Membership fuzzy

Chemical Normal Boiling Point/°C Volatile (Fuzzy Membership) Volatile (Crisp Membership)... 

Table 8.1 compares the fuzzy membership and crisp membership values in the volatile set for a few liquids (Figure 8.5). 

Once the input data have been used to find fuzzy memberships, in the second step we compute the membership value for each part in the condition or antecedent of a rule. These are then combined to determine a value for the conclusion or consequent of that rule. If the antecedent is true to some degree, the consequent will be true to the same degree. 

If molecule X is not isolated, for example, if the total electron density at some point r can be regarded as a sum of electron densities px(r) and py(r) assigned to molecules X and Y, respectively, then the fuzzy membership function of points r with respect to the two molecules are determined by the relative magnitudes of the individual electron densities. 

Then, in the absence of other molecules, definition (6) of the fuzzy membership function for points r of the space belonging to molecule Xj becomes... 

This fuzzy membership function Pxi,L(r) can be written in another form ... 

For the derivation of appropriate fuzzy membership functions for the electronic densities of functional groups, first we take each individual functional group Fj, with its share PfiC1 ) °f the complete electron density of molecule X, and consider this share as a separate, individual object in the absence of all other functional groups of... 

The fuzzy membership function defined above reflects the actual electronic charge distribution of functional group within the given molecule X, without directly involving any other density contributions from other functional groups of the molecule. 

If the simultaneous presence of all functional groups Fj, F2,. .. F ,. .. Fm within molecule X is taken into account, then a new fuzzy membership function M-Fi,x(r) for points r of the space belonging to functional group Fj of molecule X can be defined as... 

This fuzzy membership function ppi,x(r) can also be written in a form of a simple density ratio ... 

Local molecular properties can be represented by the properties of functional groups Fj of molecule X, that in turn can be characterized by the fuzzy membership function PFi,x(r) f°r points r of the space. 

Variations in the local properties can be monitored by calculating the variations of the fuzzy membership function i-Fi,x(r)- If two functional groups Fj and Fj of a molecule X have the same chemical formula, then their differences must be due to their different local surroundings within the molecule X. These differences are necessarily manifested in differences of their fuzzy membership functions iFi,x(r) and PFj,x(r) >n molecule X. 

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 terms of pmax, a fuzzy membership function px(r) is defined for points r of the space, expressing the "degree" of their belonging to molecule X ... 

An alternative, equivalent expression for the fuzzy membership function PX,L(r) is given by... 

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... 

The role of operator can be described as follows. To take full advantage of the fuzzy indistinguishability of sets A and R 4 at the fuzzy level j8, operator j, resets the values of fuzzy membership functions whenever this fuzzy indistinguishability is implied by the presence of the fuzzy symmetry element R( /3) at the fuzzy level /3. 

The fuzzy folded set is the fuzzy average of thi defined by the fuzzy membership function (x) as follows ... 

FIGURE 13 Depiction of the on-screen selections of the most likely extensions using fuzzy membership values. Reprinted with permission from J. Chem. Inf. Comput. Sci. (ref. 87), 1992 American Chemical Society. 

The corresponding fuzzy membership function pFiyX (r) for points r of the space belonging to functional group F, of molecule X can be defined as... 

These fuzzy membership functions Pf,x(t) describe the degree of belonging of individual points r to various functional groups Fi, Fi,. .., F . .., Fm of molecule X. 

It is natural to decide the relative importance of various density values with reference to this maximum value Pm - With respect to Pmjx> a fuzzy membership function Pm( ) s defined for all points r of the space R, expressing the degree of r belonging to the molecule M ... 

Of course, if the molecule M is not isolated, that is other molecules also have electron density contribution to a selected point r, then this point r may partially belong to several different molecules to various degrees. This fact can also be expressed easily using fuzzy membership functions, as follows. 

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