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

Figure 14.15. Granulation of a two-dimensional I/O space.

Figure 14.16. Selection of characteristic points in a granulated space.

Figure 14.17. Graphical representation of generated fuzzy rules and the resulting crisp approximation.

Note how the generated model misses the extremes that lie far from the existing rule centers. This behavior occurs because only one pattern per rule is used to determine the outcome of this rule. Even a combined approach would very much depend on the predefined granulation. If the function to be modeled has a high variance inside one rule, the resulting fuzzy rule model will fail to model this behavior.

For practical applications it is obvious, however, that using such a predefined, fixed grid results in a fuzzy model that will either not fit the underlying functions very well or consist of a large number of rules because of small granulation. Therefore, new approaches have been introduced that automatically determine the granulations of both input and output variables based on a given data set. We will explain the basic steps for one of these algorithms using the same data set from the previous example and the graphical representation of applied procedures.

1. Initially, only one MF is used to model each of the input variables as well as the output variable, resulting in one large rule covering the entire feature space. Subsequently, new MFs are introduced at points of maximum error (the maximum distance between data points and the obtained crisp approximation). Figure 14.18 illustrates this first step in which the crisp approximation is represented with a thick line and the selected point of maximal error with a triangle.

2. For the selected point of maximum error, new triangular fuzzy values for both input and output variables are introduced. Processes of granulation, determining fuzzy rules in the form of space regions, and crisp approximation are repeated for a space, with additional input and output fuzzy values for the second step—that means two fuzzy values for both input and output variables. The final results of the second step, for our example, are presented in Figure 14.19.

3. Step 2 is repeated until a maximum number of divisions (fuzzy values) is reached, or the approximation error remains below a certain threshold value. Figures 14.20 and 14.21 demonstrate two additional iterations of the algorithm for a data set. Here granulation was stopped after a maximum of four MFs was generated for each variable. Obviously this algorithm is able to model extremes much better than the previous one with a fixed granulation. At the same time, it has a strong tendency to favor extremes and to concentrate on outliers. The final set of fuzzy rules, using dynamically created fuzzy values Ax to Dx and Ay to Dy for input and output variables, isR1:IF x is Ax, THEN y is Ay.R2:IF x is Bx, THEN y is By.R3:IF x is Cx, THEN y is Cy.R4:IF x is Dx, THEN y is Dy.

Figure 14.18. The first step in automatically determining fuzzy granulation.

Figure 14.19. The second step (first iteration) in automatically determining granulation.

Figure 14.20. The second step (second iteration) in automatically determining granulation.

Figure 14.21. The second step (third iteration) in automatically determining granulation.

14.7 DATA MINING AND FUZZY SETS

There is a growing indisputable role of fuzzy set technology in the realm of data mining. In a data mining process, discovered models, learned concepts, or patterns of interest are often vague and have non-sharp boundaries. Unfortunately, the representation of graduality is often foiled in data-mining applications, especially in connection with the learning of predictive models. For example, the fact that neural networks are often used as data-mining methods, although their learning result (weight matrices of numbers) is hardly interpretable, shows that in contrast to the standard definition the goal of understandable models is often neglected. In fact, one should recognize that graduality is not only advantageous for expressing concepts and patterns, but also for modeling the qualifying properties and relations. Of course, correctness, completeness, and efficiency are important in data-mining models, but in order to manage systems that are more and more complex, there is a constantly growing demand to keep the solutions conceptually simple and understandable. Modern technologies are accepted more readily, if the methods applied and models derived are easy to understand, and the results can be checked against human intuition.

The complexity of the learning task, obviously, leads to a problem: When learning from information, one must choose between mostly quantitative methods that achieve good performances, and qualitative models that explain to a user what is going on in the complex system. Fuzzy-set theory has the potential to produce models that are more comprehensible, less complex, and more robust. Fuzzy information granulation appears to be appropriate approach for trading off accuracy against complexity and understandability of data-mining models. Also, fuzzy-set theory in conjunction with possibility theory, can contribute considerably to the modeling and processing of various forms of uncertain and incomplete information available in large real-world systems.

The tools and technologies that have been developed in fuzzy-set theory have the potential to support all of the steps that comprise a process of knowledge discovery. Fuzzy methods appear to be particularly useful for data pre- and postprocessing phases of a data-mining process. In particular, it has already been used in the data-selection phase, for example, for modeling vague data in terms of fuzzy sets, to “condense” several crisp observations into a single fuzzy one, or to create fuzzy summaries of the data.

Standard methods of data mining can be extended to include fuzzy-set representation in a rather generic way. Achieving focus is important in data mining because there are too many attributes and values to be considered and can result in combinatorial explosion. Most unsupervised data-mining approaches try to achieve focus by recognizing the most interesting structures and their features even if there is still some level of ambiguity. For example, in standard clustering, each sample is assigned to one cluster in a