Data Mining and Knowledge Discovery Handbook, 2 Edition part 53 ppsx

500
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Swagatam Das and Ajith Abraham
24
Using Fuzzy Logic in Data Mining
Lior Rokach
1
Department of Information System Engineering, Ben-Gurion University, Israel

Summary. 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 pre-
senting 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. However this chapter will not profoundly discuss neuro-fuzzy
techniques, assuming that there will be a dedicated chapter for this issue.
24.1 Introduction
There are two main types of uncertainty in supervised learning: statistical and cognitive. Sta-
tistical uncertainty deals with the random behavior of nature and all existing data mining tech-
niques can handle the uncertainty that arises (or is assumed to arise) in the natural world from
statistical variations or randomness. While these techniques may be appropriate for measuring
the likelihood of a hypothesis, they says nothing about the meaning of the hypothesis.
Cognitive uncertainty, on the other hand, deals with human cognition. Cognitive uncer-
tainty can be further divided into two sub-types: vagueness and ambiguity.
Ambiguity arises in situations with two or more alternatives such that the choice between
them is left unspecified. Vagueness arises when there is a difficulty in making a precise dis-

tinction in the world.
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 un-
stable state. In addition there was also the problem that the representation of subjective knowl-
edge 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.
24.2 Basic Concepts of Fuzzy Set Theory
In this section we present 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
O. Maimon, L. Rokach (eds.), Data Mining and Knowledge Discovery Handbook, 2nd ed.,
DOI 10.1007/978-0-387-09823-4_24, © Springer Science+Business Media, LLC 2010
506 Lior Rokach
set theory and fuzzy logic are much broader than the narrow perspective presented here, the
interested reader is encouraged to read (Zimmermann, 2005)).
24.2.1 Membership function
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.
Definition 1. 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 member-
ship function
μ
A
which takes values in the interval [0, 1]. Where
μ
A
(u)=0 means that u is
definitely not a member of A and

μ
A
(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 24.1 presents the following membership function:
μ
Young
(u)=
⎧
⎨
⎩
0
1
32−age(u)
16
age(u) > 32
age(u) < 16
otherwise
(24.1)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7

0.8
0.9
1
10 15 20 25 30 35
Age
Young Membership
Fig. 24.1. Membership function for the young set.
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 objects membership in a set (such as ”young”) and
the sets complement (”not young”) must still sum to 1.
24 Using Fuzzy Logic in Data Mining 507
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 24.2 presents a crisp membership function
defined as:
μ
CrispYoung
(u)=

0 age(u) > 22
1 age(u) ≤ 22
(24.2)
0
0.1
0.2
0.3
0.4

0.5
0.6
0.7
0.8
0.9
1
10 15 20 25 30 35
Age
Crisp Young Membership
Fig. 24.2. Membership function for the crisp young set.
In regular classification problems, we assume that each instance takes one value for each
attribute and that each instance is classified into only one of the mutually exclusive classes. To
illustrate how fuzzy logic can help data mining tasks, we introduce the problem of modelling
the preferences of TV viewers. In this problem there are 3 input attributes:
A = {Time of Day,Age Group,Mood}
and each attribute has the following values:
• dom(Time of Day)={Morning,Noon,Evening,Night}
• dom(Age Group)={Young,Adult}
• dom(Mood)={Happy,Indifferent,Sad,Sour,Grumpy}
The classification can be the movie genre that the viewer would like to watch, such as
C = {Action,Comedy,Drama}.
All the attributes are vague by definition. For example, peoples feelings of happiness, in-
difference, sadness, sourness and grumpiness are vague without any crisp boundaries between
them. Although the vagueness of ”Age Group” or ”Time of Day” can be avoided by indicating
the exact age or exact time, a rule induced with a crisp decision tree may then have an artificial
crisp boundary, such as ”IF Age < 16 THEN action movie”. But how about someone who is
508 Lior Rokach
17 years of age? Should this viewer definitely not watch an action movie? The viewer pre-
ferred genre may still be vague. For example, the viewer may be in a mood for both comedy
and drama movies. Moreover, the association of movies into genres may also be vague. For

instance the movie ”Lethal Weapon” (starring Mel Gibson and Danny Glover) is considered
to be both comedy and action movie.
Fuzzy concept can be introduced into a classical problem if at least one of the input at-
tributes is fuzzy or if the target attribute is fuzzy. In the example described above , both input
and target attributes are fuzzy. Formally the problem is defined as following (Yuan and Shaw,
1995):
Each class c
j
is defined as a fuzzy set on the universe of objects U. The member-
ship function
μ
c
j
(u) indicates the degree to which object u belongs to class c
j
. Each at-
tribute a
i
is defined as a linguistic attribute which takes linguistic values from dom(a
i
)=
{v
i,1
,v
i,2
, ,v
i,
|
dom(a
i

