Discovering fuzzy association rules using fuzzy partition methods
Yi-Chung Hu
a, Ruey-Shun Chen
a, Gwo-Hshiung Tzeng
b,*
a
Institute of Information Management, National Chiao Tung University, Hsinchu 300, Taiwan, ROC
b
Institute of Management and Technology, National Chiao Tung University, Hsinchu 300, Taiwan, ROC Received 19 December 2000; revised 29 March 2002; accepted 3 May 2002
Abstract
Fuzzy association rules described by the natural language are well suited for the thinking of human subjects and will help to increase the flexibility for supporting users in making decisions or designing the fuzzy systems. In this paper, a new algorithm named fuzzy grids based rules mining algorithm (FGBRMA) is proposed to generate fuzzy association rules from a relational database. The proposed algorithm consists of two phases: one to generate the large fuzzy grids, and the other to generate the fuzzy association rules. A numerical example is presented to illustrate a detailed process for finding the fuzzy association rules from a specified database, demonstrating the effectiveness of the proposed algorithm. q2002 Elsevier Science B.V. All rights reserved.
Keywords: Data mining; Fuzzy partition; Association rules; Decision making
1. Introduction
Relational databases have been widely used in data processing and support of business operations, and there the size has grown rapidly. For the activities of decision-making and market prediction, knowledge discovery from a database is very important for providing necessary information to a business. Association rules are one of the ways of representing knowledge, having been applied to analyze market baskets to help managers realize which items are likely to be bought at the same time [1,2]. For example, rule {P} ) {Q} represents that if a customer bought P, then he should buy Q at the same time. The left-hand side of ‘ ) ’ is the antecedence of rule, and the right-hand side is the consequence. We call {P} and {Q} itemsets. Two important parameters are required to generate effective association rules; one is support and the other is confidence [2]. Support is the number of transactions with all the items in the rule; and confidence is the ratio of the number of transactions with all the items in the rule to the number of transactions with just the items in the condition [1]. Hence, the support of {P, Q} can be described as (Number of transactions which
contains both P and Q )/(Number of transactions in the database); and the confidence of {P} ) {Q} can be described as (Number of transactions which contains both P and Q )/(Number of transactions contains P ).
Generally, there are two phases for mining association rules [3]. In the first phase, we first find all the large itemsets. The supports of large itemsets are larger than the minimal supports specified by users. If there are k items in a large itemset, then we call it a large k-itemset. We can find that a subset of a large itemset must also be large. Subsequently, we use the large itemsets generated in the first phase to generate effective association rules. If the confidence of an association rule is larger than or equal to the minimum confidence specified by users, then it is effective. The key work for finding the association rules is to find all the large itemsets.
Initially, Agrawal et al.[4]proposed a method to find the large itemsets. Subsequently, Agrawal et al. [5] also proposed the Apriori algorithm. However, these algorithms must scan a database many times to find the large itemsets. Moreover, when they generated a candidate itemset, the apriori-gen function must have wasted much time to check if its subsets are large or not.
Wur and Leu [6] proposed the Boolean algorithm to scan a database only once, not wasting much time reading data from disk. Moreover, it used Boolean operations (AND, OR and XOR) on the table structure 0950-7051/03/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved.
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* Corresponding author. Tel.: 5712121x57505; fax: þ886-3-5753926.
E-mail addresses: ghtzeng@cc.nctu.edu.tw (G.H. Tzeng), ychu. iim89g@nctu.edu.tw (Y.C. Hu).
they proposed to generate the large itemsets and the association rules. For discovering association rules, it seems that Boolean algorithm works more efficient than other algorithms.
By partitioning quantitative attributes, Srikant and Agrawal [7] proposed the partial completeness to be the criterion for finding association rules. Fukuda et al.[8]also proposed concepts of optimized association rules. In the rule representation, the consequent part was required to be a fixed value, and the antecedent part was composed of one or two quantitative attributes. However, it seems that such representations were restricted.
A clustering method named CLIQUE was further proposed by Agrawal et al. [9]. In general, data clusters were distributed in the feature subspaces which were constructed by some quantitative attributes, and CLIQUE could efficiently find the subspaces where data clusters were really distributed. To do this, CLIQUE divided each quantitative attribute into many partitions with equal length, and viewed each partition as a candidate 1-itemset. There-fore, a k-itemset (k $ 1) is an itemset that consisted of k partitions distributed in k quantitative attributes. Finally, large k-itemsets could be found. In comparison with other clustering methods, including c-means[1]and BIRCH[10], we can find that CLIQUE worked more efficient because it cannot directly find clusters that were constructed by all quantitative attributes.
