Non-Euclidean distance

An application of (27) is the proposition that the non-Euclidean distance of two points and of a tangent space is conserved on displacement A. The non-Euclidean distance of both points is determined by... 

Methods Based on Selected Wavelengths Use of Non-Euclidean Distance Measures. . 308... 

Thus we can reduce our problem to the investigation of the random walk on the Lobachevsky plane, where the non-Euclidean distance between ends is the topological invariant. 

In non-metric MDS the analysis takes into account the measurement level of the raw data (nominal, ordinal, interval or ratio scale see Section 2.1.2). This is most relevant for sensory testing where often the scale of scores is not well-defined and the differences derived may not represent Euclidean distances. For this reason one may rank-order the distances and analyze the rank numbers with, for example, the popular method and algorithm for non-metric MDS that is due to Kruskal [7]. Here one defines a non-linear loss function, called STRESS, which is to be minimized ... 

Each object or data point is represented by a point in a multidimensional space. These plots or projected points are arranged in this space so that the distances between pairs of points have the strongest possible relation to the degree of similarity among the pairs of objects. That is, two similar objects are represented by two points that are close together, and two dissimilar objects are represented by a pair of points that are far apart. The space is usually a two- or three-dimensional Euclidean space, but may be non-Euclidean and may have more dimensions. 

Because these vectors live in an -dimensional hypercubic space, the use of non-integer distance measures is inappropriate, although in this special case the square of the Euclidean distance is equal to the Hamming distance. 

An important question is whether the proximity measures are compatible with those of these references addresses the important issue of whether the proximity measure is compatible with embedding in a Euclidean space. For example, satisfying the distance axioms does not in itself guarantee that any distance matrix associated with a given set of molecules be compatible as the distance axioms are still satisfied in a non-Euclidean space. Gower has written extensively on this important issue, and his work should be consulted for details (89-91). Benigni (92), and Carbo (67) have also contributed interesting approaches in this area. 

We have presented here only the non-zero values of the coordinates. In both cases the Euclidean distance between the variants inside these pairs, [If,1(A/ i - N )2]112, is equal to 21/2 bx - b2 and tends to zero at b4 - b2. For the other pairs of vertices the situation is different. 

CA is commonly used to investigate and display compound similarity, however, it can also be used for descriptor selection from a larger set. CA relies on the fact that similarity and dissimilarity among two points in multi-dimensional space can be quantified by calculating their inter-point distance. The most common measure being the Euclidean distance. Both hierarchical and non-hierarchical approaches exist. CA is often used complementary to PCA. 

The spacing is a measure of the relative distance between consecutive (nearest neighbor) solutions in the non-dominated set. The maximum spread is the length of the diagonal of the hyper-box formed by the extreme function values in the non-dominated set. For two-objective problems, this metric refers to the Euclidean distance between the two extreme solutions in the /-space. It is given by... 

Euclidean distance is essentially a measure of positive, linear correlations however, other similarity measures may be used for clustering. For example, mutual information, an information theoretic measure, may be used to capture positive, negative, and non-linear correlations all at the same time. A pictorial explanation of the concept of mutual information along with instructions on doing calculations can be found in [18]. Mutual information is based on Shannon entropy (H = — Lpi log2 pp see above explanation of entropy) and is calculated as follows M(X, Y) = H(X) + H(Y) — Ff(X, Y),... 

The alternative non-Euclidean geometry bears the same relation to Euclidean geometry as the geometry of curved surfaces bears to the geometry of the flat plane. The distance between two infinitesimally close points in the plane is given by the equation... 

Recognition of non-Euclidean geometries creates the new problem of describing the position and motion of finite objects in curved space, a problem which does not occur in Euclidean spaces that extend uniformly to inhnity. An object such as the distance between two points. 

Geometry alone could not produce a theory of gravity, free of action at a distance, until physics managed to catch up with the ideas of Riemann. The development of special relativity, after discovery of the electromagnetic held, is described. It requires a holistic four-dimensional space-time, rather than three-dimensional Euclidean space and universal time. Accelerated motion, and therefore gravity, additionally requires this space-time to be non-Euclidean. The important conclusion is that relativity, more than a theory, is the only consistent description of physical reality at this time. Schemes for the unihed description of the gravitational and electromagnetic helds are briehy discussed. 

We use the Bulienkov parametric bound water model to describe the protein water shell. Water molecules are bonded into H-bond network. This H-bond network can be performed as a system of hexacycles in twist-boat conformation. The twist-boat hexacycles provide non-Euclidean geometry parameters as it should be in crystal stracture such an ice [18]. In ice stracture the internal parameters of all hexacycles are equal-intermolecular distances, valence and torsion angles are constant and can vary only by a thermal motion. So if any hexacycle system is constructed using only twist-boat pattern then geometrical parameters must distort [23]. 

Figure 10.5 Cluster analysis, (a) A combination of unsupervised clustering and heatmap visualization. The Euclidean distance measure and Ward linkage are used. Peptide intensities are log-transformed and normalized to zero mean unit variance (row by row). The profiles of 27 non-small-cell lung cancer patients are intermingled with those of 13 healthy controls (columns) (b) Supervised analysis using 11 peptides with Benjamini-Hochberg adjusted p-values <0.001 results in two distinctive branches at the root of the tree. Two cancer profiles are grouped with those of the healthy controls. All but one of the peptides are upregulated in cancer samples.

A number of different methods may be described as looking for nearest neighbours, e.g., cluster analysis (see Section 5.4), but in this book the term is applied to just one approach, k-nearest-neighbour. The starting point for the fe-nearest-neighbour technique (KNN) is the calculation of a distance matrix as required for non-linear mapping. Various distance measures may be used to express the similarity between compounds but the Euclidean distance, as defined in eqn (4.2) (reproduced below), is probably most common ... 

It can be observed from this equation that, compared to the discrete vector distance, the Euclidean distances 5 s ) are weighted with the widths of the intervals A s. In case of equidistant measurement frequencies the interval widths and thus the weighting is constant. In case of non-equidistant measurement frequencies the weights vary depending on the interval widths and thus on the distribution of the measurement points. In the limiting case, for infinitesimal small interval widths, the discretized distance measure approaches the functional distance measure. 

Lastly, it is well to point out that similarity values that lie on the unit interval [0,1] of the real line can be obtained by transforming Euclidean distances, d, or non-Euclidean geodesic distances, d, using any one of a nnmber of different mathematical expressions, one possibility being... 

The aim of non-linear Sammon s mapping (NLM) is to represent a p-dimensional space, containing n objects, by a two-dimensional map that preserves the distances between the objects optimally. This optimization problem is not easy and requires extensive computing time because of the large number (2n) of parameters. The starting point is a distance matrix for the p-dimensional space with the Euclidean distance most widely applied. The initial 2n map co-ordinates can be chosen randomly but the scores of the first and second principal component may be used favourably. A mapping error E that reflects the differences of the distances between two objects i and J in p-space (c/,/) and in the 2-dimensional map (dij) has to be minimized (equation 35) ... 

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