Weighted Euclidean metric

A weighted Euclidean metric is defined by the weighted scalar product ... 

Fig. 32.3. Effect of a weighted metric on distances, (a) representation of a circle in the space 5 defined by the usual Euclidean metric, (b) representation of the same circle in the space S defined by a weighted Euclidean metric. The metric is defined by the metric matrix W.

Distances with C = 1 are especially useful in the classification of local data as simple as in Fig. 5-12, where simply d( 1, 2) = a + b. They are also known as Manhattan, city block, or taxi driver metrics. These distances describe an absolute distance and may be easily understood. With C = 2 the distance of Eq. 5-7, the EUCLIDean distance, is obtained. If one approaches infinity, C = oo, in the maximum metric the measurement pairs with the greatest difference will have the greatest weight. This metric is, therefore, suitable in outlier recognition. 

No other a priori assumptions about the form or the structure of the function will be made. For a given choice of g. Kg) in Eq. (1) provides a measure of the real approximation error with respect to the data in the entire input space X. Its minimization will produce the function g (x) that is closest to G to the real function, /(x) with respect to the, weighted by the probability P(x,y) metric p.. The usual choice for p is the Euclidean distance. Then 1(g) becomes the L -metric ... 

The Euclidean distance is the best choice for a distance metric in hierarchical clustering because interpoint distances between the samples can be computed directly (see Figure 9.6). However, there is a problem with using the Euclidean distance, which arises from inadvertent weighting of the variables in the analysis that occurs... 

Metrical MDS operates on an input matrix of dissimilarities, or distances, between pairs of samples, giving as a result a matrix of coordinates whose configuration minimizes a loss function. This method presents an optimization phase, which can be performed with a variety of loss functions to be considered, and other possible variations of the methods concern the input distance matrix, which can be calculated according to different weights and criteria. When the Euclidean distance is considered, the classical MDS, also known as the principal coordinates analysis, consists in performing a PC A on the double-centred distance matrix and then rotating the solution so that the stress criterion S is minimized ... 

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