Euclidean distance definition

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

The manner in which sample-to-sample resemblance is defined is a key difference between the various hierarchical clustering techniques. Sample analyses may be similar to one another in a variety of ways and reflect interest in drawing attention to different underlying processes or properties. The selection of an appropriate measure of similarity is dependent, therefore, on the objectives of the research as set forth in the problem definition. Examples of different similarity measures or coefficients that have been used in compositional studies are average Euclidean distance, correlation, and cosine. Many others that could be applied are discussed in the literature dealing with cluster analysis (15, 18, 19, 36, 37). 

In this case the distance cannot be calculated using the Euclidean distance (13.1) because the number of dimensions might not be the same for all points under test. Since the number of dimensions influences the value of the distance, it would require the definition of several sets of membership functions, one per possible number of dimensions. The number of dimensions cannot be predicted beforehand because it will depend on the maximum number of references, the distance to the references and the sensitivity of the mobile receiver. To cope with this, the distance will be calculated using (13.2) ... 

A definitive account of similarity coefficients is provided by Sneath and Sokal. 9 They describe four main classes of similarity coefficients distance, association, correlation, and probabilistic coefficients. The most common distance measure is the Euclidean distance, and many nearest-neighbor searching algorithms use this to measure the degree of resemblance between pairs Euclidean distance is given by... 

For a formal definition of the Hausdorff distance, first we shall review some relevant concepts. We assume that A and B are subsets of a set X, and for points of W a distance function is already defined. For example, if X is the ordinary, three-dimensional Euclidean space and if the points a and b of A are represented by their three Cartesian coordinates [, 2) 3) and respectively, where a and b can be written as... 

After the new inner product definition is introduced, the related quantities, the length of a vector and the distance between the vectors, ate defined in exactly the same way as in the Euclidean space. Also, the definitions of the orthogonality and the Schwartz inequality remain unchanged. 

One must be aware that nowadays the mathematical concept of distance has evolved into an intricate labyrinth of alternative definitions and variants however, one can safely rely on the classic Euclidean concepts for practical QS purposes. From the QS point of view, any DF can be studied as a function belonging to a vector semispace. " Furthermore, DF can be seen as vectors belonging to infinite-dimensional Hilbert semispaces and thus can be also subject to comparative measures of distances and angles between the two of them. A pair of DF may, in this way, be considered as vectors subtending an angle a and situated in a plane... 

One of the most intuitive ways to describe how cluster analysis works in practice is by referring to the agglomerative hierarchical cluster analysis (HCA) method. Beside the common preliminary steps already discussed, that is definition of the metric (Euclidean, Mahalanobis, Manhattan distance, etc.) and calculation of the distance matrix and the corresponding similarity matrix, the analysis continues according to a recursive procedure such as... 

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