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An Approach for the Extraction of Classification Rules - ICMC
Instituto de Ciências Matemáticas e de Computação
Computer Science and Mathematics Institute
ISSN - 0103-2569
An Approach for the Extraction of Classification Rules
from Fuzzy Formal Contexts
Marcos Evandro Cintra
Trevor P. Martin
Maria Carolina Monard
Heloisa de Arruda Camargo
RELATÓRIOS TÉCNICOS DO ICMC
ICMC TECHNICAL REPORTS
São Carlos
August/2011
An Approach for the Extraction of Classification
Rules from Fuzzy Formal Contexts
Marcos Evandro Cintra
Maria Carolina Monard
Trevor P. Martin
Heloisa de Arruda Camargo
1
Universidade de São Paulo – USP
Instituto de Ciências Matemáticas e de Computação – ICMC
Laboratório de Inteligência Artificial – LABIC
Av. Trabalhador São-carlense, 400 – Centro
Caixa Postal: 668 – CEP: 13560-970 – São Carlos – SP – Brasil
cintra,[email protected]
[email protected], [email protected]
Abstract. This work describes experiments carried out using
fuzzy formal concept analysis for the generation of fuzzy classification rules to be used by a genetic process. These rules are simply
the intention of the formal concepts extracted from a fuzzy-based
formal context. The motivation we have is the need for a method to
generate fuzzy classification rules to be used as the search space
of the genetic process with reasonable computational cost. The hypothesis we work on is that the extraction of classification rules
from a fuzzy formal context can be a suitable alternative to generate the search space of a genetic process due to its low complexity and straightforwardness/simplicity, generating not only good
rules, due to the fact that the support of these rules is a subproduct
of the extraction of them, but also better interpretable rules, with a
varying number of conjunctions in their antecedent part. In order
to reduce the complexity of the extraction of the formal concepts,
besides the generation of all existing formal concepts representing
classification rules, an alternative approach is also proposed and
studied: the use of the lattice in order to sequentially extract classification rules. Both ideas are detailed together with the preliminary
experiments and results.
Contents
1 Introduction
1
2 Genetic Fuzzy Systems
3
3 Methods for the Generation of Rules to Form the Search Space
of Genetic Fuzzy Systems
5
3.1 Use of Heuristic Criteria
. . . . . . . . . . . . . . . . . . . . .
5
3.2 Association Rules Extraction . . . . . . . . . . . . . . . . . . .
6
4 Formal Concept Analysis
7
4.1 The Extraction of Formal Concepts Using the NextNeighbor
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Adaptation of the NextClosure Algorithm for Formal Concept
Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3 Proposal for Extracting Fuzzy Rules from a Formal Context . 18
5 Preliminary Experiments
19
5.1 Experiments and Results . . . . . . . . . . . . . . . . . . . . . 19
5.2 Additional Experiments on Attribute Exploration to Reduce
the Number of Extracted Rules . . . . . . . . . . . . . . . . . 22
6 Conclusions and Future Work
25
References
28
ii
1. Introduction
Fuzzy systems, which are basically systems with variables based on fuzzy
logic, have been successfully used for the solution of problems in many
areas, including pattern classification, optimization, and control of processes (Dumitrescu et al., 2000; Pedrycz, 1996). Recently, fuzzy systems
have also been used in conjunction with Formal Concept Analysis (FCA),
a theory of data analysis which identifies conceptual structures among
data sets.
The focus of this work is the automatic definition of fuzzy systems,
especially those known as Rule Based Fuzzy Systems (RBFS), which usually have two main components: a Knowledge Base (KB) and an Inference
Mechanism (IM). The KB comprises the Fuzzy Rule Base (FRB), i.e., a
set of fuzzy rules that represents a given problem, and the Fuzzy Data
Base (FDB), which contains the definitions of the fuzzy sets related to the
linguistic variables used in the FRB. The IM is responsible for carrying
out the required computation that uses inferences to derive the output
(or conclusion) of the system, based on both the KB and the input to the
system.
The special term “Genetic Fuzzy System” (GFS) was coined by the
community to refer to fuzzy systems that use a genetic algorithm to create
or adjust one or more of their components (Cordon et al., 2004). Specifically, the classification of GFSs, according to Herrera (2008), takes into
account if the goal is:
• the genetic tuning of an existing knowledge base;
• the genetic learning of components of the knowledge base.
This work is focused on the genetic learning of the rule base, one
of the components of the knowledge base.
Regarding the generation of rules to form the search space of a
Genetic Algorithm (GA), Ishibuchi and Yamamoto (2004) proposed an approach based on the rules confidence and support to preselect rules. A
predefined number of rules with 0, 1, 2, and 3 antecedent conjunctions
were generated for the wine dataset (Frank and Asuncion, 2010) and used
with a GA. A similar approach, named D O C-B ASED, is proposed in (Cintra
et al., 2007), in which, after the exhaustive generation of possible rules, a
subset of them is selected to form the search space of a GA according to
the degree of coverage of the rules. However, since the task of generating
1
all possible rule combinations has exponential complexity, depending on
the number of fuzzy variables and sets, the number of possible rules can
be very large, interfering with the codification of the chromosomes, and
overloading the whole genetic learning process. Thus, for datasets described by many features, a preselection of the most relevant features is
important. Also, preselecting the most important possible rules to form
the seach space might be essential for datases described by many features.
