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

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′:yμ(y) 1200.3 1501.0 1801.0 2101.0

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 borrowed from communication theory). The encoding activities occur at the transmitter while the decoding take place at the receiver. Figure 14.12 illustrates encoding and decoding with the use of the codebook A. The channel functions as follows. Any input information, whatever its nature, is encoded (represented) in terms of the elements of the codebook. In this internal format, encoded information is sent with or without processing across the channel. Using the same codebook, the output message is decoded at the receiver.

Figure 14.12. Fuzzy-communication channel with fuzzy encoding and decoding.

Fuzzy-set literature has traditionally used the terms fuzzification and defuzzification to denote encoding and decoding, respectively. These are, unfortunately, quite misleading and meaningless terms because they mask the very nature of the processing that takes place in fuzzy reasoning. They neither address any design criteria nor introduce any measures aimed at characterizing the quality of encoding and decoding information completed by the fuzzy channel.

The next two sections are examples of the application of fuzzy logic and fuzzy reasoning to decision-making processes, where the available data sets are ambiguous. These applications include multifactorial evaluation and extraction of fuzzy rules-based models from large numeric data sets.

14.5 MULTIFACTORIAL EVALUATION

Multifactorial evaluation is a good example of the application of the fuzzy-set theory to decision-making processes. Its purpose is to provide a synthetic evaluation of an object relative to an objective in a fuzzy-decision environment that has many factors. Let U = {u1, u2, … , un} be a set of objects for evaluation, let F = {f1, f2, … , fm} be the set of basic factors in the evaluation process, and let E = {e1, e2, … , ep} be a set of descriptive grades or qualitative classes used in the evaluation. For every object u ∈ U, there is a single-factor evaluation matrix R(u) with dimensions m × p, which is usually the result of a survey. This matrix may be interpreted and used as a 2-D MF for fuzzy relation F × E.

With the preceding three elements, F, E, and R, the evaluation result D(u) for a given object u ∈ U can be derived using the basic fuzzy-processing procedure: the product of fuzzy relations through max–min composition. This has been shown in Figure 14.13. An additional input to the process is the weight vector W(u) for evaluation factors, which can be viewed as a fuzzy set for a given input u. A detailed explanation of the computational steps in the multifactorial-evaluation process will be given through two