=Paper=
{{Paper
|id=Vol-162/paper-2
|storemode=property
|title=A Formal Concept Analysis Approach to Discover Association Rules from Data
|pdfUrl=https://ceur-ws.org/Vol-162/paper2.pdf
|volume=Vol-162
|dblpUrl=https://dblp.org/rec/conf/cla/Maddouri05
}}
==A Formal Concept Analysis Approach to Discover Association Rules from Data==
https://ceur-ws.org/Vol-162/paper2.pdf
A Formal
A FormalConcept
ConceptAnalysis Approach
Analysis to Discover
Approach to
Association Rules from Data
Discover
Association Rules from Data
Mondher Maddouri
Department Mondher Maddouri
of Mathematics & Computer Sciences,
National Institute of Applied Sciences & Technology of Tunis – INSAT
Department of Mathematics
University ofand Computer Sciences,
Carthago,
National Centre
Institute of Applied
Urbain Sciences
Nord, B.P. andTUNIS
676, 1080 Technology of TUNISIE
CEDEX, Tunis - INSAT
University of Carthago,
mondher.maddouri@fst.rnu.tn
Centre Urbain Nord, B.P. 676, 1080 TUNIS CEDEX, TUNISIE
mondher.maddouri@fst.rnu.tn
Abstract. The Discovery of association rules is a non-supervised task of data
mining. Its mostly hard step is to look for the frequent itemsets embedded into
large amounts of data. Based on the theory of Formal Concept Analysis, we
suggest that the notion of Formal Concept generalizes the notion of itemset,
since it takes into account the itemset (as the intent) and the support (as the
cardinality of the extent). Accordingly, we propose a new approach to mine
interesting item-sets as the optimal concepts covering a binary table (concept
coverage). This approach uses a new quality-criteria of a rule: the gain that
generalizes the support criteria.
1 Introduction
The volume of stored data increases rapidly. In the past, it doubles each three
centuries. Nowadays, it doubles each 72 days. The analysis of huge volumes of data
by computers leads to the emerging field of Data-Mining and Knowledge-Discovery.
This includes techniques to create knowledge from data sets serving for classification,
clustering, prediction, estimation and optimization tasks [1].
The notion of “Concept” as a couple of intent and extent (serving to represent
nuggets of knowledge) was introduced and formalized many decades ago [2, 3, 4].
Recently, its use in Data-Mining and Knowledge-Discovery is in growth [5]. Our
ConceptMiner project (a development environment in MS-Visual C++) aims to design
new heuristic algorithms for Data-Mining based on Formal Concept Analysis.
In this paper, we study the clustering problem that we resolve by the induction of
association rules. In fact, the experts (biologists) want to know which sub-sequences
of proteins that appear together in Tall Like Receptor (TLR) sequences or even in
Similar TLR sequences [8].
In section 2, we resume the basic concepts of association rules and we outline the
well-known techniques. In section 3, we introduce the basic definitions of Formal
Concept Analysis that we use later. In section 4, we give two heuristic algorithms to
calculate the optimal Item-sets from rows of data. Illustrative examples are given
throughout the paper. Finally, we present our concluding remarks and the future
perspectives of this work.
Radim Bělohlávek, Václav Snášel (Eds.): CLA 2005, pp. 10–21, ISBN 80–248–0863–3.
2 A FCA Approach to Discover Association Rules from Data
Mondher Maddouri 11
2 Basics in Association Rules
Discovering association rules is a non-supervised data-mining task that aims to induce
symbolic conditional rules from a data file [1]. The conditional rule has the following
syntax: “IF conditions THEN results”. The combination of conditions leads to a
descriptive rule like: “IF A AND NOT B THEN C AND D”. It can lead, also, to a
predictive rule including the time dimension like: “IF condition at time_1 THEN
result at time_2”. This looks like: “IF he_bays(TV, today) THEN he_bayes(recorder,
in_one_year)”.
The association rules are used in a wide range of applications, due to their
comprehensive format. Mainly, it is used to analyze the supermarket tickets. This
helps in optimal stock management, in merchandizing and marketing [1].