The theory of FCA was introduced in 1982 by Wille (1982) and
has since then grown rapidly. FCA has successfully been applied to
many fields, such as medicine and psychology, musicology, linguistic
databases, library and information science, software re-engineering, civil
engineering, ecology, and others. A strong feature of FCA is its capability
of producing graphical visualizations of the inherent structures among
data. Another strong feature of FCA is related to its mathematical lattices since they can be interpreted as classification systems and even
used for the extraction of association and classification rules. For the extraction of association rules, these rules are composed of the attributes
present in the formal concepts extracted from the formal context, i.e.,
the intention of the formal concept. Their support is calculated directly
from the process of extracting the formal concept, with no extra computation cost. Regarding the classification rules, they can be extracted
from the formal concepts containing a class in their intention, discarding
their extensions. Several methods for the extraction of formal concepts
have been proposed (Ganter, 1984, 2002; Krajca et al., 2010). The one
by (Krajca et al., 2010), is particularly interesting because it proposes a
new method for the structuring of the formal context that reduces considerably the processing time for the extraction of the formal concepts
and allows the parallelization of the process.
The integration of FCA with fuzzy logic has also produced several
proposals (Zheng et al., 2009; Sigmund et al., 2010; Wolff, 2002; Belohlávek et al., 2005). One of the reasons for this integration is due to
the fact that, since FCA requires discrete attributes, the fuzzy theory can
be used to transform continuous attributes into binary ones using fuzzy
sets, in order to define the formal context.
Our interest in FCA is related to its simplicity in the extraction of
classification rules from data. These rules are particularly interesting be2
cause they present a variable number of conjunctions in their antecedent
and are extracted from examples. Thus, our hypothesis is that we can
extract fuzzy rules from a fuzzy FCA to populate the search space of a
genetic process that, in turn, will be used to induce a fuzzy rule base.
We intend to use the genetic rule selection approach (Herrera, 2008) representing each chromosomes as complete rule bases with a predefined
number of rules.
The remainder of this work is organized as follows. Section 2
presents the basic concepts of Genetic Fuzzy Systems. Section 3 describes some of the most frequently used methods in the literature for
the generation of genetic search spaces. Section 4 introduces the theory
of Formal Concept Analysis and fuzzy scaling, as well as our proposal for
the use of FCA to extract fuzzy rules. Section 5 presents the preliminary
experiments. Section 6 presents the conclusions and future work.
2. Genetic Fuzzy Systems
Accordingly to Herrera (2008), GFSs can be classified as:
1. Genetic tuning: if there exists a knowledge base, a genetic tuning
process for improving the rule-based fuzzy system performance is
applied without changing the existing fuzzy rule base;
2. Genetic learning: a component of the knowledge base is learnt,
either the fuzzy rule base, fuzzy data base, or both. An adaptive
inference mechanism can also be included in the process.
The genetic tuning can be further divided according to the taxonomy provided in Figure 1(a), extracted and adapted from Herrera (2008),
into: i) genetic tuning of KB parameters; ii) genetic adaptive inference
engine. Similarly, the genetic learning can also be further divided into:
i) genetic KB learning; and ii) genetic learning of DB components and
inference engine parameters.
Our work is related to the genetic learning of the FRB, which is
a part of the KB. Figure 1(b) provides the complete classification of the
genetic KB learning, according to Herrera (2008), which is also listed with
some references next.
2.1 Genetic rule learning with a predefined fuzzy data base (Thrift, 1991);
2.2 Genetic rule selection with a priori rule extraction (Cintra and Camargo, 2007);
3
2.3 Genetic fuzzy data base learning Cordon et al. (2001);
2.4 Simultaneous genetic learning of knowledge
nents (Homaifar and McCormick, 1995).
base
compo-
Figure 1. Genetic Fuzzy Systems Classification Herrera (2008).
More specifically, our work focuses on methods to form the search
space of a GA in order to build a GFS, which is part of item 2.2 from the
previous list. Next section surveys some of the key existing approaches
for this task, i.e., the task of generating the search space for a GFS.
4
3. Methods for the Generation of Rules to Form the Search
Space of Genetic Fuzzy Systems
In this section we describe some of the most frequently used methods in
the literature for the generation of genetic search spaces, presenting and
discussing their pros and cons.
3.1. Use of Heuristic Criteria
Ishibuchi and Murata (1995) propose the generation of classification
rules to form the search space of a GA using the following steps:
1. generation of all possible rule antecedent combinations;
2. calculation of the degree of certainty of each antecedent combination with each possible class using a set of training examples;
3. definition of the consequent of each rule antecedent combination
as the class with highest degree of certainty.
Although this approach suits low-dimensional domains, it is not
scalable to larger domains. Thus, Ishibuchi and Yamamoto (2004) propose the extensive generation of rule antecedent and class calculation
using the degree of certainty with preselection of candidate rules as a
more feasible alternative to larger domains. The criteria used to preselect
rules are the confidence and support measures, as defined in the data
mining context for association rules.
Similarly to the approach proposed by Ishibuchi and Yamamoto
(2004), Cintra et al. (2007) propose the generation of the search space
by extensively generating all possible rules combining all attributes, then
calculating their degree of coverage as a criterium to preselect a subset
of them. This way, all rules were composed by a combination of valid
linguistic values for each attribute. The Degree of Coverage (DoC) was
used as an indication of the classification power of these rules.
In both approaches described previously, the main issue is related
to the exponential complexity of the exhaustive generation of all possible
rules. While these approaches are feasible for domains described by a
small number of attributes combined with a reduced set of linguistic values for each attribute, they are not directly scalable for larger domains
or when attributes are described by many linguistic values. Another issue is the computational cost required to calculate the degree of certainty
and degree of coverage of these rules when combined with the substantial
computational effort and processing time required by GAs.
