Fuzzy metric distances

For all fuzzy sets, including three-dimensional functions of electron density-like continua provided with suitable membership functions, the differences between the corresponding fuzzy sets can be expressed by a metric based on a generalization of the Hausdorff distance. The basic idea is to take the ordinary Hausdorff distances h a) for the a-cuts of the fuzzy sets for all relevant a values, scale the Hausdorff distance h)a according to the a value, and from the family of the scaled Hausdorff distances, the supremum determines the fuzzy metric distance f A,B) between the fuzzy sets A and B. If, in addition, the relative positions of the fuzzy sets A and B are allowed to change, then the infimum of the f(A, B) values obtained for the various positionings determines a fuzzy metric of the dissimilarities of the intrinsic shapes of the two fuzzy sets. 

As can be easily proven by a simple modification of the proof presented here for the unsealed fuzzy Hausdorff metric, this scaled fuzzy Hausdorff distance is also a metric in the space of fuzzy subsets of the underlying set X. A proof is given in subsequent text. 

Of all works cited here that more close to our idea is the work by Chakraborty Qiakraborty (2006) which proposes a fuzzy distance for fuzzy numbers, which preserves a distance xm-certeinties. The authors put a natural question " if we do not know the numbers exactly how can the distance between them be an exact value ". At the same time, they criticise the use of the supreme, of the minimal or of any other candidate as absolute representative of the distance between two fuzzy numbers. They also consider the distance between two fuzzy numbers as a fuzzy number, saying that the distance between two numbers with xmcertainties must be a number with uncertainty. In addition, for fuzzy sets, Grzegorzewski (2004) uses a Hausdorff metric in the construction of a fuzzy metric, that unfortunately it does not preserve uncertainty. In their works they use metric spaces and topological spaces. 

We now detail a second approach to the evaluation of the degree of symmetry deficiency, from a different point of view, namely the fuzzy sets approach. Fuzzy set methods are especially suitable to reflect the Heisenberg uncertainty relation and other aspects of quantum chemistry " and computational chemistry. Symmetry in molecules relies on the concepts of distance and metric. For the approximate symmetries of fuzzy electron densities, fuzzy set methods, fuzzy distance or fuzzy metric, and fuzzy symmetry are of importance. In Section 3.3 below, we compare and generalize the two approaches. 

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... 

For the description of shape differences between fuzzy objects, such as molecular electron density clouds, it is useful to generalize the Hausdorff metric for fuzzy sets. The ordinary Hausdorff distance, a formal dis-... 

Take three fuzzy sets A, B, and C and their a-cuts G (a), Gg(a), and G ia), respectively, for each membership function value a. Assume that the a-cuts GJ.a), Gg(a and G(-(a) depend at least piecewise continuously on the a parameter from the unit interval [0,1], where the intervals of continuity have nonzero lengths and where continuity is understood within the metric topology of the underlying space X. For the three pairs formed from these three fuzzy sets, the ordinary Hausdorff distances h(G (a),Gg(a)), h(Gg(a), Gcia)), and h GJ,a ... 

For two molecules A and B, represented by fuzzy sets of electron densities, their inherent dissimilarities can be better measured by the scaled fuzzy Hausdorff-type distance f(A, B), where the relative positions of the molecules correspond to maximum superposition. The new variant fg (A,B) of the scaled fuzzy Hausdorff-type metric f(A,B) is defined as... 

For applications of the scaled fuzzy Hausdorff-type metric f p(A,B) for assessing the similarity of molecules, the f p(A,B) distance can be used as a dissimilarity measure. 

This measure df (A,B) of fuzzy dissimilarity is in fact a metric for fuzzy shapes. With the provision that volume of ordinary sets is replaced with the mass of fuzzy sets, all the steps in the proof of the metric properties of the ordinary SNDSM measure, described in Section VII, can be repeated for FSNDSM df A,B) in identical form, proving that the fuzzy set version FSNDSM of the SNDSM measure is, indeed, a metric. The fuzzy scaling-nesting dissimilarity measure FSDNSM dfJ A, B) provides a useful definition for distance between fuzzy sets, interpreted as a metric expressing dissimilarity in a formal space of fuzzy shapes, such as electronic densities of molecules. 

If the clusters are close it is possible that the hyperplane of equal membership will intersect the greater cluster. Some points of the greater cluster will be captured by its neighbor, as shown in Fig. 4. This represents a pathological situation that may be avoided by using a data-dependent (or adaptive) distance.With an adaptive metric the apparent sizes of clusters become equal. An adaptive distance may be induced by the radius or by the diameter of each fuzzy class A,. The diameter 6 of the fuzzy class Ai is defined as... 

Grzegorzewski, P. (2004). Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the hausdorff metric. Fuzzy Sets and Systems (148) 319-328. 

Pattern recognition techniques based on fuzzy objective function minimization use objective functions particular to different cluster shapes. Ways to approach the problem of correctly identifying the cluster s shape are the use of adaptive distances in a second run to change the shapes of the produced clusters so that all are unit spheres, and adaptive algorithms that dynamically change the local metrics during the iterative procedure in the original run, without the need of a second run. 

Fuzzy Principal Component Analysis was also performed on the structural features concerned with the interaction of carbon-hydrogen bonds and molyb-denum-oxo bonds. In addition to all possible atom-atom distances and the angles subtended thereof, several additional metrics were defined and tabu-lated. These include the distance R, p, and the angles ot, p, and y, as well as the dihedral MOCH, Scheme 1. Interactions between metal-oxo and carbon-hydrogen bonds are of importance with respect to microbial and industrial oxidation, and for these reasons molybdenum was the focus of this research. 

For a description of differences between fuzzy electron density clouds exhibiting approximate symmetries to various degrees, some fuzzy dissimilarity measure is needed. If in an underlying set X a metric is given, for example, the Pythagorean distance within the 3D Euclidean space then the distance between a point x of X, and a subset A of X is defined as the greatest lower bound d(x,A) = inia [d x, a)) of distances between points a of A and point x. 

For fuzzy electron density clouds, the fuzzy, commitment-weighted Hausdorff-type metric f(A,B) = sup g[o,i) ( -4 (a), Cb(q )) is a proper distance between fuzzy sets A and B ... 

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