The well-known methods (i.e. partial completeness, optimized association rules and CLIQUE) we have mentioned above, divided the quantitative attributes into many crisp partitions. There were no intersections between the partitions. However, crisp partitions may be unreasonable for some situations. For example, if we tried to divide the range (170, 180 cm) of the attribute ‘height’ into two partitions, then the separable point was not different between 175.01 and 174.99 cm. Hence, intersection between any of the neighborhood partitions can be promised. Moreover, we considered that the fuzzy association rules described by the natural language are well suited for the thinking of human subjects and will help to increase the flexibility for users in making decisions or designing the fuzzy systems. The fuzzy partition methods are thus used to find the fuzzy association rules.
In this paper, an effective algorithm named fuzzy grids based rules mining algorithm (FGBRMA) is proposed. For the proposed algorithm, both quantitative and categorical attributes are divided into various linguistic values. Large fuzzy grids and effective fuzzy association rules can be determined by the proposed fuzzy support and the fuzzy confidence, respectively. Like Boolean algorithm, FGBRMA uses the proposed table structures to generate both large fuzzy grids and fuzzy association rules. It seems that the proposed algorithm is also an efficient algorithm because it also scans a database only once and applies Boolean
operations on the proposed table structures to generate both large fuzzy grids and fuzzy association rules.
In the following sections, the cases for fuzzy partitioning in quantitative and qualitative attributes are introduced in Section 2. In Section 3, the definitions of the fuzzy support and the fuzzy confidence are proposed. We present the proposed algorithm in Section 4. In Section 5, a numerical example is presented to illustrate a detailed process for finding the fuzzy association rules from a specified database relation, demonstrating the effectiveness of the proposed algorithm. Discussions and conclusions are presented in Sections 6 and 7, respectively.
2. Fuzzy partition method
Notations used in this paper are stated as follows: K: prespecified number of linguistic values in a
linguistic variable;
d: number of attributes of a database relation, where 1 # d;
k: dimension of a fuzzy grid, where 1 # k # d; Axm
K;im: imth linguistic value of K various linguistic value
defined in xm, where 1 # im# K;
mxm
K;im: membership function of A
xm
K;im;
tp: pth tuple of a database relation, where tp¼
ðtp1; tp1; …; tpdÞ and p $ 1.
Fuzzy set was proposed by Zadeh[16], and the division of the features into various linguistic values has been widely used in pattern recognition and fuzzy inference. From this, various results have been proposed, such as application to pattern classification by Ishibuchi et al.[10 – 12], the fuzzy rules generated by Wang and Mendel[13], and methods for partitioning feature space were also discussed by Sun[14]
and Bezdek[15].
In this paper, we view each attribute as a linguistic variable, and the variables are divided into various linguistic values. A linguistic variable is a variable whose values are linguistic words or sentences in a natural language[17 – 20]. For example, the values of the linguistic variable ‘Age’ may be ‘close to 30’ or ‘very close to 50’, and referred to as linguistic values. Triangular membership functions are used for each linguistic value defined in each quantitative attribute for simplicity. Hence, each linguistic value is a fuzzy number, which is a fuzzy subset in the universe of discourse that is both convex and normal[20,21].
The cases for fuzzy partitioning in quantitative and categorical attributes are introduced in Sections 2.1 and 2.2, respectively.
2.1. Fuzzy partitioning in quantitative attributes
A quantitative attribute can be divided into K various linguistic values ðK ¼ 2; 3; 4…Þ: For example, for
the attribute ‘Age’ (range from 0 to 60), we describe K ¼ 2; K ¼ 3 and K ¼ 4 in Figs. 1 – 3, respectively. Moreover, AAgeK;i
m can be used to represent a candidate 1-dim fuzzy grid.