5
Regarding the advantages of these methods, and others based on
heuristic criteria, we can state the following:
1. Since these approaches preselect only relevant rules (according to
criteria related to their classification power), the search space can
be substantially reduced;
2. Due to the fact that the rules are previously selected according to
their classification power, the search process of the GA is reduced
in terms of time;
3. With the preselection of rules, each rule in the search space has
an index that is used in the chromosomes, thus, the chromosome
codification is simplified (each position of the chromosome has an
index to a rule or a special value to represent the absence of a rule).
This way, the whole genetic process is optimized saving processing
time;
4. Although the approach is not scalable to larger domains, or when
a large number of fuzzy terms is used, one might argue that it is,
nevertheless, totally feasible and produces good results for a large
number of real domains. Also, the use of a feature subset selection
algorithm reduces the computational cost and time required by this
approach, making it scalable to larger domains.
3.2. Association Rules Extraction
Another option for the generation of the rules to form the search space of
a GA is to use an association rule extraction algorithm, such as Apriori
(Agrawal and Srikant, 1994). Given a set of examples described by a set
of attributes, the idea is to find rules based on associated items. The support and confidence values of these rules are usually considered for their
evaluation. Although association rules are not concerned with supervised
domains, association rule algorithms can be used to extract associative
classification rules, or simply classification rules, from a set of examples,
by simply fixing the consequent of the rules as the class attribute.
In the literature, it is possible to find proposals to generate fuzzy
rule-based classifiers using fuzzy association rules (Hu et al., 2002; Pach
et al., 2008). Hu et al. (2002) propose the generation of large fuzzy grids
from training examples by fuzzy partitioning each attribute; these grids
are then use to generate fuzzy association rules for classification. Pach
et al. (2008) use the Fuzzy APriori algorithm to search for frequent fuzzy
6
item sets to form classification rules. The set of these rules is then pruned
using the complexity, importance, and generality measures of the rules,
forming a fuzzy classifier.
While approaches for the generation of classifiers using fuzzy association rules are abundant in the literature, to the best of our knowledge,
the closest proposal using association rules and GAs to a GFS can be
found in (Ishibuchi et al., 2006), in which the authors propose the use of
association rule mining to define classifiers using a technique to search
for Pareto-optimal rule sets. The authors first mine all possible classification rules using a minimum support and confidence values. These
rules are then used by an evolutionary multiobjective optimization algorithm to search for Pareto-optimal rule sets. The three objectives used
are: i) number of correctly classified training patterns; ii) number of selected rules (number of rules in the classifier); and iii) total number of
antecedent conditions over the selected rules in the classifiers. It is important to notice that the rules generated in (Ishibuchi et al., 2006) are
not fuzzy, but classic association rules.
An advantage of this approach based on association rules is related
to the fact that the support of these rules can be used in a selection
process, discarding rules with very low support, which would not help
improve the final rule base, speeding up the genetic search process, and
contributing to a better interpretability of the final FRB.
The disadvantages might include the fact that extracting all possible rules is an exponentially complex task. Furthermore, the number
of attributes to be included in the extraction process must be defined
in advance and, thus, the total number of extracted rules might not be
sufficient to form the search space, creating a dilemma: the number of
combined attributes to be used versus the number of possible extracted
rules.
Next, we briefly introduce the topic of Formal Concept Analysis
(FCA) and present our proposed approach for the generation of the genetic
search space using FCA.
4. Formal Concept Analysis
Formal Concept Analysis (FCA) is a mathematical technique for extracting
concepts and structure from data introduced in Wille (1982), which is
7
becoming increasingly popular, especially for allowing the visualization of
structures in data.
FCA transforms a formal context into a concept lattice. A formal
context is a representation of the relation between objects and their attributes. The basic data structure in FCA is the context, which is normally represented in a table form where the columns represent the attributes and the rows represent the objects. The context table contains 1
(true) in cell (i, j) if object i has attribute j, and 0 (false) otherwise. Formally, a context is a triple k = (G, M, I), where G is a set of objects, M is
a set of attributes, and I is a binary relation I ⊆ G × M . Given a set of
objects A ⊆ G, the shared image of A in M is defined as:
A↑ = {m ∈ M |(g, m) ∈ I ∀ g ∈ A}
(1)
Consider the formal context present in Table 1. The set of objects,
called G or extension, is formed by {O1 , O2 , O3 , O4 }. The set of attributes,
called M , is formed by {At1 , At2 , At3 , At4 }. Let us define A as set {O1 , O2 , O4 }.
The shared image of A in M , noted as A↑ , is defined by the set containing
all attributes shared by all elements of A. This way, A↑ = {At2 }.
Table 1. Toy Example.
O1
O2
O3
O4
At1
1
0
1
1
At2
1
1
1
1
At3
0
1
0
0
At4
0
0
0
1
Similarly, for a set of attributes B ⊆ M , its shared image in G is:
B ↓ = {g ∈ G|(g, m) ∈ I ∀ m ∈ B}
(2)
Consider the formal context present in Table 1. Let us define B
as set {At1 , At2 }. The shared image of B in G, noted as B ↓ , is defined
by the set containing all objects sharing all attributes of B. This way,
B ↓ = {O1 , O2 , O3 }. Let us now define B as set {At3 , At4 }. The shared image
of B in G is the empty set, since there are no objects that share attributes
At3 and At4 at the same time. Thus , B ↓ = ∅.
The pair (A, B) ∈ G × M is a formal concept of (G, M, I) if and only if
8
A ⊆ G, B ⊆ M and A = B ↓, B = A ↑. A is called the extent of the concept
and B is called the intent of the concept (Wille, 1982). In other words,
Equation 1 defines the collection of all attributes shared by all objects
from A, and Equation 2 defines the collection of all objects sharing all the
attributes from B. A ↑ and B ↓ are also commonly noted as A0 and B 0 by
the FCA community, and this notation is adopted in the remaining of this
work.