During the few last years, a wide variety of algorithms have been proposed. The
most popular algorithm is certainly the iterative one: Apriori [9]. It consists in three
steps. First aboard, it considers all the attributes/properties describing the set of data
as the initial itemsets. Then, it combines the couples of item-sets in order to obtain
larger item-sets. Finally, it searchs for the rules that can be derived from each itemset.
MaxEclat and MaxClique [10] use graph theory to decompose the lattice into sub-
lattices in order to mine maximal frequent item-sets. MaxMiner [11] is an exploring
algorithm for the maximal frequent item-sets with a look-ahead pruning strategy.
PincerSearch [12] follows both a bottom-up and a top-down search in the lattice at
the same time while looking for the maximal frequent patterns. GenMax [13] uses a
backtracking search to enumerate all maximal frequent item-sets and eliminates
progressively non maximal ones. Mafia [14] works with a depth first search and uses
pruning strategies to mine maximal frequent item-sets. DepthProject [15] follows
dynamic re-ordering to reduce the search space. Salleb & al. [16] uses a boolean
algebra approach that loads the data base in main memory as a binary decision
diagram, and searchs for maximal frequent item-sets by making iteratively the product
of vectors representing the items. The reader can find further description on these
algorithms in [13, 16].
Generally, the number of association rules (induced by these algorithms) grows in
an exponential manner with the number of data-rows (or the attributes). This can
reach hundreds’ of thousands using only some thousands of data rows [17]. So, it
becomes very hard to read and interpret them. The induced rules can be classified in
three categories:
- Useful rules: that a human expert can understand and use.
- Trivial rules: that represents evidence. They are valid but never used.
- Weak rules: those are not acceptable by the expert and not understandable.
To remedy to this problem, many methods were proposed [17]. The basic one
consists of calculating a minimal set of rules that allows us to re-produce the others.
This can be done while calculating the rules [18], or after that using the concept lattice
[19, 20]. Another method searches only the rules where the condition or the result part
matches a particular group of terms [21]. Agrawal et al. [9] maintains a partial order
relation between the rules using two measures: the support and the confidence.
Consider the rule: X → Y , where X (conditions) and Y (results) are sets of atomic
propositions:
- Support (X → Y): the percentage of examples satisfying both X and Y.
12 A Mondher Maddouri
Formal Concept Analysis Approach to Discover Association Rules from Data 3
- Confidence (X → Y): the number of examples satisfying both X and Y,
devided by the number of examples satisfying only the condition X.
The authors introduce two thresholds defined by the end-user:
- MinSup: the minimal value of the rule support that can be accepted.
- MinConf: the minimal value of the rule confidence that can be accepted.
The method discards all the item-sets with a support less than MinSup, and all the
rules with a confidence less than MinConf.
To filter more efficiently the association rules, some authors [22] introduce the
conviction measure and [17] propose five different measures like the benefit (interest)
and the satisfaction.
In this paper, we introduce a new mesure: the gain. It combines the support of the
rule with the length of the itemset. This is done thanks to the theory of Formal
Concept Analysis [3, 4], since we suggest that an item-set is completely represented
by a formal concept as a couple of intent (the classic item-set) and extent (its support).
3. Basics in Formal Concept Analysis
Let O be a set of objects (or examples), P a set of properties (or attributes) and R a
binary relation defined between O and P [3, 4].
Definition 1 [3]: A formal context (O, P, R) consists of two sets O and P and a
relation R between O and P. The elements of O are called the objects and the elements
of P are called the properties of the context. In order to express that an object o is in a
relation R with a property p, we write oRp or (o, p)∈R and read it as "the object o has
the property p".
Definition 2 [3]: For a set A⊆O of objects and a set B⊆P of properties, we define :
The set of properties common to the objects in A :
A4={p∈P | oRp for all o∈A} (1)
The set of objects which have all properties in B :
B3={o∈O | oRp for all p∈B} (2)
The couple of operators (4,3) is a Galois Connection.
Definition 3 [3]: A formal concept of the context (O, P, R) is a pair (A, B) with
A⊆O, B⊆P, A4=B and B3=A. (3)
We call A the extent and B the intent of the concept (A, B).
