Euclidean distance measures

It should be noted that we have to envisage two types of distances at two levels (of the two-story structure) one on the level of the primordial network and the other on the level of the web of hazy lumps. Normally, if we consider the Euclidean distance measure, this distance tends to zero as the centers of the lumps approach each other. On the other hand, if we take, for instance, the... 

Other similarity coefficients used in similarity studies include the cosine coefficient, and the Hamming and Euclidean distance measures [7], Similarity coefficients can also be applied to vectors of attributes where the attributes are real numbers, for example, topological indices or physiochemical properties. 

For radial basis function networks, each hidden unit represents the center of a cluster in the data space. Input to a hidden unit in a radial basis function is not the weighted sum of its inputs but a distance measure a measure of how far the input vector is from the center of the basis function for that hidden unit. Various distance measures are used, but perhaps the most common is the well-known Euclidean distance measure. 

The most common data distance matrix is the EucUdean distance matrix, that is, the matrix obtained by using the Euclidean distance measure. A very important Euclidean distance matrix is the —> geometry matrix, where rows and columns represent the molecule atoms, and matrix elements are interatomic distances calculated from the (x, y, and z) spatial atomic coordinates. 

To analyse a simple plot of this type in order to distinguish between circles and squares , one would calculate an Euclidean distance measurement between clusters. Known samples would be a training set (calibration set) of samples in order to locate each cluster. An unknown sample can be identified based on its response to the two sensors and the relative distance to each of the known clusters. Already, with just the addition of one sensor, several analytes can be characterized using a bivariate plot. 

Figure 10.5 Cluster analysis, (a) A combination of unsupervised clustering and heatmap visualization. The Euclidean distance measure and Ward linkage are used. Peptide intensities are log-transformed and normalized to zero mean unit variance (row by row). The profiles of 27 non-small-cell lung cancer patients are intermingled with those of 13 healthy controls (columns) (b) Supervised analysis using 11 peptides with Benjamini-Hochberg adjusted p-values <0.001 results in two distinctive branches at the root of the tree. Two cancer profiles are grouped with those of the healthy controls. All but one of the peptides are upregulated in cancer samples.

Haddad et al. [8] developed a metric for odorant comparison based on a chemical space constracted from 1664 molecular descriptors. A refined version of this metric was devised following the elimination of redundant descriptors. The study included the comparison with models previously reported for nine datasets. The final, so-called multidimensional metric, based on Euclidean distances measured in a 32-descriptor space, was more efficient at classifying odorants cf. reference models previously reported. Thus, this study demonstrated the use of structural similarity for the classification of odors in multidimensional space. 

Methods Based on Selected Wavelengths Use of Non-Euclidean Distance Measures. . 308... 

The competitive learning algorithm, involving the computation of the Euclidean distance measure (z - for each cell, exploits the same mechanism. The monostable unit is used to generate a time period T oc (z-w). The result, according to eqn. (1), is the required squared difference (plus an offset that can be easily subtracted). 

Vector Quantization (VQ) technique is used in speaker recognition system. This technique is easy to implement through the use of the Euclidean distance measure. However, it provides less accuracy in terms of performance if only this algorithm was used for the speaker verification system. This algorithm for VQ is base on [3] with large data training required. They used 100 speakers and able to achieve 56% performance improvement using text dependent mode compared to independent. 

Table 3 Component Identification in the Ultraviolet Spectral Range by Using a Fuzzy Criterion According to equation (24) and by Use of the Euclidean Distance Measure (from Ref. 5)...

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