In traditional FCA, the attributes is binary, although multi-valued
contexts are much more common than binary-valued ones. For attributes
that can take a range of values, the idea of “conceptual scaling” that
transforms a many-valued attribute (e.g. a number) into a symbolic
attribute can be used. For example, an attribute such as “height in
centimetres”, given an integer or real value between 0 and 200, could
be transformed into attributes “height-less-than-50”, “height-from-50-to99”, etc. These derived attributes have true/false values and can thus be
treated within the FCA framework.
A toy example is illustrated in Table 2, which presents an attribute × value table with the name, age, sex, and hair colour of six
people.
Table 2. Attribute × value table of a toy example.
Name
Age Sex Hair Colour
Andy
48
M
Black
Lina
29
F
Black
Mark
23
M
Brown
Martina 46
F
Blonde
Mike
18
M
Brown
Suzy
17
F
Blonde
In order to create the formal context, once FCA only admits binary
attributes, attribute Age is discretized into three attributes, Age "‘≤ 20"’,
"‘> 20 & ≤ 30"’, and "‘> 30"’. Once attributes Sex and Hair Colour already
present nominal values, these values are used to create binary attributes.
Table 3 is the resulting table after the scaling process.
Using the formal context, it is possible to generate a conceptual
lattice that presents the information in a friendly visual way. Figure 2
shows the generated conceptual lattice of the formal context presented
in Table 3. In the lattice structure, formal concepts are represented by
nodes. Attributes are noted slightly above nodes while objects are noted
9
Name
Andy
Lina
Mark
Martina
Mike
Suzy
Table 3. Formal context generated by the scaling of Table 2.
Age
Sex
Hair Colour
≤ 20 > 20 & ≤ 30 > 30 Male Female Black Blonde Brown
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
slightly under nodes. This way, each node will contain the labels of the
attributes and/or objects it shares. The positioning of the nodes can be
arranged in a variety of ways. In the lattice presented in Figure 2, the
nodes were arranged in order to minimize intersections, thus, attributes
are not displayed in the order they occur in the formal context. Another option would be to arrange the nodes respecting the order of the
attributes or the order of the examples in the formal context. In order to
retrieve extensions, one must simply trace all paths leading down from
the node. To retrieve intentions, on the other hand, one must trace all
paths leading up from the node. For example, the intention of the formal
concept represented by the node with the label Mark is {> 20 & ≤ 30,
Brown, Male}. The extension of the formal concept represented by the
node with the label Blonde is {Martina, Suzy}.
Figure 2. Conceptual lattice.
10
As stated previously, the transformation of continuous attributes
into binary ones is commonly called scaling by the FCA community. The
fuzzy theory has also contributed to this task. Wolff (2002) presents an
introduction to the use of the fuzzy theory for the task of FCA scaling.
Some of the advantages of using the fuzzy theory for FCA scaling include:
• the partitioning assume linguistic values, which are easily interpretable by humans, such as young, old, tall, short, etc. This way,
the formal concepts extracted will convey this highly desirable interpretability characteristic;
• unnatural divisions are avoided, such as the division in Table 3 for
the 30 year-old people that are borderlines but must be place in
only one category. The fuzzy logic easily avoids this problem with
the use of the membership degrees that allow one object to belong
to different categories with different degrees of membership;
• it is a natural choice when the interest in FCA is for the extraction
of fuzzy rules, since these rules will reflect the fuzzy data base, also
used by the inference mechanism of the induced classifier.
4.1. The Extraction of Formal Concepts Using the NextNeighbor
Algorithm
For the automatic extraction of formal concepts, several algorithms have
been proposed in the literature (Ganter, 2002; Krajca et al., 2010). In the
experiments presented in this work, we used the NextNeighbor approach
(Ganter, 2002), which optimizes the automatic extraction of formal concepts by finding neighboring concepts. Next, we explain this approach in
details based on the author’s notes (Ganter, 2006).
As stated previously, for a formal context (G, M, I) and given A ⊆ G,
we define:
A0 = {m ∈ M |(g, m) ∈ I ∀ g ∈ A}
(3)
Similarly, given a set of attributes B ⊆ M , the shared image of B in
G is:
B 0 = {g ∈ G|(g, m) ∈ I ∀ m ∈ B}
(4)
These two operators, A0 and B 0 , are the derivation operators for
(G, M, I). The derivation operators can be combined so that, starting with
a set A ⊆ G, we obtain that A0 is a subset of M . By applying the second
operator on this set, we get (A0 )0 , or A00 for short, both constituting sets of
11
objects. It is possible to continue with the process, obtaining A000 , A0000 ,and
so on. But these sets are not all different, as shown next.
Proposition 4.1. For subsets A, A1 , A2 ⊆ G, we have:
1. A1 ⊆ A2 ⇒ A02 ⊆ A01
2. A ⊆ A00
3. A0 ⊆ A000
Dually, for subsets B, B1 , B2 ⊆ M , we have:
1. B1 ⊆ B2 ⇒ B20 ⊆ B10
2. B ⊆ B 00
3. B 0 ⊆ B 000
By combining the derivation operators, we obtain two operators of
the form X ⇒ X 00 , one on G, the other on M . For A ⊆ G, we have that
A00 ⊆ G. The set A00 is called the extent closure of A. Dually, when B ⊆ M ,
then also B 00 ⊆ M , and B 00 is called the intent closure of B. Note that A00
and B 00 have a clear meaning for the data:
• Whenever all objects from a set A ⊆ G have a common attribute m,
then also all objects from A00 have that attribute.
• Whenever an object g ∈ G has all attributes from B ⊆ M , then this
object also has all attributes from B 00 .
