Euclidean metric

This corresponds with a choice of factor scaling coefficients a = 1 and p = 0, as defined in Section 31.1.4. Note that classical PCA implicitly assumes a Euclidean metric as defined above. Let us consider the yth coordinate axis of column-space, which is defined by a p-vector of unit length of the form ... 

The same idea can be developed in the case of a non-Euclidean metric such as the city-block metric or L,-norm (Section 31.6.1). Here we find that the trajectories, traced out by the variable coefficient kj are curvilinear, rather than linear. Markers between equidistant values on the original scales of the columns of X are usually not equidistant on the corresponding curvilinear trajectories of the nonlinear biplot (Fig. 31.17b). Although the curvilinear trajectories intersect at the origin of space, the latter does not necessarily coincide with the centroid of the row-points of X. We briefly describe here the basic steps of the algorithm and we refer to the original work of Gower [53,54] for a formal proof. 

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 in these spaces should be based upon an Zj or city-block metric (see Eq. 2.18) and not the Z2 or Euclidean metric typically used in many applications. The reasons for this are the same as those discussed in Subheading 2.2.1. for binary vectors. Set-based similarity measures can be adapted from those based on bit vectors using an ansatz borrowed from fuzzy set theory (41,42). For example, the Tanimoto similarity coefficient becomes... 

The general line-element expression (9.28) allows one to envision possible geometries with fto/i-Euclidean metric [i.e., failing to satisfy one or more of the conditions (9.27a-c)] or with variable metric [i.e., with a matrix M that varies with position in the space, M = M( i )> a Riemannian geometry that is only locally Euclidean cf. Section 13.1]. However, for the present equilibrium thermodynamic purposes (Chapters 9-12) we may consider only the simplest version of (9.28), with metric elements (R R,-) satisfying the Euclidean requirements (9.27a-c). 

It is a remarkable fact that properties (13.4a-c) are necessary and sufficient to give Euclidean geometry. In other words, if any rule can be found that associates a number (say, (X Y)) with each pair of abstract objects ( vectors X), Y)) of the manifold in a way that satisfies (13.4a-c), then the manifold is isomorphic to a corresponding Euclidean vector space. We introduced a rather unconventional rule for the scalar products (X Y) [recognizing that (13.4a-c) are guaranteed by the laws of thermodynamics] to construct the abstract Euclidean metric space Ms for an equilibrium state of a system S, characterized by a metric matrix M. This geometry allows the thermodynamic state description to be considerably simplified, as demonstrated in Chapters 9-12. 

This generalized gradient is based on a Riemannian metric defined on the interior of the concentration space x > 0 X x = 1, fc = 1,. . . , n, which replaces the conventional Euclidean metric. We compare the definitions of the two inner products ... 

It should be noted that the form of the objective function presented in Eq. (2) is not limited to this Euclidean metric and can accommodate almost any pairwise similarity measure. 

The k nearest neighbor routine is conceptually the simplest. An unknown pattern is assigned to the class to which the majority of its nearest neighbors belong. The metric that is used to determine proximity is ordinarily the Euclidean metric, but any metric can be used. 

The mean value for each variable j, for all objects in cluster L is denoted by Bl,p L K). The number of objects residing in cluster L is Rl-The distance, between the I th object and the centre or average of each cluster is given by the Euclidean metric. 

The distance, Di between the fth object and the centre or average of each cluster is given by the Euclidean metric,... 

To specify the directions of two different vectors at nearby points it is necessary to define tangent vectors at these points. Stated in different terms, at each point of space-time, known as the contact point, there is an associated tangent Minkowski space. The theory of these spaces together with the underlying space becomes a Riemannian geometry if a Euclidean metric is introduced in each tangent space by means of a differential quadratic form. ... 

Euclidean Metric of a Five-dimensional Associated Space. 61... 

In an underlying space the tensor g j gives a Riemannian metric. Since this metric is invariantly connected with our quadric we suspect that the Riemannian metric is exactly the non-Euclidean metric that appears as absolute image of our quadric. 

A non-Euclidean or Caleyan metric is readily defined by means of a "tangential" Euclidean metric. The general concept of a tangential metric according to E. Cartan (Bibl. 1928, 1, chap.IV) is as follows Two types of measurement with the same gij in a given point are called tangential in the point concerned. 

In each point q of our space there exists a Euclidean metric with respect to which the quadric is exactly a sphere with radius 1 and centre q. The infinitely remote hyperplane of this Euclidean space is the polar hyperplane of q with respect to the quadric. 

With respect to the Euclidean metric with arc element... 

Each Riemann metric is known to produce a Euclidean metric in each tangent space. With respect to the mode of measurement (3), the surface... 

Klein, Y., Efrati, E., Sharon, E. Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315(5815), 1116-1120 (2007)... 

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