Similarity Euclidean distance

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

So far we have been considering leverage with respect to a point s Euclidean distance from an origin. But this is not the only measure of distance, nor is it necessarily the optimum measure of distance in this context. Consider the data set shown in Figure E4. Points C and D are located at approximately equal Euclidean distances from the centroid of the data set. However, while point C is clearly a typical member of the data set, point D may well be an outlier. It would be useful to have a measure of distance which relates more closely to the similarity/difference of a data point to/from a set of data points than simple Euclidean distance.The various Mahalanobis distances are one such family of such measures of distance. Thus, while the Euclidean distances of points C and D from the centroid of the data set are equal, the various Mahalanobis distances from the centroid of the data set are larger for point D than for point C. 

In the same way, in Fig. 30.4b, clusters G1 and G2 are closer together than G3 and G4 although the Euclidean distances between the centres are the same. All groups have the same shape and volume, but G1 and G2 overlap, while G3 and G4 do not. G1 and G2 are therefore more similar than G3 and G4 are. 

Fig. 30.5. The point i is equidistant to i and i" according to the Euclidean distances (Z) - and Z) ) but much closer to i (cos 0, ) than to i" (cos 0 ), when a correlation-based similarity measure is applied.

Manhattan distances can be used also for continuous variables, but this is rarely done, because one prefers Euclidean distances in that case. Figure 30.6 compares the Euclidean and Maiihattan distances for two variables. While the Euclidean distance between i and i is measured along a straight line connecting the two points, the Manhattan distance is the sum of the distances parallel to the axes. The equations for both types of distances are very similar in appearance. In fact, they both belong to the Minkowski distances given by ... 

The similarities between all pairs of objects are measured using one of the measures described earlier. This yields the similarity matrix or, if the distance is used as measure of (dis)similarity, the distance matrix. It is a symmetrical nx matrix containing the similarities between each pair of objects. Let us suppose, for example, that the meteorites A, B, C, D, and E in Table 30.3 have to be classified and that the distance measure selected is Euclidean distance. Using eq. (30.4), one obtains the similarity matrix in Table 30.4. Because the matrix is symmetrical, only half of this matrix needs to be used. 

Similarity matrix (based on Euclidean distance) for the objects from Table 30.3... 

The Euclidean distance from the model is then obtained, similarly to eq. (33.14)... 

The technology of proximity indices has been available and in use for some time. There are two general types of proximity indices (Jain and Dubes, 1988) that can be distinguished based on how changes in similarity are reflected. The more closely two patterns resemble each other, the larger their similarity index (e.g., correlation coefficient) and the smaller their dissimilarity index (e.g., Euclidean distance). A proximity index between the ith and th patterns is denoted by D(i, j) and obeys the following three relations ... 

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. 

Once a dimensionality for the map and the type of local measure to be used have been chosen, training can start. A sample pattern is drawn at random from the database and the sample pattern and the weights vector at each unit are compared. As in a conventional SOM, the winning node or BMU is the unit whose weights vector is most similar to the sample pattern, as measured by the squared Euclidean distance between the two. 

In 1980, Carbo et al. were the first to express molecular similarity using the electron density [17]. They introduced a distance measure between two molecules A and B in the sense of a Euclidean distance in the following way ... 

A different class of indices is the D indices, where similarity is expressed as a distance. With these indices, perfect similarity is characterized by a zero distance. The best known is the Euclidean distance, introduced in Equation 16.5. Again, different connections have been found to exist between C- and D-class indices [59-62]. 

Euclidean distance City block (Manhattan) distance Minkowski distance Correlation coefficient (cos a), similarity... 

FIGURE 2.10 Euclidean distance and city block distance (Manhattan distance) between objects represented by vectors or points xA and xB. The cosine of the angle between the object vectors is a similarity measure and corresponds to the correlation coefficient of the vector... 

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). 

The distance between object points is considered as an inverse similarity of the objects. This similarity depends on the variables used and on the distance measure applied. The distances between the objects can be collected in a distance matrk. Most used is the euclidean distance, which is the commonly used distance, extended to more than two or three dimensions. Other distance measures (city block distance, correlation coefficient) can be applied of special importance is the mahalanobis distance which considers the spatial distribution of the object points (the correlation between the variables). Based on the Mahalanobis distance, multivariate outliers can be identified. The Mahalanobis distance is based on the covariance matrix of X this matrix plays a central role in multivariate data analysis and should be estimated by appropriate methods—mostly robust methods are adequate. 

Spectral similarity search is a routine method for identification of compounds, and is similar to fc-NN classification. For molecular spectra (IR, MS, NMR), more complicated, problem-specific similarity measures are used than criteria based on the Euclidean distance (Davies 2003 Robien 2003 Thiele and Salzer 2003). If the unknown is contained in the used data base (spectral library), identification is often possible for compounds not present in the data base, k-NN classification may give hints to which compound classes the unknown belongs. 

A cluster analysis of the amino acid structures by PCA of the A -matrix is shown in Figure 6.5a note that PCA optimally represents the Euclidean distances. The score plot for the first two principal components (preserving 27.1% and 20.5% of the total variance) shows some clustering of similar structures. Four structure pairs have identical variables 1 (Ala) and 8 (Gly), 5 (Cys) and 13 (Met), 10 (He) and 11 (Leu), and 16 (Ser) and 17 (Thr). Objects with identical variables of course have identical scores, but for a better visibility the pairs have been artificially... 

Instead of using the Euclidean distance, also other distance measures can be considered. Moreover, another power than 2 could be used for the membership coefficients, which will change the characteristics of the procedure (degree of fuzzification). Similar to fc-means, the number of clusters k has to be provided as an input, and the algorithm also uses cluster centroids Cj which are now computed by... 

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... 

Similar compounds were selected for further testing using BCUT descriptors and Euclidean distance to identify the untested compounds closest to the initial hit (24). 

The statistical similarity analysis was performed based on determination of Euclidean distance between hypothetical and catalyst profile according to the following formula ... 

Initially cluster analysis defines a measure of simUarity given by a distance or a correlation or the information content Distance can be measured as euclidean distance or Mahalanobis distance or Minkowski distance. Objects separated by a short distance are recognized as very similar, while objects separated by a great distance are dissimilar. The overall result of cluster analysis is reported as a dendrogram of the similarities obtained by many procedures. 

Once the classification space has been defined, it is then necessary to define the distance in the space that will be used to assess the similarity of prediction samples and calibration samples. The most straightforward distance that can be used for this purpose is the Euclidean distance between two vectors (Dab), which is defined as ... 

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