Some properties of these operators are listed in the next proposition. They are simple consequences of Proposition 4.1.
Proposition 4.2. For subsets A, A1 , A2 ⊆ G, we have:
1. A1 ⊆ A2 ⇒ A001 ⊆ A002
2. A ⊆ A00
3. (A00 )00 ⊆ A00
Dually, for subsets B, B1 , B2 ⊆ M , we have:
1. B1 ⊆ B2 ⇒ B100 ⊆ B200
2. B ⊆ B 00
3. (B 00 )00 ⊆ B 00
The technical term for operators satisfying the properties given in
Proposition 4.2 is closure operators. The sets which are images of a
closure operator are the closed sets. This way, in the case of a closure
operator X ⇒ X 00 , the closed sets are the sets of the form X 00 .
12
Proposition 4.3. If (G, M, I) is a formal context and A ⊆ G, then A00 is an
extent. Conversely, if A is an extent of (G, M, I), then A = A00 . Dually if
B ⊆ M , then B 00 is an intent, and every intent B satisfies B 00 = B.
Preposition 4.3 follows from the fact that for each subset A ⊆ G, the
pair (A00 , A0 ) is a formal concept, and that similarly (B 00 , B 0 ) is a concept
whenever B ⊆ M .
This way, the closed sets of the closure operator A ⇒ A00 , A ⊆ G
are precisely the extents of (G, M, I), and the closed sets of the operator
B ⇒ B 00 , B ⊆ M , are precisely the intents.
In order to understand the NextNeighbor algorithm, also called
NextClosure, let us consider that a base set M has an arbitrary linear
order:
M = {m1 < m2 < ... < mn }
Every subset S ⊆ M can conveniently be described by a row of
crosses and blanks. For example, if M = {a < b < c < d < e < f < g}, the
subset S = {a, c, d, f }, can be written as shown in Figure 3:
Figure 3. Vector representation of subset S.
Using this notation it is easy to visualize if a given set is a subset
of another given set.
It is now possible to define the lectic order, i.e., a linear order of
the subsets of M , as follows.
Definition 4.1. Let A, B ⊆ M be two distinct subsets. We say that A is
lectically smaller than B, if the smallest element in which A and B differ
belongs to B. Formally
A < B ⇔ ∃i (i ∈ B, i ∈
/ A, ∀j<i (j ∈ A ⇔ j ∈ B)).
For example, {a, c, e, f } < {a, c, d, f }, because the smallest element
in which the two sets differ is d, and this element belongs to the larger
set. Figure 4 shows the vector representation of the previous sets.
It is important to notice that the lectic order extends the subsetorder, i.e.,
13
Figure 4. Vector representation of sets {a, c, e, f } and {a, c, d, f } .
A ⊆ B → A ≤ B.
To lectically compare two sets, the following notation is helpful:
A <i B → (i ∈ B, i ∈
/ A, ∀j<i (j ∈ A ⇔ j ∈ B)).
The previous equation can be read as: A <i B if and only if i is the
smallest element in which A and B differ, and i ∈ B.
Proposition 4.4.
1. A < B if and only if A <i B for some i ∈ M .
2. If A <i B and A <j C with i < j , then C <i B.
For the task of finding closures, and considering the closure operator A → A00 on the base set M , to each subset A ⊆ M its closure is defined
as A00 ⊆ M . In order to find the list of all these closures, the naive algorithm is “for each A ⊆ M , check if the resulting formal concept is already
listed”, which requires an exponential number of lookups in a list that
may have exponential size. A better idea is to generate the closures in
lectic order.
Following, we demonstrate how to compute, given a closed set, the
lectically next closed set. Using this approach, no lookups in the list of
found solutions is necessary. In fact, it is not even necessary to store the
list of solutions. For many applications, it will suffice to generate the list
of elements on demand. Therefore, we do not have to store exponentially
many closed sets. Instead, only one closed set, the current one, is stored.
To find the next closure we define, for A ⊆ M and mi ∈ M :
A ⊕ mi = ((A ∩ {m1 , ..., mi−1 }) ∪ mi )00 .
For example, let A = {a, c, d, f } and mi = e. We first remove all
elements that are greater or equal to mi from A, then we insert mi , and
form the closure. These steps are shown in Figure 5.
14
Figure 5. Finding the next closure.
Proposition 4.5.
1. If i ∈
/ A then A < A ⊕ i.
2. If B is closed and A <i B then A ⊕ i ⊆ B, in particular, A ⊕ i ≤ B.
3. If B is closed and A <i B then A <i A ⊕ i.
Using the statements of Proposition 4.5, it is easy to characterize
the “next”closed set:
Theorem 4.1. The smallest closed set larger than a given set A ⊂ M , with
respect to the lectic order, is:
A⊕i
with i being the largest element of M with A <i A ⊕ i.
Algorithms 1, 2, and 3 implement the steps of computing closed
sets to extract formal concepts.
Input
: A closure operator X → X 00 on a finite set M .
Output : All closed sets in lectic order.
F irst_Closure;
repeat
Output A;
N ext_Closure;
until not success ;
Algorithm 1: All Closures Algorithm.
Notice that the so-called lectic order is simply the lexicographic
order of the formal concepts (Distel, 2010). Algorithm 1 basically repeatedly evokes the formal concept extraction method, i.e., the Next_Closure
method, until no more formal concepts can be found.
The First_Closure method simply returns an empty set representing the union of all attributes.
15
Input
: A closure operator X → X 00 on a finite set M .
Output : The closure A of the empty set.
begin
A := ∅;
end
Algorithm 2: First Closure Algorithm.
