Fuzzy similarity measures

Because x, as well as w are normalized, represents the cosine or correlation coefficient between the two vectors. In a variant of ART, Fuzzy ART, a fuzzy similarity measure is used instead of the cosine similarity measure [14]. 

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. 

A fuzzy set A is said to have the fuzzy symmetry element R( jS) corresponding to the symmetry operation R at the fuzzy level )3 of the fuzzy Hausdorff-type similarity measure if and only if the fuzzy similarity measure measure R and A is greater than or equal to )3 ... 

If in relationship (124) the fuzzy similarity measure Sg is replaced by either one of the unsealed similarity measures tg and Zg or the scaled similarity measures s, tf, and Zf, then the fuzzy symmorphy element 5( )3) is specified in terms of the corresponding fuzzy Hausdorff-type similarity measure. 

Assume that R is the ordinary symmetry operator corresponding to the fuzzy symmetry element Rip ) present for fuzzy set A according to the FSNSM fs g. A fuzzy symmetry operator R(fs) of fuzzy symmetry element Rip ) present for fuzzy set A at the fuzzy level p of the fuzzy similarity measure fs is defined by its action on the fuzzy set A ... 

A modified approach was used to characterize the performance of a fuzzy similarity measure in comparison with the estimation of the correlation coefficient. Here the variance weighted relative cardinality was superior to the classical method for identification of UV spectra. ... 

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... 

Considering a collection of functional groups of the same chemical formula but located in a series of different molecules, local similarity measures among these functional groups can be based on their fuzzy membership functions (for a discussion of local similarity measures see sections 5 and 6). These similarity measures provide clues for the similarities and differences in their reactivities, caused by the similarities and differences in their molecular environments. 

In some of the preceding representations, a natural interrelationship between fuzziness and resolution is used, leading to resolution-based chirality, symmetry and similarity measures. ... 

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 ... 

These similarity measures have the value of 1 for identical fuzzy sets, and they give 0 for two fuzzy sets of infinite value for fuzzy generalizations of Hausdorff-type distances. 

Various fuzzy subsets B of 4 may have the fuzzy symmetry element B(/3) at different fuzzy levels /3 of the fuzzy Hausdorff-type similarity measure s. Here the concept of fuzzy subset is interpreted in the usual way for the fuzzy subset B of fuzzy set A the condition... 

Note that if a fuzzy set A has the fuzzy symmetry element / ()3) corresponding to the symmetry operator R at the fuzzy level j8 of the fuzzy Hausdorff-type similarity measure then the application of R on 4 generates a set R indistinguishable from set A at the fuzzy level (i. For the fuzzy set A the application of symmetry operator R of fuzzy symmetry element R(p ) present at the fuzzy level is completed by a formal recognition of the indistinguishability of set R/1 and set A at the given fuzzy level. This additional step, for which the sufficient and necessary condition is the presence of fuzzy symmetry element Ri ) at the fuzzy level /S, involves operator setting the membership functions of elements of the fuzzy set R/1 equal to those of fuzzy set A. 

A fuzzy set A has the / fuzzy symmetry group G Sg, /3 ) at fuzzy level if has the fuzzy symmetry element R( j8 ) at the fuzzy level j8 of the fuzzy Hausdorff similarity measure for each symmetry operation R of the crisp symmetry group G. The )3 fuzzy symmetry group G(s, at the fuzzy level /3 that is the supremum of the levels j8 of all )3 fuzzy symmetry groups G(Sg, (3 ) of A is the fuzzy symmetry group G s, of the fuzzy set A ... 

Analogous fuzzy symmetry groups Git13), G(z, )3), Gisf, ), G(tf,p), and GizpP) of fuzzy set A are defined for the unsealed similarity measures t and and scaled similarity measures Sp tf, and Zf, respectively. 

A measure 5(/l,R,gg) of the symmetry deficiency of fuzzy set A in symmetry element R according to the fuzzy Hausdorff-type similarity measure s can be defined as... 

The fuzzy Hausdorff-type similarity measures can be used for a direct comparison of a fuzzy set A and its various R sets and / -deficient sets, providing alternative fuzzy symmetry deficiency measures. 

Let Sfj(A,B) denote any one of the fuzzy Hausdorff-type similarity measures, for example, any one of the examples... 

Various generalizations of syntopy are possible by replacing the configuration space distance by some alternative similarity measures. In fact, any of the similarity measures and symmetry deficiency measures may serve as a suitable parameter in the definition of fuzzy syntopy membership functions. 

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