)
|
}. Each linguistic value v
i,k
is also a fuzzy set defined on U. The mem-
bership
μ
v
i,k
(u) specifies the degree to which object u’s attribute a
i
is v
i,k
. Recall that the
membership of a linguistic value can be subjectively assigned or transferred from numerical
values by a membership function defined on the range of the numerical value.
Typically, before one can incoporate fuzzy concepts into a data mining application, an
expert is required to provide the fuzzy sets for the quantitative attributes, along with their
corresponding membership functions. Alternatively the appropriate fuzzy sets are determined
using fuzzy clustering.
24.2.2 Fuzzy Set Operations
Like classical set theory, fuzzy set theory includes operations union, intersection, complement,
and inclusion, but also includes operations that have no classical counterpart, such as the
modifiers concentration and dilation, and the connective fuzzy aggregation. Definitions of
fuzzy set operations are provided in this section.
Definition 2. The membership function of the union of two fuzzy sets A and B with membership
functions
μ
A
and

μ
B
respectively is defined as the maximum of the two individual membership
functions:
μ
A∪B
(u)=max{
μ
A
(u),
μ
B
(u)} (24.3)
Definition 3. The membership function of the intersection of two fuzzy sets A and B with mem-
bership functions
μ
A
and
μ
B
respectively is defined as the minimum of the two individual
membership functions:
μ
A∩B
(u)=min{
μ
A
(u),
μ
B

(u)} (24.4)
Definition 4. The membership function of the complement of a fuzzy set A with membership
function
μ
A
is defined as the negation of the specified membership function:
μ
A
(u)=1 −
μ
A
(u). (24.5)
To illustrate these fuzzy operations, we elaborate on the previous example. Recall that
John has a degree of youth of 0.875. Additionally John’s happiness degree is 0.254. Thus, the
membership of John in the set Young ∪ Happy would be max(0.875, 0.254)=0.875, and its
membership in Young ∩ Happy would be min(0.875,0.254)=0.254.
24 Using Fuzzy Logic in Data Mining 509
It is possible to chain operators together, thereby constructing quite complicated sets. It
is also possible to derive many interesting sets from chains of rules built up from simple
operators. For example John’s membership in the set
Young ∪ Happy would be max(1 −
0.875,0.254)=0.254
The usage of the max and min operators for defining fuzzy union and fuzzy intersection,
respectively is very common. However, it is important to note that these are not the only
definitions of union and intersection suited to fuzzy set theory.
Definition 5. The fuzzy subsethood S(A,B) measures the degree to which A is a subset of B.
S(A,B)=
M(A ∩B)
M(A)
(24.6)

where M(A) is the cardinality measure of a fuzzy set A and is defined as
M(A)=
∑
u∈U
μ
A
(u) (24.7)
The subsethood can be used to measure the truth level of the rule of classification rules.
For example given a classification rule such as ”IF Age is Young AND Mood is Happy THEN
Comedy” we have to calculate S(Hot ∩Sunny,Swimming) in order to measure the truth level
of the classification rule.
24.3 Fuzzy Supervised Learning
In this section we survey supervised methods that incoporate fuzzy sets. Supervised meth-
ods are methods that attempt to discover the relationship between input attributes and a target
attribute (sometimes referred to as a dependent variable). The relationship discovered is repre-
sented in a structure referred to as a model. Usually models describe and explain phenomena,
which are hidden in the dataset and can be used for predicting the value of the target attribute
knowing the values of the input attributes.
It is useful to distinguish between two main supervised models: classification models
(classifiers) and Regression Models. Regression models map the input space into a real-value
domain. For instance, a regressor can predict the demand for a certain product given its char-
acteristics. On the other hand, classifiers map the input space into pre-defined classes. For
instance, classifiers can be used to classify mortgage consumers as good (fully payback the
mortgage on time) and bad (delayed payback).
Fuzzy set theoretic concepts can be incorporated at the input, output, or into to backbone
of the classifier. The data can be presented in fuzzy terms and the output decision may be
provided as fuzzy membership values. In this chapter we will concentrate on fuzzy decision
trees.
24.3.1 Growing Fuzzy Decision Tree
Decision tree is a predictive model which can be used to represent classifiers. Decision trees

are frequently used in applied fields such as finance, marketing, engineering, medicine and
security (Moskovitch et al. (2008)). In the opinion of many researchers decision trees gained
popularity mainly due to their simplicity and transparency. Decision tree are self-explained.
There is no need to be an expert in data mining in order to follow a certain decision tree.