Euclidean distance calculation

The problem lies in the model. The Euclidean distance calculation is inappropriate for use with correlated variables because it is based only on pairwise comparisons, without regard to the elongation of data point swarms along particular axes. In effect, Euclidean distance imposes a spherical constraint on the data set (18). When correlation has been removed from the data, (by derivation of standardized characteristic vectors) Euclidean distance and average-linkage cluster analysis return the three groups. 

A set of molecules will thus appear as a set of curves drawn across the plot (Figure M2). This transformation preserves the means, the variances and the Euclidean distances calculated from the original variables x. However, the curves are not invariant with respect to the order of the variables in general, the low frequencies (xi, X2, X3) are distinguished more readily on the plot than the high frequencies (Xp, Xp i, Xp 2). For this reason, it is better to associate the most important variables with the low frequencies [Todeschini, Consonni et al, 1998]. 

The main structural parameters of the fully optimized geometries of the eight (NH3)3M"(H20) o>mptex studied are given in Table 8. Table 9 gathers the Euclidean distances calculated using Eq. (25) between the eight metal-substituted models of CA. 

Figure 1. Result of MDS employing Euclidean distance calculation.

This is done by calculating the Euclidean distance between the input data vector Xc and the weight vectors Wj of all neurons ... 

For n = 2, this is the familiar -space Euclidean distance. Similarity values, are calculated as... 

A mathematically very simple classification procedure is the nearest neighbour method. In this method one computes the distance between an unknown object u and each of the objects of the training set. Usually one employs the Euclidean distance D (see Section 30.2.2.1) but for strongly correlated variables, one should prefer correlation based measures (Section 30.2.2.2). If the training set consists of n objects, then n distances are calculated and the lowest of these is selected. If this is where u represents the unknown and I an object from learning class L, then one classifies u in group L. A three-dimensional example is given in Fig. 33.11. Object u is closest to an object of the class L and is therefore considered to be a member of that class. 

The degree to which one animal is like another can then be measured by calculating the Euclidean distance between their sets of properties. 

Calculate how similar the sample pattern is to the weights vector at each node in turn, by determining the Euclidean distance between the sample pattern and the weights vector. 

Each value in the chosen sample pattern is compared in turn with the corresponding weight at the first node to determine how well the pattern and weights vector match (Figure 3.9). A numerical measure of the quality of the match is essential, so the difference between the two vectors, dpq, generally defined as the squared Euclidean distance between the two, is calculated ... 

Points with a constant Euclidean distance from a reference point (like the center) are located on a hypersphere (in two dimensions on a circle) points with a constant Mahalanobis distance to the center are located on a hyperellipsoid (in two dimensions on an ellipse) that envelops the cluster of object points (Figure 2.11). That means the Mahalanobis distance depends on the direction. Mahalanobis distances are used in classification methods, by measuring the distances of an unknown object to prototypes (centers, centroids) of object classes (Chapter 5). Problematic with the Mahalanobis distance is the need of the inverse of the covariance matrix which cannot be calculated with highly correlating variables. A similar approach without this drawback is the classification method SIMCA based on PC A (Section 5.3.1, Brereton 2006 Eriksson et al. 2006). 

Similarity and Distance. Two sequences of subgraphs m and n such as those in Table 1 have the property that there is a built-in one-to-one correspondence between the elements of one sequence (m,) and those of the other (/i,). Accordingly, it is straightforward to calculate various well-known (17) measures of the distance d between the sequences, e.g. Euclidean distance [2,( Wi city block distance... 

The distance d between two patterns i and j in the multidimensional feature space is calculated according to the Euclidean distance definition ... 

Two otiKTpoints must be made relative to how the distances are measured. The first is fet distance needs to foe defined. A common metric used in Cartesian crardinatc sj stems is Euclidean distance, where the distance between twofgjints (.Vp j/p jTj) and ix, y, in three-dimensional space is calculated as... 

The infomsdon submitted to the computer program includes the pre-processed daa, the metric for classification (e.g.. Euclidean distance), and the choice of cisatering method (e.g., single link). Intercluster distances are calculated and used to construct a dendrogram. 

Single-link clustering and Euclidean distance are used in the calculations. 

With KNN, the predicted class of an unknown sample is assigned as the class of the sample(s) nearest to it in multidimensional space. In this book, Euclidean distances are used to measure the nearness between samples in row space. The Euclidean distance between samples x and j is calculated in nvars dimensions as... 

Prior to analysis, the Raman shift axes of the spectra were calibrated using the Raman spectrum of 4-acetamidophenol. Pretreatment of the raw spectra, such as vector normalization and calculation of derivatives were done using Matlab (The Mathworks, Inc.) or OPUS (Bruker) software. OPUS NT software (Bruker, Ettlingen, Germany) was used to perform the HCA. The first derivatives of the spectra were used over the range from 380 cm-1 to 1700 cm-1. To calculate the distance matrix, Euclidean distances were used and for clustering, Ward s algorithm was applied [59]. 

Fig. 4.3. Dendrogram resulting from cluster analysis containing 91 spectra from 15 tree species (see also Table 4.2). Cluster analysis was done on first derivatives over the spectral range 380 cm-1 to 1700 cm-1). The distance matrix was calculated using Euclidean distance and Ward s algorithm was applied for clustering. Spectra were measured after decomposition of carotenoid molecules with 633 nm irradiation. For example, spectra of each species are shown in Fig. 4.1. Reprinted with permission from [52]...

Finally the EUCLIDean distances are calculated from the z values rather than from the raw x values. At the end of the process the mutual distances of all n objects are arranged in a squared array called a distance matrix D. This (n, w)-matrix is normally symmetrical with zero values on the main diagonal ... 

The diversity between structures has been calculated by the formula I as Euclidean Distance (D). 

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