Fuzzy Sets And Fuzzy Membership Functions

The fuzzy sets were made to represent each of the variable's intensities. These sets were associated to the linguistic variables “Black” and “white”. The created membership functions for the fuzzy sets are associated to the input and to the output were triangles, as shown below.

Membership functions of fuzzy set associated to the input and to the output

FUZZY SETS AND CRISP SETS

In many aspects fuzzy sets are simultaneously qualitative and quantitative and also they incorporate both kinds of distinctions in the measurements of degree of set membership. So these fuzzy sets have many of the virtues of conventional interval scale variable. But at the same time they allow set theoretic operations. These operations are outside the scope of conventional variable-oriented analysis.

FUZZY SETS

conventional Boolean is developed really for the analysis of configurations of crisp set memberships . With crisp sets these each case is assigned one of two possible membership scores in each set included in a study: 1 i.e membership in the set or 0 i.e non-membership in the set. An object or element (e.g., a country) within a domain (e.g., members of the United Nations) is either in or out of the various sets within this domain (e.g., membership in the U.N. Security Council). Crisp sets establish distinctions among cases that are wholly qualitative in nature (e.g., membership versus non-membership in the Security Council). Categorical-level measurement, the foundation of crisp sets, is considered by many social scientists to be inferior to interval-level measurement. However, crisp sets are often better anchored in substantive and theoretical knowledge than conventional interval-scale measures. For example, it is a relatively simple matter to identify an interval-scale indicator of country wealth, which in turn provides a simple tool for evaluating the relative positions of countries on this dimension. By contrast, it is more challenging to define the set of "rich countries" in a qualitative manner and then specify which countries are fully in this set and which are not. The key difference is that qualitative distinctions are explicit and must be grounded in substantive and theoretical knowledge, while the relative rankings of an interval scale can be pegged simply to scores on a crude indicator of the underlying construct (e.g., GNP per capita as an indicator of country wealth).

The generalization is performed as follows: For any crisp set A, it is possible to define a characteristic function X = {0,1 }. i.e. the characteristic function takes either of the values 0 or 1 in the classical set. For a fuzzy set, the characteristic function can take any value between zero and one.

The elements which have been assigned the number 1 can be interpreted as the elements that are in the set A and the elements which have assigned the number 0 as the elements that are not in the set A. This concept is sufficient for many areas of applications, but it can easily be seen, that it lacks in flexibility for some applications like classification of remotely sensed data analysis. For example, water shows low interferometric coherence g in SAR images. Since g starts at 0, the lower range of this set ought to be clear. The upper range, on the other hand, is rather hard to define. As a first attempt, we set the upper range to 0.2. Therefore we get B as a crisp interval B= [0,0.2]. But this means that a g value of 0.20 is low but a g value of 0.21 not. If the upper boundary moved for the range from g =0.20 to an arbitrary point,a more natural way to construct the set B would be to relax the strict separation between low and not low. This can be done by allowing not only the (crisp) decision Yes/No, but more flexible rules like ” fairly low”.

A fuzzy set allows us to define such a notion. The aim is to use fuzzy sets in order to make computers more 'intelligent', therefore, the idea above has to be coded more formally. In the example, all the elements were coded with 0 or 1. A straight way to generalize this concept is to allow more values between 0 and 1. In fact, infinitely many alternatives can be allowed between the boundaries 0 and 1, namely the unit interval I = [0, 1]. The interpretation of the numbers, now assigned to all elements is much more difficult. Of course, again the number 1 assigned to an element means that the element is in the set B and 0 means that the element is definitely not in the set B. All other values mean a gradual membership to the set B. This is shown in Fig. 5. The membership function is a graphical representation of the magnitude of participation of each input. It associates a weighting with each of the inputs that are processed, define functional overlap between inputs, and ultimately determines an output response. The rules use the input membership values as weighting factors to determine their influence on the fuzzy output sets of the final output conclusion. The membership function, operating in this case on the fuzzy set of interferometric coherence g, returns a value between 0.0 and 1.0. For example, an interferometric coherence g of 0.3 has a membership of 0.5 to the set low coherence (see Fig. 5). It is important to point out the distinction between fuzzy logic and probability. Both operate over the same numeric range, and have similar values: 0.0representing False (or non-membership), and 1.0 representing True (or full-membership). However, there is a distinction to be made between the two statements: The probabilistic approach yields the natural-language statement, “There is an 50% chance that g is low,” while the fuzzy terminology corresponds to”g's degree of membership within the set of low interferometric coherence is 0.50.” The semantic difference is significant: the first view supposes that g is or is not low; it is just that we only have an 50% chance of knowing which set it is in. By contrast, fuzzy terminology supposes that g is ”more or less” low, or in some other term corresponding to the value of 0.50.

OPERATIONS ON FUZZY SETS

Crisp-set causal conditions can be included along with fuzzy-set causal conditions in a fuzzy-set analysis.

The three common operations on fuzzy sets: negation, logical and, and logical or. These three operations provide important background knowledge for understanding how to work with fuzzy sets.

Negation. Like conventional crisp sets, fuzzy sets can be negated. With crisp sets, negation switches membership scores from “1” to “0” and from “0” to “1.” The negation of the crisp set of democracies, for example, is the crisp set of not-democracies. This simple mathematical principle holds in fuzzy algebra as well, but the relevant numerical values are not restricted to the Boolean values 0 and 1, but extend to values between 0 and 1. To calculate the membership of a case in the negation of fuzzy set A (i.e., not-A), simply subtract its membership in set A from 1, as follows: (membership in set not-A) = 1 - (membership in set A) or a = 1 - A (Lower-case letters are used to indicate negation.) Thus, for example, if the U.S. has a membership score of 0.9 in the set of “democratic countries,” it has a score of 0.1 in the set of “not-democratic countries.”

Logical and. Compound sets are formed when two or more sets are combined, an operation commonly known as set intersection. A researcher interested in the fate of democratic institutions in relatively inhospitable settings might want to draw up a list of countries that combine being “democratic” with being “poor.” Conventionally, these countries would be identified using crisp sets by cross tabulating the two dichotomies, poor versus not-poor and democratic versus not-democratic and seeing which countries are in the democratic/poor cell of this 2 X 2 table. This cell, in effect, shows the cases that exist in the intersection of the two crisp sets. With fuzzy sets, logical and is accomplished by taking the minimum membership score of each case in the sets that are combined. For example, if a country's membership in the set of poor countries is 0.7 and its membership in the set of democratic countries is 0.9, its membership in the set of countries that are both poor and democratic is the smaller of these two scores, 0.7. A score of 0.7 indicates that this case is more in than out of the intersection.

Logical or. Two or more sets also can be joined through logical or--the union of sets. For example, a researcher might be interested in countries that are “developed” or “democratic” based on the conjecture that these two conditions might offer equivalent bases for some outcome (e.g., bureaucracy-laden government). When using fuzzy sets, logical or directs the researcher's attention to the maximum of each case's memberships in the component sets. That is, a case's membership in the set formed from the union of two or more fuzzy sets is the maximum value of its memberships in the component sets. Thus, if a country has a score of 0.3 in the set of democratic countries and a score of 0.9 in the set of developed countries, it has a score of 0.9 in the set of countries that are “democratic or developed.”

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