Membership linguistic variables

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. 

One of the most important tools in applications of fuzzy set theory is the concept of linguistic variables (LV). These are groups of fuzzy sets with (partially) overlapping membership functions over a common (crisp) basic variable x. To represent several classes within an LV, the membership functions should cover all the relevant definition space of the basic variable x. The overlaps of these functions define the fuzziness. A linguistic variable L, classified by n fuzzy sets Aj, can be defined as... 

FIGURE 3 Membership functions for linguistic variables describing molecular shape. 

In such rules, X and Y and Z are linguistic variables whose values, e.g., small, medium, and large, are words rather than numbers. In effect, the values of linguistic variables are labels for fuzzy sets. It is understood that the membership functions of these sets must be specified in context. Usually, the membership functions are assumed to be triangular or trapezoidal. 

Figure 8 0 Membership functions for the linguistic variable high" according to Eq. (8.40) together with different modifiers.

Here, the premise is described by a membership function for the linguistic variable high and the function for the detection limit is the sum of the blank signal, y, and three times the standard deviation of the blank signal, Sg, (cf. Eq. (4.3)). Optimization of the parameters in the premise part of the rules is adaptively done by combining the fuzzy rule-based system with a neural network. Consider an adaptive neuro-fuzzy system with two inputs, and... 

Table 1. Linguistic variables of risk parameters and their fuzzy memberships (Yang et al., 2009).

Risk parameters Linguistic variables Fuzzy memberships... 

In this layer, the inputs are the filtered data, and each of these inputs is classified to fuzzy set membership functions. The inputs of fuzzy inference system are averaged measured distances to an obstacle information from sensor 1, sensor 2, sensor 3, and sensor 4, which are described by three linguistic variables near, far, and very far. The domain of functions is from 0 (minimum) to 2.5 m (maximum) for each sensor. 

The two linguistic variables near and far are described by triangular membership functions, whereas very far is described by trapezoidal membership function as shown in Fig. 3.23. Indeed, the input values of 2 < Xs < 2.5 are regarded as there are no detected obstacles neither at far nor near, thus they were interpreted as very far (see developed fuzzy rules). 

To get a feel of the fuzzy definition set, detection likelihood, as shown in Fig. lV/2.2.3-1, has been transformed into a fuzzy definition. A typical fuzzy membership is shown in Fig. lV/2.6.4-2. Actual fuzzy values are derived based on the fuzzy rule set. Fuzzy inputs are evaluated using a rule-based set, so that criticality and RPN calculations can be made. In the fuzzification process, with help of crisp ranking, set S O D is converted into fuzzy representation so that these can be matched with the rule base. Here, the if then rule has two parts an antecedent (which is compared to input) and consequence (which is the result). On the other hand, in the defuzzification process, the reverse takes place. It is possible to automate FMEA using fuzzy logic and rule-based systems. The rule allows quantitative data such as occurrence to be easily combined with judgmental and quantitative data (such as severity and detectability) very easily and uniformly. The rule based on the linguistic variables is more expressive and useful (for further reading see Ref. [11]). 

As shown in Figure 2, a point where the TFN intersects a linguistic term is circled and the corresponding degree of membership (i.e. Z ) is presented in Table 4. As illustrated in Table 4, Zp is normalised to obtain Z (i.e. linguistic variables linguistic terms with their membership degrees). 

Our models will most often assume that the membership functions of linguistic variable values have a trapezoidal shape and that a typical membership function with the parameters (a,b,c,d) is as follows ... 

The values assumed by the Detectability linguistic variable are very small, small, medium, big and very big. It is assumed that the membership functions of given fuzzy sets will have a trapezoidal shape. To establish them we will use a standard procedure for the evaluation of the device that is performed during every start-up. This calls for using a standard test sample that help verify if the device meets the minimum values related to set requirements (European Commission 2010) ... 

Figure 3. Membership functions of the Detectability linguistic variable values.

Figure 4. Membership funetions of the TIP Number linguistic variable.

Figure 7. Membership functions for the Number of detection lines linguistic variable.

Table 2. The (a,h,c,d) parameters of trapezoidal membership functions of input linguistic variables for Training model.

Each neuron in this layer corresponds to a linguistic label. The crisp inputs xj and X2 are fuzzified by using membership functions of the linguistic variables, and fi,. Usually, triangular, trapezoid or Gaussian membership curves are used. For example, the Gaussian membership function is defined by ... 

Each membership function defines a fuzzy set, also referred to as a linguistic variable. For example, the fuzzy set Hot consists of the values of T and the membership function jXHot- Thus, it can be expressed as Hot = T, iiHot(T). ... 

Fuzzy logic control calculations are executed by using both membership functions of the inputs and outputs and a set of rules called a rule base, as shown in Fig. 16.21. Typical membership functions for the inputs, e and deldt, are shown in Fig. 16.23, where it is assumed that these inputs have identical membership functions with the following characteristics three linguistic variables which are negative (N), positive P), and zero (Z) with trapezoidal, triangular and trapezoidal membership function forms respectively. Input variables e and deldt have been scaled so that the membership functions overlap for the range from -1 to +1. Furthermore, Fig. 16.24 shows the membership functions of the output Aw(r), which are... 

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. ... 

Suppose we have linguistic variables temperature (f) and yield (y). The discrete membership functions are given in Table 1, rows /Z/i(f), Ms(> )- The conditional fuzzy proposition... 

Step 1 Reading the physical variables and finding the membership degrees of the linguistic variables... 

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