Fuzzification

Instead of using the Euclidean distance, also other distance measures can be considered. Moreover, another power than 2 could be used for the membership coefficients, which will change the characteristics of the procedure (degree of fuzzification). Similar to fc-means, the number of clusters k has to be provided as an input, and the algorithm also uses cluster centroids Cj which are now computed by... 

In order to combine the conclusions of several such Fuzzy Inference Systems, they will be translated into belief structures, according to the method proposed in [22]. Each Fuzzy Inference System presented Figure 12 is the association of fuzzification functions (providing a numerical evaluation of the membership of a variable to fuzzy sets) and of rules linking these observations to different classes which can be states or disjunctions of states of the process. 

It is possible to write, from the fuzzification function of each input variable, a belief structure defined on the size Xi of the variable VI... 

This belief structure represents the belief into the state of the system of the set of rules associated to the fuzzification functions. 

The issue of fuzzification of similarity search has been addressed by Horvath et The first fuzzy similarity metric suggested relies on... 

No doubt Professor Zadeh s enthusiasm for fuzziness has been reinforced by the prevailing political climate in the U.S.—one of unprecedented permissiveness. Fuzzification is a kind of scientific permissiveness it tends to result in socially appealing slogans unaccompanied by the discipline of hard scientific work and patient observation. (Rudolf E. Kalman, 1972)... 

It thus appears that proper application of the terms chiral and achiral to real (chemical) systems requires fuzzification. To place this concept in proper perspective, we need to take a look at the role of inexactitude in scientific communication. ... 

Fuzzification of crisp basic variables into linguistic variables (LVs)... 

Essentially, the neurofuzzy architecture is a neural network with two additional layers for fuzzification and defuzzification. The fuzzification and input weighting are illustrated graphically in Fig. 9, adapted from the thesis of Bossley. It can be seen that there are similarities with the RBF network, except now the radial functions are replaced by the multivariate membership functions. 

The architecture of an ANFIS model is shown in Figure 14.4. As can be seen, the proposed neuro-fuzzy model in ANFIS is a multilayer neural network-based fuzzy system, which has a total of five layers. The input (layer 1) and output (layer 5) nodes represent the descriptors and the response, respectively. Layer 2 is the fuzzification layer in which each node represents a membership. In the hidden layers, there are nodes functioning as membership functions (MFs) and rules. This eliminates the disadvantage of a normal NN, which is difficult for an observer to understand or to modify. The detailed description of ANFIS architecture is given elsewhere (31). 

Figure 57.17 shows the basic configuration of an FLC system, comprising four components a fuzzification interface, a knowledge base, a decision-making logic (control algorithm), and a defuzzification interface. 

In Fig. 2, the fuzzy inference flow from linguistic variable fuzzification to defuzzification of the aggregate output is shown it proceeds up from the inputs in the lower left, then across each row, or rule, and then down the rule outputs to end in the lower right. 

Fuzzification is the process of mapping crisp input xElU into fuzzy set f G U. This is achieved with three different types of fuzzifier, including singleton fuzzifiers, Gaussian fuzzifiers, and trapezoidal or triangular fuzzifiers. These fuzzifiers map crisp input x into fuzzy set with different membership functions pfix) listed below. 

The last step in a fiizzy logic system is defuzzification. As the name suggests, defuzzification is the opposite of fuzzification, which produces crisp output f for a fuzzy logic system from the aggregated output of fuzzy set B. A number of defuzzifiers have been developed the most popular is the centroid defuzzifier, which finds a vertical line and divides an aggregated set into two equal portions. Mathematically the center of gravity (COG) can be defined by ... 

The crisp output Wo from the fuzzy set Q is obtained by center of gravity de-fuzzification method. Since Q w) can have K possible values, Q(l), <2(2),..., Q K), its centre of gravity is given by ... 

Fuzzy control is reahzed within 3 processes fuzzification, inference and defuzzification. The fuzzification process describes variables by means of input values. Thus a fuzzy set is represented by a membership function. The membership function is constructed based on the variable s input values (laanineh Maijohann 1996). Accordingly every input value obtains a membership degree between one and zero. If the input value is clearly assigned to the description of the variable, it receives a membership degree one. It is a fiiU membership to a related fuzzy set. The membership degree zero means that an input value does not belong to the fuzzy set. Membership values between zero and one indicate a partial membership of an input value to the certain fuzzy set. 

As for the other algorithms, the first step, after acquiring the current value of the received power, is to determine the distance in the signal domain between the current location and all the points that make part of the fingerprint map. The next step is to transform these distance values into grades of membership, i.e., it is made the fuzzification. For this phase a set of membership functions, such as the one presented in Fig. 13.1, must be used. 

Taheri Shahraiyni (2007) developed new heuristic search, fuzzification and defuzzification methods for ALM algorithm. In the next sections of this chapter, ALM algorithm with these modifications is explained and the ALM abilities and applications are illustrated. 

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