Fuzzy set

Fuzzy sets were introduced as generalizations of the classical crisp sets, in order to represent and manipulate imprecise data. However, not all the properties valid for operations on crisp sets are valid for fuzzy sets, and the inability to deal with this may result in improper use of fuzzy sets. The basic concepts are defined below  

In what follows, some of the properties of crisp sets will be analyzed from the point of view of applicability with fuzzy sets. We have listed these properties in Table 1 along with examples of two widely used t-norms and 

Property 1 in Table 1 is satisfied only by the standard t-norm and -conorm. 

For any t-norm and t-conorm that posses the properties 2 and 3, the fuzzy sets have only crisp values, that is, they reduce to crisp sets. 

In real life, information is often fuzzy. Consider the statement  

In a worst-case scenario, the run-off from agricultural land may contain levels of fertilizers or animal waste that might seriously contaminate the water supply. 

This water will kill you if you drink more than 100 ml of it. gets the point across rather more directly. 

A phrase such as seriously contaminate is known as a linguistic variable it gives an indication of the magnitude of a parameter, but does not provide its exact value. Subjective knowledge is expressed by statements that contain vague terms, qualifications, probabilities, or judgmental data. Objects described by these vague statements are more difficult to fit into crisp sets. 

Semantic uncertainty is the type of uncertainty for which we shall need fuzzy logic. Expressed by phrases such as acidic or much weaker, this is imprecision in the description of an event, state, or object rather than its measurement. Fuzzy logic offers a way to make credible deductions from uncertain statements. We shall illustrate this with a simple example. 

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Fuzzy sets and fuzzy logic. Fuzzy sets differ from the normal crisp sets in the fact that their elements have partial membership (represented by a value between 0 an 1) in the set. Fuzzy logic differs from the binary logic by the fact that the truth values are represented by fuzzy sets. 

All the three techniques mentioned above may make use of fuzzy sets and fuzzy logic (for fuzzy classification, fuzzy rules or fuzzy matching) but this does not effect the discussion of the applicability to NDT problems in the next section. 

To know about fuzzy sets and fuzzy logic... 

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

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). 

Fuzzy logic and fuzzy set theory are applied to various problems in chemistry. The applications range from component identification and spectral Hbrary search to fuzzy pattern recognition or calibrations of analytical methods. 

An overview over different applications of fuzzy set theory and fuzzy logic is given in [15] (see also Chapter IX, Section 1.5 in the Handbook). 

If a spectrum lacks certain Lines or contains extra lines from additional unknown components, or if the true line positions are blurred, fuzzy set theory can improve the matching. 

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-... 

Duhois, D., H. Prade, and R. R. Yager (eds.) Readings in Fuzzy Sets for Intelligent Systems, Morgan Kaufmann (1993). 

APPLICATION OF FUZZY SETS THEORY TO SOLVING TASKS OF MULTICOMPONENT QUALITATIVE ANALYSIS... 

Data collected by modern analytical instalments are usually presented by the multidimensional arrays. To perform the detection/identification of the supposed component or to verify the authenticity of a product, it is necessary to estimate the similarity of the analyte to the reference. The similarity is commonly estimated with the use of the distance between the multidimensional arrays corresponding to the compared objects. To exclude within the limits of the possible the influence of the random errors and the nonreproductivity of the experimental conditions and to make the comparison of samples more robust, it is possible to handle the arrays with the use of the fuzzy set theory apparatus. 

Fuzzy logic control systems 10.2.1 Fuzzy set theory... 

Fuzzy logie was first proposed by Zadeh (1965) and is based on the eoneept of fuzzy sets. Fuzzy set theory provides a means for representing uneertainty. In general, probability theory is the primary tool for analysing uneertainty, and assumes that the... 

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. 

Fuzzy sets represented by symmetrical triangles are commonly used because they give good results and computation is simple. Other arrangements include non-symmetrical triangles, trapezoids, Gaussian and bell shaped curves. 

Let the fuzzy set medium temperature be called fuzzy set M. If an element u of the universe of discourse U lies within fuzzy set M, it will have a value of between 0 and 1. This is expressed mathematically as... 

When the universe of discourse is discrete and finite, fuzzy set M may be expressed as... 

In equation (10.2) / is a delimiter. Hence the numerator of each term is the membership value in fuzzy set M associated with the element of the universe indicated in the denominator. When = 11, equation (10.2) can be written as... 

Let A and B be two fuzzy sets within a universe of diseourse U with membership funetions /ta and /tb respeetively. The following fuzzy set operations ean be defined as... 

Equality. Two fuzzy sets A and B are equal if they have the same membership funetion within a universe of diseourse U. 

Union The union of two fuzzy sets A and B eorresponds to the Boolean OR funetion and is given by... 

Find the union and interseetion of fuzzy set low temperature L and medium temperature M shown in Figure 10.4. Find also the eomplement of fuzzy set M. Using equation (10.2) the fuzzy sets for = 11 are... 

Fuzzifieation is the proeess of mapping inputs to the FLC into fuzzy set membership values in the various input universes of diseourse. Deeisions need to be made regarding... 

The number and shape of fuzzy sets in a partieular universe of diseourse is a tradeoff between preeision of eontrol aetion and real-time eomputational eomplexity. In this example, seven triangular sets will be used. 

Figure 10.9 assumes that the output window contains seven fuzzy sets with the same linguistic labels as the input fuzzy sets. If the universe of discourse for the control signal u(t) is 9, then the output window is as shown in Figure 10.10. 

In Figure 10.8 and equation (10.20) the fuzzy sets that were hit in the error input window when e(t) = 2.5 were PS and PM. In the rate of change input window when ce = -0.2, the fuzzy sets to be hit were NS and Z. From Figure 10.9, the relevant rules that correspond to these hits are... 

Figure 10.38 shows an input window with three triangular fuzzy sets NB, Z and PB. Each set is positioned in its regime of operation by the centre parameter c so that, for example, NB can only operate on the negative side of the universe of discourse. The width of each set is controlled by parameter ri . 

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