Hausdorff metric

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-... 

Nevertheless, an unsealed fuzzy Hausdorff metric provides important insight into the generalization of concepts originally proposed and proved for crisp sets to fuzzy sets, and some features of the proof can be utilized in the proof of the metric properties of the scaled fuzzy Hausdorff-type metric described here. For this purpose, first a proof is presented for the metric properties of an unsealed fuzzy Hausdorff metric, equivalent to the metric proposed by Puri and Ralescu, followed by a proof of the metric properties of an a-scaled fuzzy Hausdorff-type metric, suitable for fuzzy electron densities. 

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. 

A comparison of sets A and A R,c) provides a measure of the symmetry aspect R for set A with respect to center c. In principle, any similarity measure is applicable for this comparison, for example, the Hausdorff metric hi A, A R,c)) provides a valid measure of the symmetry aspect R for set A with respect to center c. If the center of mass is chosen for c, then the simpler notations A R) and hi A, yd(R)) are used. 

XIII. TWO GENERALIZATIONS OF THE ZPA FOLDING-UNFOLDING CONTINUOUS SYMMETRY MEASURES FOR CONTINUA USING THE SNDSM METRIC AND THE HAUSDORFF METRIC... 

For crisp continuum sets and fuzzy sets, the crisp and fuzzy versions of the SNSM metric, as well as the crisp and fuzzy versions of the Hausdorff metric, provide generalizations of the ZPA approach. The generalization we shall describe requires a computational technique applicable for generating a crisp average of a family... 

TTie dissimilarity of A and Af i p p provides a symmetry deficiency measure analogous to the ZPA continuous symmetry measure of discrete point sets. As a dissimilarity measure, both the SNDSM metric and the Hausdorff metric, or any other dissimilarity measure suitable for continua, are applicable. 

Using the Hausdorff metric for the dissimilarity of sets A and Af f p p, one obtains another generalization of the ZPA continuous symmetry measure of discrete point sets to crisp continuum sets, leading to a new symmetry deficiency measure /Jzpa( > that is a valid measure of... 

A measure of the symmetry aspect R for set A is obtained with respect to center c if one compares sets A and (R,c,int). Using the Hausdorff metric... 

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. 

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. 

If F is a bounded measurable function on C (the set of closed subsets of [0,1] equipped with the Hausdorff metric) then we have... 

This metric is equivalent to the Matheron one (restricted to Ci). We point out that it is also possible to define the Hausdorff metric Coo by adding the point oo to the set and by performing the standard compactification... 

If the topology T is chosen as the metric topology, that is, if the T-open sets are precisely those which are open in some metric d introduced into the set X, then one obtains the metric topological space (X, T). Note that the metric topological space (X, T) is a Hausdorff space and also a normal space. 

Frechet [63] made an abstract formulation of the notion of distance in 1906. Hausdorff [64] proposed the term metric space, where he introduced the function d that assigns a nonnegative real number d p, q) (the distance between p and q) to every pair ip. q) of elements (points) of a nonempty set S. A metric space is a pair (S,d) if the function d satisfies several conditions, such as triangle inequality. In 1942, Menger [65] proposed that if we replace d(p, q) by a real function Fpq whose value is Fpq(x) for any real number x, this can be interpreted as the probability that the distance between p and q is less than x. Since probabilities can be neither negative nor greater than 1, we have... 

The Hausdorff distance h A,B) is a proper metric within any family of compact sets, for example, / (/ , B) is zero if and only if the two sets are the same. 

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. 

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 ... 

Both the unsealed and scaled fuzzy Hausdorff-type metrics g(A,B) and f(A, B) serve as the basis for various choices for similarity measures between fuzzy sets, for example ... 

Both the unsealed fuzzy Hausdorff-type metric g A,B) and the scaled fuzzy Hausdorff-type metric f(A,B) have been shown in the preceding text to fulfill the conditions for proper metric if the mutual locations of fuzzy sets A, B, and C are fixed. However, in some applications, various translated, rotated, or reflected versions of these fuzzy sets can be regarded as equivalent. For example, the inherent dissimilarities between... 

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... 

Combining inequalities (74) and (75), one obtains the triangle inequality for the optimum position version fop(A, B) of the scaled fuzzy Hausdorff-type metric ... 

Note that the versions of fuzzy sets taken into account in the set /(A, B,.) of definition (70) can be restricted to translated versions only. In this case the proof follows the same steps as before, and the translation-restricted fgp (A,B) scaled fuzzy Hausdorff-type metric is obtained. Alternatively, the allowed rotations can be confined to some angle interval A a, leading to another scaled fuzzy Hausdorff-type metric /op,tr,Aa(" Furthermore, if in addition to translated and rotated versions, reflected versions are also included among the versions in the set /(B .), then one obtains a new version of scaled fuzzy Hausdorff-type metric, f p (A, B). For these metrics, the following general relations hold ... 

By contrast to these new, optimized metrics involving free or constrained repositioning of sets A and B, the fuzzy Hausdorff-type metrics defined by Eqs. (43) and (63)-(68) do not involve any repositioning of either set A or B and are referred to as direct fuzzy Hausdorff-type metrics. 

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. 

Consider a set of molecules A,B,C,... with electron densities p (r), pg(r), pc(r)> and level sets G (fl),Gg(fl),Gc(a),respectively, for each density threshold value a. By choosing an appropriate definition for fuzzy membership function describing the fuzzy assignment of points r of the three-dimensional space to each molecule, such as the membership function /x (r) = p. (r)/p of Eq. (20) or = 1 - exp(- rpj(r)) of Eq. (21), the density-scaled fuzzy Hausdorff-type metric f p(A,B) applies... 

These fuzzy similarity measures of molecules are based on electronic density and fuzzy generalizations of Hausdorff-type metrics. These metrics and similarity measures are applicable for any collection of molecules. 

IV. FUZZY SYMMETRY DEFICIENCY MEASURES, FUZZY CHIRALITY MEASURES, AND FUZZY SYMMETRY GROUPS BASED ON THE MASS OF FUZZY SETS AND FUZZY HAUSDORFF-TYPE METRICS... 

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