Types of Membership Function

In principle a membership function associated with a fuzzy set A depends not only on the concept to be represented, but also on the context in which it is used. The graphs of the functions may have very different shapes and may have some specific properties (e.g. continuity). Whether a particular shape is suitable or not can be determined only in the application context (Klir and Yuan (1995)). In many practical instances, fuzzy sets can be represented explicitly by families of parameterised functions, the most common being  

Fuzzy sets can be characterised in more detail by referring to the features used in characterising the membership functions that describe them (Kandel (1986), Dubois et al. (1993)). 

The Support of a fuzzy set A, denoted by Supp(A), means that all elements of X belong to A to a non-zero degree (Kruse et al. (1994)). Formally, this is given by  

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The whole implementation has been programmed within MATLAB, using functions from the fuzzy inference toolbox. Different types of membership functions and defuzzification methods has been tested, with minor changes of the results. 

The number of parameters in the fuzzy sub-models is quite large. One rule of the fuzzy model for the net growth rate, for example, contains about 10 parameters, depending on the type of membership function that is used. Owing to the curse of dimensionahty the number of parameters increases exponentially for systems with higher dimensions. Therefore, only the consequence parameters have been optimized. 

The structure of this article is as follows. In a theoretical part, basic notions of fuzzy theory are explained, such as types of membership function, operations with fuzzy sets, definitions of fuzzy numbers, and the way to perform arithmetic operations with them, the concept of linguistic variables and widely used reasoning schemes of fuzzy logic. The application section refers to examples of utilization of fuzzy theory in chemistry. As an introduction to the mathematical theory Refs. 2-4 can be recommended and overviews with respect to chemical applications have been made. Recently a collection of papers from the sixth conference - devoted to Fuzzy Logic in Chemistry - in a series of Mathematical Chemistry Conferences was published. ... 

Figure 4 Algebraic operations with fuzzy numbers on the basis of the LR representation. The type of membership function is described by /( r) = max(0, I — jc ). As parameters were chosen m = 5 a = 0.5 d = 0.3 n =2 y = 0.2 5 = 0.1 and the scalar crisp number A. = 3. The fuzzy numbers M and N are characterized by dotted triangles and the results by solid triangles...

The Membership Manager communicates with the outside world via an API over which can be transmitted keys and suitably encoded forms of the modeled types. There is a set of inquiry functions (not shown) for reading the database. 

Limited data are available on the salaries that computational chemists earn. The ACS conducts an annual survey of a sampling of their membership. The ACS is the world s largest scientific society with more than 163,000 members. Of these, almost 10,000 domestic members respond to the survey a different random sample is used each year. The ACS reports the data in terms of type of employer, work function, discipline (the major traditional ones, but not computational chemistry), degree level, years of experience, age, and the other orthodox ways of looking at certain groups identified by gender, race, and ethnicity. 

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... 

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. 

The most simple filters are calculated directly from the molecular formula, from the element composition, or from an extended description of the atom types of a molecule. Extended atom types are notations of atoms that include more information than just the element name (e.g., level of hybridization, membership in a certain functional group, etc.). Many molecular properties are usable as filter descriptors [24-27]. 

For operations with exponential functions, truncation at a certain spread by setting the membership value, m x), to zero is recommended. Some membership function types provide a more natural truncation, for example, a function of quadratic type ... 

Figure 6 shows a structure of applying fuzzy logic in control. First, two types of inputs must be obtained numerical inputs and human knowledge or rule extraction from data (i.e., fuzzy rules). Then the numerical inputs must be fuzzified into fuzzy numbers. The fuzzy rules consist of the fuzzy membership functions (knowledge model) or so-called fuzzy associative memories (FAMs). Then the... 

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. 

All these uncertainties translate into uncertainties about fuzzy sets membership functions [28]. Type-1 fuzzy Logic caruiot fully handle these uncertainties because type-1 fuzzy logic membership functions are totally precise which means that all kinds of uncertainties will disappear as soon as type-1 fuzzy set membership functions are used [11]. The existence of uncertainties in the majority of... 

In the second stage, this chemical reaction algorithm will be applied to optimize the parameters of the membership functions of a Type-1 and Type-2 fuzzy logic controller for the tracking of a unicycle mobile robot. With this work, we pretend to demonstrate that this novel optimization paradigm is able to perform well in these specific topics and will encourage its use in different soft computing approaches. 

Table 6.11 shows the values of the input/output membership functions for the Type-1 FLC found by the chemical reaction algorithm. 

Table 6.11 Parameters of the membership functions for Type-1 FLC for Case 1...

A Type-2 fuzzy logic controller was developed using the parameters of the membership functions found for the ILC of Case 2. The parameters searched with the chemical reaction algorithm were for the footprint of uncertainty (FOU). Table 6.17 shows the parameters used in the simulations. 

In the third stage of the fuzzy controller application, the idea was to take the type-1 fnzzy and use it as the primary membership functions. The secondary membership fnnctions were fonnd by means of the CRA and then, a disturbance was apphed to the tracking system in order to compare the performance of both controllers (type-1 and type-2). 

More specifically, a Mamdani-type Fuzzy Inference System, consisted of two inputs and one output, was developed. The system receives as inputs the displacement (m) and the velocity (u), while gives as output the increment of the control force (z). Triangular membership functions (trimf) were chosen both for inputs and output. These are shown in the following figures. 

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