Fuzzy Membership Function

Figure 9-25. Membership function for the fuzzy set of numbers close to 3.

On the other hand, if we want to characteri2e objects which are described by the rather fuzzy statement "numbers dose to three", we then need a membership function which describes the doseness to three. An adequate membership function could be the one plotted in Figure 9-25 m x) has its maximum value of m x) = 1 for value x = 3. The greater the distance from x to 3 gets, the smaller is the value of m x). until it reaches its minimum m x) = 0 if the distance from x to 3 is greater than say 2, thus for x > 5 or x < 1. 

An important property of a fuzzy set is its cardinality. While for crisp sets the cardinality is simply the number of elements in a set, the cardinality of a fuzzy set A, CardA, gives the sum of the values of the membership function of A, as in Eq. (9). 

The principle of applying fuzzy logic to matching of spectra is that, given a sample spectrum and a collection of reference spectra, in a first step the reference spectra are unified and fuzzed, i.e., around each characteristic line at a certain wavenumber k, a certain fuzzy interval [/ o - Ak, + Afe] is laid. The resulting fuzzy set is then intersected with the crisp sample spectrum. A membership function analogous to the one in Figure 9-25 is applied. If a line of the sample spec-... 

The central concept of fuzzy set theory is that the membership function /i, like probability theory, can have a value of between 0 and 1. In Figure 10.3, the membership function /i has a linear relationship with the x-axis, called the universe of discourse U. This produces a triangular shaped fuzzy set. 

Square nodes in the ANFIS structure denote parameter sets of the membership functions of the TSK fuzzy system. Circular nodes are static/non-modifiable and perform operations such as product or max/min calculations. A hybrid learning rule is used to accelerate parameter adaption. This uses sequential least squares in the forward pass to identify consequent parameters, and back-propagation in the backward pass to establish the premise parameters. 

Zadeh [1975] extended the classical set theory to the so-called fuzzy set theory, introducing membership functions that can take on any value between 0 and 1. As illustrated by the intersection of the (hard) reference data set (A) and the fuzzed test data set (C), the intersection (E) shows an agreement of about 80%. Details on application of fuzzy set theory in analytical chemistry can be found in Blaffert [1984], Otto and Bandemer [ 1986a,b] and Otto et al. [1992],... 

The x-axis in a plot of a membership function represents the universe of discourse. This is the complete range of values that the independent variable can take the y-axis is the membership value of the fuzzy set. 

Suppose that we have defined a membership function for the "Low pH" set. Most acid solutions would be, to some degree, members of this fuzzy set. We may want to be able to qualify the description by adding modifiers, such as "very," "slightly," or "extremely" whose use allows us to retain close ties to natural language. The qualifiers that modify the shape of a fuzzy set are known as hedges. We can see the effect of the hedges "very" and "very very" in Figure 8.11. 

The rules that the fuzzy system uses are expressed in terms such as a "high" or a "medium" pH, while the experimental input data are numerical quantities. The first stage in applying these rules is to transform the input data into a degree of membership for each variable in each class through the use of membership functions. 

This presumed membership of 0.5 in the "low" set must now be combined with the output from Rl, which was that the rate is "high" to a degree of 0.2. To combine these, we require a membership function that relates the actual reaction rate to the fuzzy descriptors "low" and "high" (Figure 8.16). 

Membership functions to convert between experimental reaction rates and fuzzy sets. 

The relationships described form the concept (Fig. 5.68) of the FUZZY RULE SET. The membership-functions of the linguistic variables are derived from the experiments that have already been mentioned however, the plausibilities of the RULE SET have not been touched on so far. 

Step 2. Based on the importance of different objective functions and the acceptable ranges for objective values, subjectively select suitable lower/upper bounds, 4 < 4 < 4 < for minimizing objective and < 4 < 4 < 4 for maximizing objective. Define membership functions for multiple fuzzy objectives as given in Eqs. (12) and (13). 

A different approach to threshold selection is based on fuzzy logic [14], By adopting this approach, the value that represents the crisp discriminant between faults and disturbances is replaced by a fuzzy set, characterized by a membership function. Hence, a yes-no decision is replaced by a continuous indication of the faulty level. 

Fuzzy logic is based on the generalization of theory of sets characteristic function that Zadeh defined as membership function , //U), (Zadeh, 1965)... 

However, the identification of the fuzziness associated with single parameter characterizing foodstuffs is not enough. The authentication process is not usually restricted to a single parameter but in fact there are often several of them. If we want to operate with their membership function (e.g. low linoleic and high 24-methylene cycloarthanol), we need to define operations on the fuzzy set. Thus, the classical rule R = IF (input) A, THEN (output) B can be extended... 

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 ... 

The above fuzzy electron density membership functions reflect the relative contributions of the fuzzy, three-dimensional charge clouds of the various molecular electron density distributions to the total electronic density of molecular family L. 

With minor modifications, the fuzzy electron density membership function formalism of molecular families can also be applied to a family of functional groups within a molecule. Consider a molecule X and some electron density threshold a within the functional group range of density. Consider the functional groups appearing as separate density domains... 

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. 

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