Weighted Euclidean distance

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

It can be regarded as a special case of the squared weighted Euclidean distance (Section 30.2.2.1), A property of (weighted) Euclidean distance functions is that the distances between row-items D are invariant under column-centering of the table X ... 

The Chi-square distance can be seen as a weighted Euclidean distance on the transformed data ... 

The Mahalanobis distance is simply a weighted Euclidean distance, where each of the dimensions is inversely weighted by its overall variance in the calibration data. As a result, deviations in a dimension that has high variance in the calibration data will not be weighted as much as deviations in a dimension that has low variance in the calibration data. With this in mind, it becomes clear that the Mahalanobis distances of the two unknowns shown in Figure 12.15 from the origin are not equal. Furthermore, it becomes qualitatively apparent that unknown 2 becomes closer to the class mean than unknown 1, if Mahalanobis distance is used instead of Euclidean distance. 

If the variables have been measured in different units, then it may be necessary to scale the data to make the values comparable. " An equivalent procedure is to compute a weighted Euclidean distance. 

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

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 ... 

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.

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 ... 

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. 

The new unit must be inserted between um and one of the units to which it is connected because the structure of the network as a series of connected simplexes must be maintained. To determine which of the neighbors this is, the weights at each of the direct neighbors are checked and the one whose weights are most different from the weights at um, as measured by the Euclidean distance between them, is selected. The new node is inserted into the network and joined ... 

However, the Euclidean distance is only one of many distance measures that can be used. For example, it may be desirable to weight certain measurement variables (dimensions) that are more reliable. This can be achieved by applying different weights to the terms in parentheses in Equation 4.1. 

Although Euclidean and Mahalanobis distances are the ones most commonly used in analytical chemistry applications, there are other distance measures that might be more appropriate for specific applications. For example, there are standardized Euclidean distances, where each of the dimensions is inversely weighted by the standard deviation of that dimension in the calibration data (standard deviation-standardized), or the range of that dimension in the calibration data (range-standardized). 

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. 

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... 

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 Euclidean distance from input vector (0,1,1,0) and the weights to Group A node is... 

Recently, a number of modifications of the classical methods have appeared that incorporate the spatial distance among pixels as an addihonal criterion in the clustering schemes. Thus, similarity measures based on spectral distances, such as p, can be weighted incorporating pixel neighboring informahon for example, the Euclidean distance can be redefined as ... 

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