4.2. Adaptation of the NextClosure Algorithm for Formal Concept
Extraction
As stated earlier, the algorithm we adopted was the NextClosure whose
details are described in (Ganter, 2002), and presented in Algorithm 1,
Algorith 2, and Algorithm 3. In order to illustrate how to modify this
algorithm for our purpose, the modified version of its core is presented in
Algorithm 4.
Notice that the so-called lectic order is simply the lexicographic order of the formal concepts (Distel, 2010). The First_Closure method simply returns an empty set representing the union of all attributes. This
way, Algorithm 1 basically repeatedly evokes the formal concept extraction method, i.e., the Next_Closure method, until no more formal concepts can be found. The Next_Closure method is presented next (Algorithm 3).
: A closure operator X → X 00 on a finite set M , and a subset
A ⊆ B.
Output : A is replaced by the lectically next closed set.
i := largest element of M ;
success := false;
repeat
if i ∈
/ A then
A := A ∩ {1, 2, ..., (i − 1)} ∪ {i};
B := A00 ;
if (B − A) contains no element < i then then
A := B (A is an extracted concept);
success := true;
end
else
A := A−{i};
end
Input
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
end
i :=pred(i);
3.17 until success or i = smallest element of M ;
3.15
3.16
Algorithm 3: Next Closure Algorithm.
This algorithm can be used to extract formal concepts by analyzing
16
the attributes or the objects. When it uses the attributes, it is called Next
Intent, and it is called Next Extent when used with the objects. This way,
the word element is used instead of attributes or objects.
Algorithm 4 presents the modified version of the NextClosure algorithm to extract formal concepts including a class. The process is quite
direct: once the algorithm looks for neighboring concepts, we just need
to verify whether a found concept has a class or not. If it does, then this
formal concept can be stored in a data structure including the extracted
concepts, otherwise, the process discards the found formal concept and
carries on looking for the next neighboring concept. This modification is
represented by line 4.9 of Algorithm 4.
: A closure operator X → X 00 on a finite set M , and a
subset A ⊆ B.
Output : A is replaced by the lectically next closed set.
i := largest element of M ;
success := false;
repeat
if i ∈
/ A then
A := A ∩ {1, 2, ..., (i − 1)} ∪ {i};
B := A00 ;
if (B − A) contains no element < i then
if B contains a class then
A := B (A is an extracted concept);
end
success := true;
end
else
A := A− {i};
end
Input
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
end
i :=pred(i);
4.19 until success or (i > (numElements - numClasses)) ;
4.17
4.18
Algorithm 4: Adaptation of the Next Closure Algorithm.
Another interesting possibility for the restriction of formal concepts
is the evaluation of their support, which, for a formal context, is just the
number of objects a given formal concept shares divided by the number
of all objects. This way, considering the balancing of the classes, formal
concepts with low support could be discarded, reducing the number of
extracted formal concepts.
17
4.3. Proposal for Extracting Fuzzy Rules from a Formal Context
As discussed in the previous sections, a formal context is the base for
the extraction of formal concepts. These formal concepts can be seen as
associations between attributes based on the existence of objects sharing
these attributes. It is also important to notice that in the process of
extracting formal concepts, their support is calculated automatically.
Our proposal is based on the fuzzy definition of a problem in an
attribute × value table to create a formal context and then obtain the
classification rules. Considering a general dataset for classification purposes with N examples and M attributes, the fuzzy scaling procedure of
our proposal is performed by the following steps:
1. Define the fuzzy database, i.e., the partitions that will define the
fuzzification of the continuous attributes;
2. Create a binary attribute defined by each fuzzy set of each continuous attribute and each value of each discrete attribute;
3. Calculate the membership degree of the input values for each example in each binary attribute (notice that this step is only required
for continuous attributes, once discrete ones will be automatically
defined as true of false);
4. Define a minimum value Amin to guide the scaling of the real values
so that if the membership degree of a certain value for a particular
fuzzy set is equal or higher than Amin , the corresponding attribute
is set to true in the formal context;
5. Use an algorithm to extract formal concepts to extract all existing
classification rules from the formal context;
6. Define a minimum support value to select a subset of the extracted
rules;
The final rule set can then be used as the search space of a GA
to generate a fuzzy system, according to the genetic rule selection with a
priori rule extraction approach, described in Section 2.
In order to reduce the number of possible formal concepts, the
fuzzy sets defining each attribute can be evenly distributed in the partition, so the maximum possible membership degree in the intersections is
0.5. This way, if Amin is set to 0.5, for each original attribute only one of
its binary attributes will be activated. Notice that ties must also be handled, thus, in our implementation we used a random variable to activate
one of 2 tied fuzzy sets.
18
An important issue regarding the extraction of the formal concepts
when using a binary fuzzification of the formal context is related to the
increase in the number of attributes. This total number of attributes will
be equal to the sum of all fuzzy sets describing each attribute, each value
of each discrete attribute, and the number of classes. This increase in the
number of attributes leads to an increase in the total number of formal
concepts that can be extracted. Nevertheless, it is important to notice
that for our purpose, the total number of formal concepts is much larger
than the number of formal concepts we are interested in extracting: since
we want to extract classification rules, we are only interested in extracting
formal concepts that have a class in their intention.
Regarding the total number of formal concepts existing in a formal
context, this number can be estimated using the Metropolis-Hastings algorithm for sampling formal concepts described by Boley et al. (2010).
The process of extracting formal concepts and, consequently, classification rules, can be done by adapting any algorithm for the extraction
of formal concepts. In our experiments we used the NextClosure algorithm described by Ganter (2002), as presented in the previous section.
Since the NextClosure can be used to extract formal concepts by analyzing either the attributes or the objects, when it uses the attributes, it is
called Next Intent, and it is called Next Extent when it uses the objects.