Definition 4 [3]: The set of all concepts of the context (O, P, R) is denoted by ³(O,
P, R). An ordering relation (<<) is easily defined on this set of concepts by :
(A1, B1) << (A2, B2) ⇔ A1⊆A2 ⇔ B2⊆B1 (4)
4 A FCA Approach to Discover Association Rules from Data
Mondher Maddouri 13
P A B C D
O
o1 1 1 0 0
o2 1 1 0 0
o3 0 1 1 0
o4 0 1 1 1
o5 0 0 1 1
1.a Formal context (O, P, R)
FC1
∅
o1,o2, o3, o4, o5
FC2 FC3
B C
o1, o2, o3, o4 o3, o4, o5
FC4 FC6
FC5
A, B B, C C, D
o1, o2 o3, o4 o4, o5
B, C, D FC7
o4
FC8
A, B, C, D
∅
1.b Concept Lattice of the context (O, P, R)
Fig. 1. An example of formal context (O, P, R) and its corresponding Concept Lattice.
Basic Theorem for Concept Lattices [3] :
(³(O, P, R), <<) is a complete lattice. It is called the concept lattice (or GALOIS
lattice) of (O, P, R), for which infimum and supremum can be described as follow:
Sup (A , B )=((∪ A )43, (∩ B )) (5)
i∈I i i i∈I i i∈I i
Inf (A , B )=( ∩ A , (∪ B )34) (6)
i∈I i i i∈I i i∈I i
Example : Figure 1 (graphic 1.a) is used to illustrate the notion of formal context (O,
P, R). This context includes five objects {o1, o2, o3, o4, o5} described by four
properties {A, B, C, D}. The concept lattice of this context is drawn in Figure 1
(graphic 1.b). It contains eight formal concepts.
14 A Mondher Maddouri
Formal Concept Analysis Approach to Discover Association Rules from Data 5
Definition 5 [4]: Let (o, p) be a couple in the context (O, P, R). The pseudo-concept
PC containing the couple (o, p) is the union of all the formal concepts containing
(o,p).
Definition 6 [4]: A coverage of a context (O, P, R) is defined as a set of formal
concepts CV={RE1, RE2, ..., REn} in ³(O, P, R), such that any couple (o, p) in the
context (O, P, R) is included in at least one concept of CV.
Fig. 2. An example of pseudo concept, optimal concept and non optimal concept
containing the couple (o3,B).
Example: To illustrate the notion of pseudo-concept and coverage, we consider the
same formal context (O, P, R) from figure 1. Figure 2 (graphic 2.a) represents the
pseudo-concept containing the couple (o3, B). It is the union of the concepts FC2 and
FC5 (see figure 1).
A coverage of the context (O, P, R) can be formed by the three concepts: {FC4,
FC5 , FC6}. FC4 is the concept containing the items ({o1, o2}, {A, B}). FC5 is the
concept containing the items ({o3, o4}, {B, C}). FC6 is the concept containing the
items ({o4, o5}, {C, D}). The lattice constitutes also a concept coverage.
4 Discovery of Optimal Item-sets
The most computationally expensive step to discover association rules is the
calculation of the frequent item-sets [15]. Generally, this step consists of applying,
iteratively, some heuristics to calculate candidate item-sets. At the iteration i, we
combine the item-sets of the iteration i-1. After that, the support threshold (minsupp)
is used to discard non-frequent candidates. The item-sets of iteration i-1, are also
discarded. We keep the remaining item-sets of the latest iteration n (where n is the
number of properties in the formal context).
In this step, we think that many approaches use, in an implicit manner, two basic
characteristics of the association rules:
- General rules: we should ignore the item-sets with high support and low
cardinality. General rules are week, since they are valid for numerous
examples (or observations).
6 A FCA Approach to Discover Association Rules from Data
Mondher Maddouri 15
- Specific rules: we should ignore the item-sets with high cardinality and low
support. Specific rules are week, since they are valid only for few examples
(or observations).
The two characteristics are defined based on the cardinality of two special sets:
- Support: that is the cardinality of the set of examples verifying the rule. In
Formal Concept Analysis, this represents the extent of a formal concept.
