Abstract
In this chapter we discuss how fuzzy logic extends the envelop of the main data mining tasks: clustering, classification, regression and association rules. We begin by presenting a formulation of the data mining using fuzzy logic attributes. Then, for each task, we provide a survey of the main algorithms and a detailed description (i.e. pseudo-code) of the most popular algorithms. There are two main types of uncertainty in supervised learning: statistical and cognitive. Statistical uncertainty deals with the random behavior of nature and all existing data mining techniques can handle the uncertainty that arises (or is assumed to arise) in the natural world from statistical variations or randomness. Cognitive uncertainty, on the other hand, deals with human cognition. Fuzzy set theory, first introduced by Zadeh in 1965, deals with cognitive uncertainty and seeks to overcome many of the problems found in classical set theory. For example, a major problem faced by researchers of control theory is that a small change in input results in a major change in output. This throws the whole control system into an unstable state. In addition there was also the problem that the representation of subjective knowledge was artificial and inaccurate. Fuzzy set theory is an attempt to confront these difficulties and in this chapter we show how it can be used in data mining tasks.
In this chapter we discuss how fuzzy logic extends the envelop of the main data mining tasks: clustering, classification, regression and association rules. We begin by presenting a formulation of the data mining using fuzzy logic attributes. Then, for each task, we provide a survey of the main algorithms and a detailed description (i.e. pseudo-code) of the most popular algorithms.
TopBackground
Data mining is a term coined to describe the process of sifting through large and complex databases for identifying valid, novel, useful, and understandable patterns and relationships. Data mining involves the inferring of algorithms that explore the data, develop the model and discover previously unknown patterns. The model is used for understanding phenomena from the data, analysis and prediction. The accessibility and abundance of data today makes knowledge discovery and data mining a matter of considerable importance and necessity.
We begin by presenting some of the basic concepts of fuzzy logic. The main focus, however, is on those concepts used in the induction process when dealing with data mining. Since fuzzy set theory and fuzzy logic are much broader than the narrow perspective presented here, the interested reader is encouraged to read Zimmermann (2005).
In classical set theory, a certain element either belongs or does not belong to a set. Fuzzy set theory, on the other hand, permits the gradual assessment of the membership of elements in relation to a set.
Let U be a universe of discourse, representing a collection of objects denoted generically by u. A fuzzy set A in a universe of discourse U is characterized by a membership function µA which takes values in the interval [0, 1]. Where µA(u) = 0 means that u is definitely not a member of A and µA(u) = 1 means that u is definitely a member of A.
The above definition can be illustrated on the vague set of Young. In this case the set U is the set of people. To each person in U, we define the degree of membership to the fuzzy set Young. The membership function answers the question ”to what degree is person u young?”. The easiest way to do this is with a membership function based on the person’s age. For example Figure 1 presents the following membership function:
Figure 1. Membership function for the young set
(1)Given this definition, John, who is 18 years old, has degree of youth of 0.875. Philip, 20 years old, has degree of youth of 0.75. Unlike probability theory, degrees of membership do not have to add up to 1 across all objects and therefore either many or few objects in the set may have high membership. However, an object’s membership in a set (such as “young”) and the set’s complement (“not young”) must still sum to 1.
The main difference between classical set theory and fuzzy set theory is that the latter admits to partial set membership. A classical or crisp set, then, is a fuzzy set that restricts its membership values to {0,1}, the endpoints of the unit interval. Membership functions can be used to represent a crisp set. For example, Figure 2 presents a crisp membership function defined as:
Figure 2. Membership function for the crisp young set
Key Terms in this Chapter
Fuzzy Logic: A type of logic that recognizes more than simple true and false values. With fuzzy logic, propositions can be represented with degrees of truthfulness and falsehood thus it can deal with imprecise or ambiguous data. Boolean logic is considered to be a special case of fuzzy logic
Instance: A single object of the world from which a model will be learned, or on which a model will be used
Association Rules: Techniques that find in a database conjunctive implication rules of the form “X and Y implies A and B.”
Attribute: A quantity describing an instance. An attribute has a domain defined by the attribute type, which denotes the values that can be taken by an attribute
Classifier: A structured model that maps unlabeled instances to finite set of classes.
Data Mining: The core of the KDD process, involving the inferring of algorithms that explore the data, develop the model, and discover previously unknown patterns
Clustering: The process of grouping data instances into subsets in such a manner that similar instances are grouped together into the same cluster, while different instances belong to different clusters
Knowledge Discovery in Databases (KDD): A nontrivial exploratory process of identifying valid, novel, useful, and understandable patterns from large and complex data repositories