Next, we present preliminary experiments.
5. Preliminary Experiments
Our main goals when carrying on the experiments were threefold:
1. to verify whether the resulting rule set would contain a suitable
number of rules to form the search space of a GA;
2. to evaluate the time taken to extract the formal concepts;
3. to evaluate the idea of using the support of the rules for a preselecting process.
Next, we present the experiments carried out, discuss the results,
and present additional experiments on attribute exploration in order to
reduce the number of extracted rules.
5.1. Experiments and Results
To evaluate the proposed modification of the NextClosure algorithm we
used 10 datasets from the UCI - Machine Learning Repository (Frank
19
and Asuncion, 2010) in order to analyze the number of concepts. Table 4
summarizes the dataset characteristics giving the total number of examples (Examples); total number of features (Features), including the number of continuous and discrete features in brackets; number of classes
(Classes); the majority error (ME); and the number of fuzzy sets for each
of the attributes (FS) predefined using the Fuzzy–DBD algorithm (Cintra et al., 2009). Examples with missing values were removed from the
datasets.
Table 4. General characteristics of the datasets.
Dataset
Examples
Features
Classes
ME
Credit
653
15
(6
9)
2
45.33
Cylinder
277
32 (19 13)
2
35.74
Dermatology
358
34 (33
1)
6
68.99
Diabetes
769
8
(8
0)
2
34.90
Glass
220
9
(9
0)
7
65.46
Heart
270
13 (13
0)
2
44.44
Ionosphere
351
34 (34
0)
2
35.90
Iris
150
4
(4
0)
3
66.60
Vehicle
846
18 1(8
0)
4
74.23
Wine
178
13 (13
0)
3
59.74
FS
2
2
2
2
7
2
3
3
2
3
Table 5 presents the total number of formal concepts (TNFC), the
total number of formal concepts with a class (FCwC), i.e., the total number of classification rules, and the percentage it represents of the total
number of formal concepts. In order to allow further comparisons, Table 5 also presents the total number of formal concepts for 50%, 20%,
10%, and 5% support values and the percentage they represent of the
total number of formal concepts with a class (FCwC). The bigger the support, the smaller the number of rules. Support values greater than 50%
were not considered as the number of rules was not enough to form the
search space of a GA.
Table 5. Extracted Formal Concepts Information
Dataset
TNFC
FCwC
50%
20%
10%
5%
Credit
20,083
9,843 49.01
4 0.04
825 8.38
2,410 24.48
4,235 43.03
Cylinder
7,041,110 1,944,271 27.61 884 0.05 113,474 5.84 546,265 28.10 1,236,470 63.60
Dermatology 21,896,570 312,177 1.43
0 0.00
553 0.18
38,141 12.22 140,502 45.01
Diabetes
2,172
1,279 58.89 32 2.50
308 24.08
669 52.31
758 59.27
Glass
4,054
1,863 45.95
0 0.00
38 2.04
301 16.16
670 35.96
Heart
81,935
36,942 45.09
9 0.02
1,648 4.46
8,342 22.58
19,854 53.74
Ionosphere 102,641,179 2,076,229 2.02 12 0.00 1,187,827 57.21 1,197,307 57.67 1,649,406 79.44
Iris
93
65 69.89
0 0.00
11 16.92
27 41.54
41 63.08
Segmentation
10,785
1,437 13.32
0 0.00
0 0.00
162 11.27
54 3.76
Vehicle
86,918
28,979 33.34
0 0.00
91 0.31
1,780 6.14
6,625 22.86
Wine
21,000
9,802 46.68
0 0.00
423 4.32
3,338 34.05
6,676 68.11
20
Regarding our first goal (the suitability of the rule set extracted in
terms of the number of rules), we verified that an appropriate number
of rules was extracted (FCwC). Our verification takes into consideration
an estimated number of rules to populate 50 chromosomes, which was
the total population used in previous experiments with the D O C-B ASED
algorithm (Cintra et al., 2007). Considering that the smallest rule set was
obtained with the Iris dataset (65 rules), for this specific dataset, the total
number of rules in each chromosome must be carefully defined in order
to avoid lack of diversity in the population. The number of rules obtained
for the remaining datasets, except Iris, can be considered sufficient to
populate a genetic search space.
Table 6 shows the time taken, in minutes, for the extraction of
the formal concepts (column TNFC of Table 5) and calculation of their
support. The process was executed in an Intelr Core™2 Duo T7250
(2.00GHz, 2MB L2 Cache, 800MHz FSB) machine. The time taken to
extract the rules can also be considered appropriate for this approach
to be used together with a genetic algorithm search process. The whole
process took a matter of seconds to finish for all datasets but Cylinder,
Dermatology, and Ionosphere, due to the total number of attributes and
examples for these databases. It is also important to notice that our
experiments were carried out with the NextClosure algorithm (Ganter,
2002), although a faster algorithm was recently proposed which has a
parallel search process (Krajca et al., 2010).
Table 6. Time (in minutes) taken to extract all formal concepts (column TNFC).
Dataset
Time
Dataset
Time
Credit
0.50
Heart
0.55
Cylinder 126.00 Ionosphere 144.63
Dermatology 168.73
Iris
0.02
Diabetes
0.05
Vehicle
1.90
Glass
0.12
Wine
0.18
If the support is taken into consideration for the selection of formal concepts including a class, it is possible to reduce even more the
number of extracted formal concepts, giving the user more flexibility to
decide on the number of extracted rules. The only issue one has to bear
in mind when using the support in order to reduce the number of extracted rules is related to classes with few examples, which, in turn, will
result in relative low support. In fact, as stated previously, in (Ishibuchi
and Yamamoto, 2004), the authors use the support, confidence, and the
21
product of these two measures to select rules. These measures will be
investigated in future work.
