Data Mining - Mehmed Kantardzic [242]
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 rule of inference in a traditional two-valued logic is modus ponens, according to which we can infer the truth of a proposition B from the truth of A and the implication A → B. However, in much of human reasoning, modus ponens is employed in an approximate manner. For example, if we have the rule “if the tomato is red, then it is ripe” and we know that “the tomato is more or less red,” then we may infer that “the tomato is more or less ripe.” This type of approximate reasoning can be formalized as
Fact: x is A′
Rule: If x is A then y is B
Conclusion: y is B′
where A′ is close to A and B′ is close to B. When A, A′, B, and B′ are fuzzy sets of an approximate universe, the foregoing inference procedure is called approximate reasoning or fuzzy reasoning; it is also called generalized modus ponens, since it has modus ponens as a special case.
Using the composition rule of inference, we can formulate the inference procedure of fuzzy reasoning. Let A, A′, and B be fuzzy sets on X, X, and Y domains, respectively. Assume that the fuzzy implication A → B is expressed as a fuzzy relation R on X × Y. Then the fuzzy set B′ induced by A′ and A → B is defined by
Some typical characteristics of the fuzzy-reasoning process and some conclusions useful for this type of reasoning are
1. ∀A, ∀ A′ → B′ ⊇ B ( \or μ B′(y) ≥ μ B(y))
2. If A′ ⊆ A (or μ A(x) ≥ μ A′(x)) → B′ = B
Let us analyze the computational steps of a fuzzy-reasoning process for one simple example. Given the fact A′ = “x is above average height” and the fuzzy rule “if x is high, then his/her weight is also high,” we can formalize this as a fuzzy implication A → B. We can use a discrete representation of the initially given fuzzy sets A, A′, and B (based on subjective heuristics):
μR(x, y) can be computed in several different ways, such as
or as the Lukasiewicz norm:
Both definitions lead to a very different interpretation of fuzzy implication. Applying the first relation for μR(x, y) on the numeric representation for our sets A and B, the 2-D MF will be
Now, using the basic relation for inference procedure, we obtain
The resulting fuzzy set B′ can be represented in the form of a table:
B′: y μ(y)
120 0.3
150 1.0
180 1.0
210 1.0
or interpreted approximately in linguistic terms: “x’s weight is more-or-less high.” A graphical comparison of MFs for fuzzy sets A, A′, B, and B′ is given in Figure 14.11.
Figure 14.11. Comparison of approximate reasoning result B’ with initially given fuzzy sets A’, A, and B and the fuzzy rule A → B. (a) Fuzzy sets A and A’; (b) fuzzy sets B and B’ (conclusion).
To use fuzzy sets in approximate reasoning (a set of linguistic values with numeric representations of MFs), the main tasks for the designer of a system are
1. represent any fuzzy datum, given as a linguistic value, in terms of the codebook A;
2. use these coded values for different communication and processing steps; and
3. at the end of approximate reasoning, transform the computed results back into its original (linguistic) format using the same codebook A.
These three fundamental tasks are commonly referred to as encoding, transmission and processing, and decoding (the terms have been