- Cardinality of the Item-set: that is the number of properties in the item-set.
In Formal Concept Analysis, this represents the intent of a formal concept.
Consequently, we think that an association rule should not be represented only by
the item-set (that is the intent). An association rule is completely and explicitly
represented by a formal concept (as a couple of intent and extent).
Also, we think that the selection of “good” item-sets should not be based only on
the support (that is the cardinality of the extent). A powerful selection should be
based on the cardinalities of both the extent and the intent of its formal concept.
To formalize the new criterions, we give the following definitions.
Definition 7: Let FCi= (Ai, Bi) be a formal concept. We define:
- Length of a concept FCi: the number of properties in the intent Bi of the concept.
- Width of a concept FCi: the number of examples in the extent Ai of the concept.
- Gain of a concept: is a function of the width and the length, given by:
Gain(FCi)=width(FCi)*length(FCi)–[width(FCi)+length(FCi)] (7)
In a manner that the gain function cannot be maximised when one of the two
parameters is low and the other is high. It becomes maximal only when the two
parameters are high (a lot of examples and a lot of properties). Hence, we are sure that
the rule is not a general one, neither a specific one. Note also that the Width indicates
the support of the itemset (or the corresponding association rule).
Definition 8 [4]: A formal concept FCi= (Ai, Bi) containing a couple (o, p) is said to
be optimal if it maximizes the gain function.
Definition 9 [4]: A coverage CV={ FC1, FC2, ..., FCk} of a context (O, P, R) is
optimal if it is composed by optimal concepts.
Example: Figure 2 (graphic 2.b) represents the optimal concept FC5 containing the
couple (o3, B) and having the gain value 0. Figure 2 (graphic 2.c) represents the non
optimal concept FC2 containing the couple (o3, B) and having the gain value -1.
The optimal coverage of the context (O, P, R) is formed by three optimal
concepts: {FC4., FC5 , FC6}. FC4 is the concept containing the items ({o1, o2}, {A,
B}). FC5 is the concept containing the items ({o3, o4}, {B, C}). FC6 is the concept
containing the items ({o4, o5}, {C, D}).
16 A Mondher Maddouri
Formal Concept Analysis Approach to Discover Association Rules from Data 7
4.1 Heuristic Searching for Optimal Concept
The pseudo-concept containing the couple (o, p), introduced in definition 5, is the
union of all the concepts containing (o, p). The importance of the notion of pseudo-
concept, indicated by PCF, is that it can be calculated in a simple manner as the
restriction of the relation R by the set of objects/examples described by p, so {p}4,
and the set of properties describing the object o, so {o}3. Where (4,3) is the Galois
connection of the context (O, P, R).
Once we obtain the pseudo-concept PCF, two cases are considered:
- Case 1: PCF forms a formal concept. This means that no zero is found in the
relation/matrix representing PCF. Then, PCF is the optimal concept that we
are looking for. The algorithm stops.
- Case 2: PCF is not a formal concept. This means, simply, that we can find
some zero entries in the relation/matrix representing PCF. In this case, we
will look for more restraint pseudo-concepts within the pseudo-concept PCF.
So, we consider the pseudo-concepts containing the couples like (X, p) or (o,
Y). These concepts contain, certainly, the couple (o, p). The heuristic that we
use here is that: the optimal concept is included in the optimal pseudo-
concept. So, we should calculate the Gain value of the different pseudo-
concepts containing the couples like (X, p) or (o, Y). Then we choose the
pseudo-concept having the greatest value of the Gain function to be the new
PCF.
This heuristic procedure (of case 2) is repeated until we meat the case 1: until PCF
becomes a formal concept. To calculate the gain of a pseudo-concept, we introduce
the general form of the previous function:
Definition 10: Let PCFi= (Ai, Bi, Ri) be a pseudo-concept, where Ri is the restriction
of the binary relation R, to the subsets Ai and Bi. We define the:
- Length of a pseudo-concept PCFi: the number of properties in Bi.
- Width of a pseudo-concept PCFi: the number of examples in Ai.
