Euclidean distances

Euclidean distance, see Distance, Euclidean Example data sets... 

Step 3. Similarity Matching Find the best matching prototype vector w. at time n based on the minimum-distance Euclidean criterion ... 

Partitional clustering using Euclidean distance as a measure of dissimilarity between pattern classes has been selected for the grouping of AE hits. 

To construct dissimilarity measures, one uses mismatches Here a + b is the Hamming (Manhattan, taxi-cab, city-block) distance, and a + h) is the Euclidean distance. 

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

Euclidean and Hamming distance measures of torsional similarity. 

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. 

Figure E4. Hypothetical data set illustrating that Euclidean distance is not an ideal metric for membership in a data set.

Corresponding elements in the two vectors of means are subtracted, and the differences are squared and added. The square root of the sum (15.21) is equal to the Euclidean distance in 15 dimensions separating the two points that represent the group means. This distance forms the base line in Fig. 4.20. 

Corresponding elements in the vector representing one particular sample and in the appropriate vector of means are worked up as in 2) to find the Euclidean distance between point i and its group mean (see lines marked with an asterisk ( ) in Table 4.16) this forms the second side of the appropriate triangle in Fig. 4.20. 

Interpretation Using Euclidean distances, the difference between the vendor s samples shows up nicely. (See data file SIEVEl.dat if some samples... 

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 ... 

In Euclidean space we define squared distance from the origin of a point x by means of the scalar product of x with itself ... 

In some cases, one wants to give larger weights to some variables. This leads to the weighted Euclidean distance ... 

It can be shown that the standardized Euclidean distance is the Euclidean distance of the autoscaled values of X (see further Section 30.2.2.3). One should also note that in this context the standard deviation is obtained by dividing by n, instead of... 

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. 

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