Then,mAgeK;im can be represented as follows:
mAgeK;i mðxÞ ¼ max{1 2 lx 2 a K iml=b K; 0} ð1Þ where aKim ¼ mi þ ðma 2 miÞ·ðim2 1Þ=ðK 2 1Þ ð2Þ bK ¼ ðma 2 miÞ=ðK 2 1Þ ð3Þ
where ma is the maximum value of the attribute’s domain, and mi is the minimum value. It is clear that ma ¼ 60 and mi ¼ 0 for ‘Age’. Generally, AAgeK;i
m can be described in a
linguistic sentence such as:
AAgeK;1 : young; and below 60=ðK 2 1Þ ð4Þ AAgeK;K: old; and above ½60 2 60=ðK 2 1Þ ð5Þ
AAgeK;i
m : close to ðim2 1Þ·½60 2 60=ðK 2 1Þ; and
between ðim2 2Þ·½60 2 60=ðK 2 1Þ and
im·½60 2 60=ðK 2 1Þ; for 1 , im, K ð6Þ
A high-dimensional fuzzy grid can be further generated. For example, if we divide both ‘Age’ (x1) and ‘Salary’ (x2) into
three linguistic values, then a feature space can be divided into 3 £ 3 2-dim fuzzy grids, as shown inFig. 4. For the shaded 2-dim fuzzy grid shown inFig. 4, we can use a 2-dim fuzzy grid whose linguistic value is AAge3;1 £ ASalary3;3 to stand for it. This concept is similar to the k-itemset used in CLIQUE[9].
2.2. Fuzzy partitioning in qualitative attributes
Qualitative attributes of a relational database have a finite number of possible values, with no ordering among values
(e.g. sex, color)[11]. If the distinct attribute values are n0(n0is finite), then this attribute can only be partitioned by n0 linguistic values. For example, the linguistic sentence of each linguistic value defined in ‘Sex’ can be stated as follows:
ASex2;1 : male ð7Þ
ASex2;2 : female ð8Þ
Each linguistic value distributed in either quantitative or categorical attributes is viewed as a candidate 1-dim fuzzy grid. The subsequent task is how to use these candidate 1-dim fuzzy grids to generate the other large fuzzy grids and fuzzy association rules. As we have mentioned above, the definitions of fuzzy support and the fuzzy confidence must be proposed.
3. Determine large fuzzy grids
After all candidate 1-dim fuzzy grids have been generated, we need to determine how to find the other large fuzzy grids and fuzzy association rules. The model for generating fuzzy association rules is described inFig. 5, from which we can see that large fuzzy grids and fuzzy association rules are generated by phases I and II, respectively.
Suppose each linguistic variable, xm, is divided into K
various linguistic values. Given a candidate k-dim fuzzy grid, say Ax1 K;i1£ A x2 K;i2£ · · · £ A xk21 K;ik21£ A xk
K;ik; the degree which tp
belongs to this fuzzy grid can be computed as
mx1 K;i1ðtp1Þ·m x2 K;i2ðtp2Þ· · ·m xk21 K;ik21ðtpk21Þ·m xk K;ikðtpkÞ:
To check whether this fuzzy grid is to be large or not, we define its fuzzy support FSðAx1
K;i1£ A x2 K;i2£ · · · £ A xk21 K;ik21A xk K;ikÞ
Fig. 3. K ¼ 4 for ‘Age’.
Fig. 2. K ¼ 3 for ‘Age’. Fig. 1. K ¼ 2 for ‘Age’.
as follows: FSðAx1 K;i1£ A x2 K;i2£ · · · £ A xk21 K;ik21£ A xk K;ikÞ ¼ X n p¼1 mx1 K;i1ðtp1Þ·m x2 K;i2ðtp2Þ· · ·m xk21 K;ik21ðtpk21Þ·m xk K;ikðtpkÞ 2 4 3 5=n ð9Þ When FSðAx1 K;i1£ A x2 K;i2£ · · · £ A xk21 K;ik21£ A xk K;ikÞ is larger than
or equal to the user-specified minimum fuzzy support (i.e. min FS), we can say that Ax1
K;i1£ A x2 K;i2£ · · · £ A xk21 K;ik21£ A xk K;ik
is a large k-dim fuzzy grid. For any two large grids, say Ax1 K;i1£ A x2 K;i2£ · · · £ A xk21 K;ik21£ A xk K;ik and A x1 K;i1£ A x2 K;i2£ · · · £ Axk21 K;ik21 £ A xk K;ik£ A xkþ1
Kkþ1;jkþ1; since mAx1K1;j1£Ax1K2;j2£· · ·£Axk21Kk21;jk21£ Axk Kk ;jk£A xkþ1 Kkþ1;jkþ1ðtpÞ #mA x1 K1;j1£Ax1K2;j2£· · ·£A xk21 Kk21;jk21£A xk Kk ;jkðtpÞ from Eq. (9), Ax1 K;i1£ A x2 K;i2£ · ·· £ A xk21 K;ik21£ A xk K;ik£ A xkþ1 Kkþ1;jkþ1# A x1 K;i1£ Ax2 K;i2£ ·· · £ A xk21 K;ik21£ A xk
K;ik thus holds. It is clear that any
subset of a large fuzzy grid must also be large.
