Euclidean group

For a free noninteracting spinning particle, invariance with respect to translations and rotations in three dimensional space, i.e., invariance under the inhomogeneous euclidean group, requires that the momenta pl and the total angular momenta J1 obey the following commutation rules... 

Nevertheless, there exists finite-additive probability which is invariant with respect to Euclidean group E(n) (generated by rotations and translations). Its values are densities of sets. 

If a light beam is considered propagating at c in Z, we obtain from Eqs. (770) the Lie algebra of the E(2) Euclidean group [6,11-20], which is a mathematical group with no physical meaning ... 

Equation (3.1) exhibits the symmetry of the homogeneous BZ reaction, namely equivariance under the Euclidean group SE(2) of all translations S and rotations R in the plane. The group multiplication of (Rj, Si) G SE 2) is given by... 

In section 3.2 we have seen how rigid rotations, meanders, and drifts are closely tied to equivariance with respect to the Euclidean group SE 2) of planar rotations and translations. We therefore prefer to not describe our curvature flows in terms of the position vector Z t,s) e C, directly. Instead, we work with the curvature scalar... 

B. Fiedler and D. Turaev. Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions. Arch. Ration. Mech. Anal., 145(2) 129-159, 1998. 

R. Murenzi, Wavelet Transforms Associated to the n-Dimensional Euclidean Group in Wavelets Time-Frequency Methods and Phase Space, (J.M. Combes, A. Grossman and Ph. Tchamitchian Eds), Springer-Verlag, Berlin, (1990), pp. 239-246... 

For applications to molecular spectra, the Euclidean groups will suffice. Suppose one takes the set of four members 1, -1, i, -e, where Further suppose that the operation of ordinary mul-... 

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

We first generalize the nomenclature. Consider a Euclidean d-dimensional lattice L, with translation group Gp. A frame, F, of L is defined to be a finite subset of (not necessarily contiguous) sites of L that is closed under (i) intersection, (ii) union, (iii) difference and (iv) operations g Gp. A block, Bp, is a specific assignment... 

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. 

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. 

In general, one maximizes between-cluster Euclidean distance or minimizes within-cluster Euclidean distance or variance. This really amounts to the same. As described by Bratchell [6], one can partition total variation, represented by T, into between-group (B) and within-group components (W). 

Since T and therefore also tr(T) is constant, minimizing tr(W) is equivalent to maximizing tr(B). It can be shown that tr(B) is the sum of squared Euclidean distances between the group centroids. 

A mathematically very simple classification procedure is the nearest neighbour method. In this method one computes the distance between an unknown object u and each of the objects of the training set. Usually one employs the Euclidean distance D (see Section 30.2.2.1) but for strongly correlated variables, one should prefer correlation based measures (Section 30.2.2.2). If the training set consists of n objects, then n distances are calculated and the lowest of these is selected. If this is where u represents the unknown and I an object from learning class L, then one classifies u in group L. A three-dimensional example is given in Fig. 33.11. Object u is closest to an object of the class L and is therefore considered to be a member of that class. 

A continuous connected group may be simply connected or multiply connected, depending on the topology of the parameter space. A subset of the euclidean space Sn is said to be k-fold connected if there are precisely k distinct paths connecting any two points of the subset which cannot be brought into each other by continuous deformation without going outside the subset. A schematic of four-fold connected space is shown in the lower diagram. 

Since the goal of SIMCA is to classify a new object jc, a measure for the closeness of the object to the groups needs to be defined. For this purpose, several proposals have been made in the literature. They are based on the orthogonal distance, which represents the Euclidean distance of an object to the PCA space (see Section 3.7.3). First we need to compute the score vector tj of x in theyth PCA space, and using Equation 5.19 and the group center Xj we obtain... 

HCA is a common tool that is used to determine the natural grouping of objects, based on their multivariate responses [75]. In PAT, this method can be used to determine natural groupings of samples or variables in a data set. Like the classification methods discussed above, HCA requires the specification of a space and a distance measure. However, unlike those methods, HCA does not involve the development of a classification rule, but rather a linkage rule, as discussed below. For a given problem, the selection of the space (e.g., original x variable space, PC score space) and distance measure (e.g.. Euclidean, Mahalanobis) depends on the specific information that the user wants to extract. For example, for a spectral data set, one can choose PC score space with Mahalanobis distance measure to better reflect separation that originates from both strong and weak spectral effects. 

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