- Size of a pseudo-concept PCFi: the number of couples (of values equal to 1) in
the pseudo-concept. When PCFi is a formal concept, we have:
Size(PCF)= Length(PCF)×Width(PCF) (8)
- Gain of a pseudo-concept: is a function of the width, the length and the size given
by (9) :
⎡ ⎤
Gain(PCF )= ⎢ ×[Size(PCF) −(Length(PCF) +Width(PCF) )]
Size(PCF)
⎣ Length(PCF)×Width(PCF) ⎥⎦
4.2 Algorithm for Optimal Coverage
The problem of covering a binary relation by a set of optimal concepts is known as
the problem of covering a binary matrix by a number of its complete sub-matrix. A
8 A FCA Approach to Discover Association Rules from Data
Mondher Maddouri 17
complete sub matrix is a matrix where all its entries are equal to '1'. This was found as
an NP-hard problem. Here, we use a polynomial heuristic solution.
Let R be the binary relation to cover. The heuristic solution consists of dividing R
into p packages (subsets): P1, ..., Pp. Each package represents one or more couples.
The idea is to construct step by step the optimal coverage of R. In the first step, we
cover the relation R1=P1 by CV1. In the kth step, let Rk-1=P1 ∪ ... ∪ Pk-1 and let CVk-1
be its optimal coverage. The task is to construct the optimal coverage CVk of Rk=Rk-1
∪ Pk using only the initial coverage CVk-1 and the package Pk. In the pth step, we
obtain a set of concepts covering the relation R.
Algorithm for building the Coverage
Begin
Let R be partitioned to p packages P1, ..., Pp.
Let CV0:=∅.
FOR k=1 to p DO
Sort the couples of Pk by the pertinence of
their pseudo-concepts
While (Pk≠∅) Do
-Select a couple (a, b) in Pk by the sorted
order of the gain function (9)
-Search PC : the pseudo-concept containing
(a, b) within Rk=CVk-1∪Pk
-Search FC: the optimal concept
containing (a, b) within PC
CVk:=(CV k-1-{r∈CV k-1/ r⊆FC }) ∪{FC}: Delete
all the redundant concepts from CVk
Pk:=Pk-{(X,Y)∈Pk/(X,Y)∈FC}
End While
End FOR
End.
Fig. 3. Illustration of the algorithm for concept coverage.
Example: Let R be the relation of figure 1. R is partitioned into five packages:
P1={o1}x{A, B}, P2={o2}x{A, B}, P3={o3}x{B, C}, P4={o4}x{B, C, D} and
P5={o5}x{C, D}. Initially, R is an empty relation and in each step we add a package.
Figure 3 presents the incrementation step when adding P5.
18 A Mondher Maddouri
Formal Concept Analysis Approach to Discover Association Rules from Data 9
In this step R4 encloses the four rows P1, ..., P4. The initial optimal coverage CV4
encloses the formal concepts FC4=({o1, o2}, {A, B}) and FC5=({o3, o4}, {B, C})
and FC7=({o4}, {B, C, D}) (graphic 3.a). The package P5 encloses only two couples.
The pseudo concept containing the couple (o5, C) represents the union of two formal
concepts FC3 and FC6. So, the optimal concept containing (o5, C) and (o5, D) is the
concept FC6. The concept FC7 is redundant. Thus, the final coverage of R contains
the concepts FC4, FC5 and FC6.
In conclusion, from the data set of figure 1, we discover three Item-sets : {A, B},
{B, C} and {C, D}. These Item-sets will be processed later by the Apriori algorithm,
to calculate the corresponding association rules.
5 Experiments
The experiments are done with a PC PENTIUM II, 233 MHZ, 144 Mo of RAM and
running under MS-Windows 98. The analyzed data sets are organized in two groups.
The first group is made by five data sets representing biological sequences coding
macromolecules (table 1). The first two data sets are proprietary sets for the TLR
macromolecules. The three others are taken from the machine-learning repository at
UCI University. Classic tables of data make the second group of data sets (table 2).
The first two data sets are also taken from the machine-learning repository at UCI
University. The other sets are taken from a proprietary database.