Next, we present a discussion on the number of rules to be extracted and an approach we worked on to reduce the number of extracted
rules.
5.2. Additional Experiments on Attribute Exploration to Reduce the
Number of Extracted Rules
Although the proposal for fuzzy classification rule extraction from formal
contexts extracts a limited number of rules and there is the possibility
of using the support value of the formal concepts to further reduce the
total number of extracted rules, we decided to investigate an alternative
to reduce even more the number of possible extracted rules related to the
various possibilities lattices provide.
The idea came from a visual analysis of resulting lattices. By visually analyzing a lattice and the set of extracted rules, we could observe
that some attributes (notice that an attribute in the formal context is defined by a single fuzzy set) were related to a single class. Thus, we started
analyzing the idea that if an attribute was only related to a single class,
it would be possible to generate a rule from this relation and prevent this
attribute from being used in other rules. Similarly, if two or more attributes were related to a single class, they could be used to generate a
single rule and would not be used in any other rule generated from that
moment on.
This removal of attributes from the search space is easily understood by a simple analysis of the lattice. In order to illustrate this idea,
consider the Iris dataset which contains 4 continuous attributes (petal
length, petal width, sepal length, and sepal width) and 3 classes (iris virginica, iris versicolor, iris setosa). Figure 6 presents the resulting lattice
for this domain. Each attribute was defined in fuzzy terms by 3 fuzzy sets,
named 1_Low, 1_Med (medium), 1_High,2_Low, 2_Med, 2_High,3_Low,
3_Med, 3_High, 4_Low, 4_Med, and 4_High, where 1, 2, 3, and 4 represent the sequence number of the original attribute. We used the ConExp
software1 to generate the lattices and proceed experimenting with the removal of attributes.
1
More information on this free licensed software can be found on http://conexp.
sourceforge.net/
22
Figure 6. Complete lattice for Iris dataset.
Although the total lattice is messy and confusing, the Conexp software allows the exploration of associations among classes and attributes
by presenting an updated lattice by simply removing attributes (using
a check box). By leaving single attributes and the classes, interesting
associations can be found, such as the ones presented in Figure 7.
Observe that for each figure in Figure 7, one class is connected
to one attribute, the remaining two classes have no connections. This
happens because they are not related to the only attribute selected to
build the lattice.
Figure 7 presents 4 attributes, each related to only one class. However, for some particular cases, one class might be related to two or more
attributes. This case is shown in Figure 8, where attributes 2_High and
4_Low are only related to class Iris Setosa.
In order to find associations of attributes and classes, considering
a dataset with M attributes, the following algorithm was proposed:
1. Generate the lattice using each single attribute alone and check
if an association between a single attribute and class was found;
if so, create a rule using this attribute and class, and remove the
attribute from the set of available attributes so it cannot be used in
future lattices;
2. Repeatedly generate lattices using combinations of 2, 3,..., M − 1
attributes and check if associations among the set of attributes and
a single class is found; if so, create a rule using the attribute set
23
(a) Lattice for attribute 2_High.
(b) Lattice for attribute 3_Low.
(c) Lattice for attribute 3_High.
(d) Lattice for attribute 4_Low.
Figure 7. Latices of attributes 2_High (a), 3_Low (b), 3_High (c), and 4_Low (d).
Figure 8. Attributes 2_High and 4_Low, related to class Iris Setosa.
and the class, and remove the attribute set from the set of available
attributes.
Although the cost of combining all possible attributes is high, we
expect that, in practice, the total number of combinations can be rapidly
24
reduced, due to the fact that attributes which participate in rules are
discarded from future combinations.
In order to evaluate this idea, preliminary experiments were carried out using the Iris and Diabetes datasets to further analyzing this
idea. However, results with the Iris and Diabetes datasets showed that
the final sets of rules extracted were quite small and would not suit a
genetic process. It is important to notice that these two datasets, Iris and
Diabetes, are described by 4 and 8 attributes, which can be considered a
small number. This way, we intend to use other datasets to evaluate this
idea as a means of extracting classification rules and building classifiers,
in future work.
Next, we present the conclusions and future work.
6. Conclusions and Future Work
The field of genetic fuzzy systems has produced some promising results
in the area of fuzzy systems and many new approaches have been proposed in the literature. Specifically for the genetic definition of the rule
base, a possible approach is the genetic rule selection, with the previous
definition of the fuzzy database and the generation of fuzzy rules to form
the search space of the genetic selection process. In this work we presented some approaches found in the literature for this task of forming
the search space of a genetic algorithm.
We also presented a new proposal and preliminary experiments and
results on the use of formal concept analysis for this task. The use of
formal concept analysis can be considered a new area of research, with
applications in various domains and with an increasing interest due to
its visual benefits and powerful mathematical basis.
Preliminary results show that our proposal is suitable for the task
of forming the search space of a genetic algorithm in terms of the number of rules extracted, processing time, and also the use of the support
measure to preselect a reduced number of rules if necessary.
As future work, we intend to adopt the D O C-B ASED method, proposed in (Cintra et al., 2007; Cintra and Camargo, 2007), to generate
fuzzy rule bases using the proposal presented here to form the search
space. This method includes the number of rules and the accuracy of the
rule base in the fitness calculation in order to induce a rule base with
25
high accuracy and interpretability rates. We intend to compare the formal concept extraction time taken by the NextClosure algorithm (Ganter,
2002) and the parallel approach proposed by Krajca et al. (2010). We
also intend to empirically evaluate the use of the support and confidence
measures, and the product of them, to select rules.
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