendrogram) or achieve, on the basis of a dissimilarity matrix, a partitioning of the products into a given number of groups (Duda et al, 2000). 

The starting point is the overall dissimilarity matrix among prodncts A. The ontcome is a tree (or more precisely, a graph) that connects the various products (Fig. 7.4). The shortest path from one prodnct to another reflects the dissimilarity between these two products. 

Very likely, the original dissimilarity matrix A is not a tree distance. Thereupon, we undertake to approximate it by a tree distance. Several approximation methods are discussed in the literature. For a comprehensive review, we refer to Abdi (1990). 

The sorting data from each consumer were expressed as a dissimilarity matrix among products. These individual dissimilarity matrices were averaged over consumers. 

Thereafter, the average dissimilarity matrix was submitted to a non-metric MDS. With a Stress value equal to 0.096, a three-dimensional MDS configuration is retained. The three axes of the configuration recover respectively 58%, 24% and 18% of the total variance. 

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