Data Mining by Mehmed Kantardzic (good book recommendations TXT) 📕

= x + 2. For a given fuzzy set A = 0.1/1 + 0.2/2 + 0.7/3 + 1.0/4 in X, it is necessary to find a corresponding fuzzy set B(y) in Y using the extension principle through function B = f(A). In this case, the process of computation is straightforward and a final, transformed fuzzy set is B = 0.1/3 + 0.2/4 + 0.7/5 + 1.0/6.

Another problem will show that the computational process is not always a one-step process. Suppose that A is given as

and the function f is

Upon applying the extension principle, we have

where ∨ represents the max function. For a fuzzy set with a continuous universe of discourse X, an analogous procedure applies.

Besides being useful in the application of the extension principle, some of the unary and binary fuzzy relations are also very important in a fuzzy-reasoning process. Binary fuzzy relations are fuzzy sets in X × Y that map each element in X × Y to a membership grade between 0 and 1. Let X and Y be two universes of discourse. Then

is a binary fuzzy relation in X × Y. Note that μR(x, y) is in fact a two-dimensional (2-D) MF. For example, let X = Y = R+ (the positive real axis); the fuzzy relation is given as R = “y is much greater than x.” The MF of the fuzzy relation can be subjectively defined as

If X and Y are a finite set of discrete values such as X {3, 4, 5} and Y = {3, 4, 5, 6, 7}, then it is convenient to express the fuzzy relation R as a relation matrix:

where the element at row i and column j is equal to the membership grade between the ith element of X and the jth element of Y.

Common examples of binary fuzzy relations are as follows:

1. x is close to y (x and y are numbers).

2. x depends on y (x and y are categorical data).

3. x and y look alike.

4. If x is large, then y is small.

Fuzzy relations in different product spaces can be combined through a composition operation. Different composition operations have been suggested for fuzzy relations; the best known is the max–min composition proposed by Zadeh. Let R1 and R2 be two fuzzy relations defined on X × Y and Y × Z, respectively. The max–min composition of R1 and R2 is a fuzzy set defined by

or equivalently,

with the understanding that ∨ and ∧ represent max and min, respectively.

When R1 and R2 are expressed as relation matrices, the calculation of R1 ° R2 is similar to the matrix-multiplication process, except that × and + operations are replaced by ∨ and ∧, respectively.

The following example demonstrates how to apply the max–min composition on two relations and how to interpret the resulting fuzzy relation R1 ° R2. Let R1 = “x is relevant to y” and R2 = “y is relevant to z” be two fuzzy relations defined on X × Y and Y × Z, where X = {1, 2, 3}, Y = {α, β, γ, δ}, and Z = {a, b}. Assume that R1 and R2 can be expressed as the following relation matrices of μ values:

Fuzzy relation R1 ° R2 can be interpreted as a derived relation “x is relevant to z” based on relations R1 and R2. We will make a detailed max–min composition only for one element in a resulting fuzzy relation: (x, z) = (2, a).

Analogously, we can compute the other elements, and the final fuzzy matrix R1 ° R2 will be

14.4 FUZZY LOGIC AND FUZZY INFERENCE SYSTEMS

Fuzzy logic enables us to handle uncertainty in a very intuitive and natural manner. In addition to making it possible to formalize imprecise data, it also enables us to do arithmetic and Boolean operations using fuzzy sets. Finally, it describes the inference systems based on fuzzy rules. Fuzzy rules and fuzzy-reasoning processes, which are the most important modeling tools based on the fuzzy-set theory, are the backbone of any fuzzy inference system. Typically, a fuzzy rule has the general format of a conditional proposition. A fuzzy If-then rule, also known as fuzzy implication, assumes the form

If x is A, then y is B

where A and B are linguistic values defined by fuzzy sets on the universes of discourse X and Y, respectively. Often, “x is A” is called the antecedent or premise, while “y is B” is called the consequence or conclusion. Examples of fuzzy if-then rules are widespread in our daily linguistic expressions, such as the following:

1. If pressure is high, then volume is small.

2. If the road is slippery, then driving is dangerous.

3. If a tomato is red, then it is ripe.

4. If the speed is high, then apply the brake a little.

Before we can employ fuzzy if-then rules to model and analyze a fuzzy reasoning-process, we have to formalize the meaning of the expression “if x is A then y is B,” sometimes abbreviated in a formal presentation as A → B. In essence, the expression describes a relation between two variables x and y; this suggests that a fuzzy if-then rule be defined as a binary fuzzy relation R on the product space X × Y. R can be viewed as a fuzzy set with a 2-D MF:

If we interpret A → B as A entails B, still it can be formalized in several different ways. One formula that could be applied based on a standard logical interpretation, is

Note that this is only one of several possible interpretations for fuzzy implication. The accepted meaning of A → B represents the basis for an explanation of the fuzzy-reasoning process using if-then fuzzy rules.

Fuzzy reasoning, also known as approximate reasoning, is an inference procedure that derives its conclusions from a set of fuzzy rules and known facts (they also can be fuzzy sets). The basic