The general form of a fuzzy association rule, say R, can be formulated as follows: Rule R : Ax1 K;i1£ A x2 K;i2£ · · · £ A xb K;ib ) Axbþ1 K;ibþ1£ A xbþ2 K;ibþ2£ · · · £ A xa21 K;ia21£ A xa K;ia with FCðRÞ; for 1 #a;b# d ð10Þ
where FC(R ) is the fuzzy confidence of the above-mentioned rule. The left-hand side of ‘ ) ’ is the antecedent part of R, and the right-hand side is the consequent part.
The linguistic description of this rule is that: if x1is A
x1 K;i1and x2is A x2 K;i2and…and xbis A xb K;ib; then xbþ1is A xbþ1 K;ibþ1 and xbþ2 is Axbþ2 K;ibþ2 and…and xais A xa K;ia:
Since R is generated by two large fuzzy grids (i.e. Ax1 K;i1£ A x2 K;i2£ · · · £ A xb K;ib£ A xbþ1 K;ibþ1£ · · · £ A xa21 K;ia21£ A xa K;iaand Ax1 K;i1£ A x2 K;i2£ · · · £ A xb
K;ib), we define its fuzzy confidence as
follows: FCðRÞ ¼ FSðAx1 K;i1£ A x2 K;i2£ · ·· £ A xb K;ib£ A xbþ1 K;ibþ1£ · · · £ Axa21 K;ia21£ A xa K;iaÞ=FSðA x1 K;i1£ A x2 K;i2£ ·· · £ A xb K;ibÞ ð11Þ
If its fuzzy confidence is larger or equal to min FC, then it is effective. The minimum fuzzy confidence is also user-specified. Here, we give a simple example to demonstrate how we generate the large 1-dim fuzzy grids.
3.1. Example
As shown in Fig. 6, the quantitative attribute ‘Age’ denoted by x1 was divided into three various linguistic
values. These three candidate 1-dim fuzzy grids could be described as follows:
AAge3;1 : young AAge3;2 : medium AAge3;3 : old
The degrees which tp(1 # p # 12) belong to A
Age 3;1; A
Age 3;2
and AAge3;3 are shown inTable 1. We can thus compute the fuzzy support of these three candidate 1-dim fuzzy grids: FSðAAge3;1Þ ¼X 12 p¼1 mAge3;1ðtp1Þ=12 ¼ 0:275 ð12Þ FSðAAge3;2Þ ¼X 12 p¼1 mAge3;2ðtp1Þ=12 ¼ 0:475 ð13Þ FSðAAge3;3Þ ¼X 12 p¼1 mAge3;3ðtp1Þ=12 ¼ 0:250 ð14Þ
If the user-specified minimum fuzzy confidence is 0.30, then only AAge3;2 is the large 1-dim fuzzy grid.
In the subsequent section, an effective algorithm named FGBRMA is proposed to discover the fuzzy association rules.
Fig. 6. Attribute ‘Age’ is divided into three various linguistic values. Fig. 5. The model for generating fuzzy association rules.
4. Fuzzy grids based rules mining algorithm
For the proposed algorithm, a table structure named FGTTFS is implemented to generate large fuzzy grids. This table consists of the following substructures:
(a) Fuzzy grids table (FG): each row represents a fuzzy grid, and each column represents a linguistic value Axm
K;im:
(b) Transaction table (TT): each column represents a tuple tp, while each element records the membership degree
to which tpbelongs to the corresponding fuzzy grid.
(c) Column FS: stores the fuzzy support corresponding to the fuzzy grid in FG.