Table 1: Properties of the biological data sets.
Biological Data Sets Identifier Rows Columns
2 Far a way TLR Families DB1 763 16
6 Near TLR Families DB2 619 14
Promoter gene DB3 106 11
Junction gene 3’ DB4 1150 19
Junction gene 5’ DB5 500 10
Table 2: Properties of the data sets having standard format.
Standard Data Sets Identifier Rows Columns
Tic Tac Toe DB6 958 10
Mushroom DB7 1191 23
T20i4d1k DB8 1000 9
T10I4D DB9 5000 9
T20I6D DB10 1000 9
To evaluate the performance of the proposed method (IAR: Incremental induction of
Association Rules), we compare it to the existing approach Aprior [9, 23]. The
Apriori method was programmed in C++ and integrated in our software package. We
10 A FCA Approach to Discover Association Rules from Data
Mondher Maddouri 19
use the two methods to analyze the ten data sets. We choose two sets of parameters.
The first one with MinSup=0.6 and MinConf=0.5 is used for the biological data sets
(table 1). The second one with MinSup=0.35 and MinConf=0.75 is used for the
standard data sets (table 2).
CPU time (sec)
CPU time (sec)
DB1 DB2 DB3 DB4 DB5 DB6 DB7 DB8 DB9 DB10
IAR 3930 304 9 18225 459 IAR 11480 32825 9587 71898 11983
Apriori 12451 916 12 22135 875 Apriori 17929 51268 14973 112295 18715
MinSup=0.6 and MinConf=0.5 MinSup=0.35 and MinConf=0.75
Fig. 4: Comparing the CPU time
5.1. Comparing the CPU time
Figure 4 presents the CPU time measured in seconds for the three methods. We
remark that the three methods keep the same behavior overall the data sets. The
Apriori method takes the greater time, since it is based on an exhaustive approach to
test all the possible combinations. The proposed method IAR takes always a CPU
time better than Apriori.
Similarity of Knowledge bases (%)
Similarity of Knowledge bases (%)
DB1 DB2 DB3 DB4 DB5 DB6 DB7 DB8 DB9 DB10
IAR/Apriori 0,21 0,13 0,49 0,03 0,16 IAR/Apriori 0,42 0,12 0,3 0,03 0,24
MinSup=0.6 and MinConf=0.5 MinSup=0.35 and MinConf=0.75
Fig. 5: Similarity between the knowledge bases of the two methods
20 Mondher
A Formal Maddouri
Concept Analysis Approach to Discover Association Rules from Data 11
5.2. Measuring the similarity of rules
Figure 5 presents the percentage of rules that are obtained by IAR in comparison with
APRIORI. We remark that the percentage is always less than the half (0.5). We
remark also that Apriori gives, always, a lot of rules compared to IAR.
6 Conclusion
In this paper, we presented a new Formal Concept Analysis approach to search for
optimal item-sets. We gave two heuristic algorithms to build the coverage of formal
concepts and to create optimal item-sets from data records. The association rules can
be induced from the Item-sets in the same manner as Apriori [9].
We assume that a Formal Concept generalizes the notion of Item-set, since it is
more powerful and more explicit. In fact, we note that an item-set represents only the
intent of a Formal Concept. Also, the support of an item-set is the cardinality of the
concept extent. Hence, the Formal Concept encloses more information than the item-
set itself.
The first experimentation of the proposed method is done on a variety of data sets
having different data formats (structured and unstructured). We compared the
proposed method with a known one: Apriori [9]. We measured their CPU time, the
number of association rules, and the percentage of similar rules. We remarked that
Apriori gives a great number of rules and takes a great time. The proposed method
IAR is faster than Apriori. It gives a low number of rules that have the advantage of
being the common rules.
Future work will focus on the quality of the created association rules. In fact, we
plan to study the significance of the discovered rules for human experts. For this
reason we propose to present the rules concerning the TLR macromolecules to our
biologists in order to get their points of view. Accordingly, we plan to study quality
measures other than the support and the confidence [17, 22], in order to identify the
measure that gives expert-acceptable rules.
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