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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 unique way. Consequently, the individual clusters are separated by sharp boundaries. In practice, such boundaries are often not very natural or even counterintuitive. Rather, the boundary of single clusters and the transition between different clusters are usually “smooth” rather than abrupt. This is the main motivation underlying fuzzy extensions to clustering algorithms. In fuzzy clustering an object may belong to different clusters at the same time, at least to some extent, and the degree to which it belongs to a particular cluster is expressed in terms of a membership degree.

The most frequent application of fuzzy set theory in data mining is related to the adaptation of rule-based predictive models. This is hardly surprising, since rule-based models have always been a cornerstone of fuzzy systems and a central aspect of research in the field. Set of fuzzy rules can represent both classification and regression models. Instead of dividing quantitative attributes into fixed intervals, they employ linguistic terms to represent the revealed regularities. Therefore, no user-supplied thresholds are required, and quantitative values can be directly inferred from the rules. The linguistic representation leads to the discovery of natural