An initial tabular FGTTFS is shown asTable 2 as an example, from which we can see that there are two tuples t1
and t2, and two attributes x1and x2in a given relation. Both
x1and x2are divided into two linguistic values. Since each
row of FG is a bits string consisting of 0 and 1, we can apply Boolean operations on FG[u ] (i.e. the uth row of FG) ¼ (FG[u,1], FG[u,2], FG[u,3], FG[u,4]) and FG[v ] (i.e. the vth row of FG) ¼ (FG[v,1], FG[v,2], FG[v,3], FG[v,4]) to generate some desired results. For example, FG[1] OR FG[3] ¼ (1, 0, 0, 0) OR (0, 0, 1, 0) ¼ (1, 0, 1, 0), corresponding to a candidate 2-dim fuzzy grid Ax1
2;1£ A x2
2;1;
is generated. Then, FSðAx1
2;1£ A x2 2;1Þ ¼ TT½1·TT½3 ¼ ½mx1 2;1ðt11Þ·m x2 2;2ðt12Þ þm x1 2;1ðt21Þ·m x2 2;1ðt22Þ=2 is obtained to
compare with the min FS. However, any two linguistic values defined in the same linguistic variable cannot be contained in the same candidate k-dim fuzzy grid (k $ 2). Therefore, both (1, 1, 0, 0) and (0, 0, 1, 1) are invalid.
Generally, a candidate k-dim fuzzy grid, say Ax1
K;i1£ Ax2 K;i2£ · · · £ A xk21 K;ik21 £ A xk
K;ik; is derived by joining two large
(k 2 1)-dim fuzzy grids (i.e. Ax1
K;i1£ A x2 K;i2£ · · · £ A xk22 K;ik22£ Axk K;ik; and A x1 K;i1£ A x2 K;i2£ · · · £ A xk22 K;ik22£ A xk21
K;ik21) and these two
large grids share (k 2 2) linguistic values. For example, we can use Ax1 3;2£ A x2 3;2and A x1 3;2£ A x3 3;3to generate a candidate
3-dim fuzzy grid Ax1
3;2£ A x2 3;1£ A x3 3;3 because A x1 3;2£ A x2 3;1 and Ax1 3;2£ A x3
3;3 share the linguistic value A x1 3;2: However, A x1 3;2£ Ax2 3;1£ A x3
3;3 can also be generated by joining A x1 3;2£ A x2 3;1 to Ax2 3;1£ A x3
3;3: This implies that we must select one of many
possible combinations to avoid redundant computations. The method we adopt here is that if there exist integers e1,
e2,…, ek21, ek where 1 # e1, e2, · · · , ek21, ek# d,
such that FG[u, e1] ¼ FG[u, e2] ¼ · · · ¼ FG[u,
ek22] ¼ FG[u, ek21] ¼ 1 and FG[v, e1] ¼ FG[v,
e2] ¼ · · · ¼ FG[u, ek22] ¼ FG[u, ek] ¼ 1, where FG[u ]
and FG[v ] correspond to large (k 2 1)-dim fuzzy grids, then FG[u ] and FG[v ] can be paired to generate a candidate k-dim fuzzy grid.
On the other hand, we still apply Boolean operations to obtain the antecedent part and consequent part of each rule. For example, if there exists FG[u ] ¼ (1, 0, 0, 0) and FG[v ] ¼ (1, 0, 0, 1) corresponding to large fuzzy grids Lu
and Lv, where Lv, Lu, respectively; then the antecedent
part Ax1
2;1 and the consequent part A x2
2;1 of one rule, say R,
can be obtained by computing (FG[u ] AND FG[v ]) (i.e. (1, 0, 0, 0)) and (FG[u ] XOR FG[v ]) (i.e. (0, 0, 1, 0)), respectively. FCðRÞ ¼ FSðAx1 2;1£ A x2 2;1Þ=FSðA x1 2;1Þ is further
obtained to compare with the min FC to determine whether R is effective or not. FGBRMA is described as follows.
Algorithm.Fuzzy grids based rules mining algorithm Input: a. a specified database; b. the user-specified minimum fuzzy support; c. and the user-specified minimum fuzzy confidence.
Output: phase I: Generate large fuzzy grids; phase II: Generate effective fuzzy association rules.
Method:
Phase I. Generate large fuzzy grids Step 1. Perform the fuzzy partition
Step 2. Scan the database, and construct the initial table FGTTFS
Step 3. Generate large 1-dim fuzzy grids
3-1. Set k ¼ 1 and eliminate the rows of initial FGTTFS corresponding to the candidate 1-dim fuzzy grids which are not large.
3-2. Reconstruct FGTTFS. Step 4. Generate large k-dim fuzzy grids
Set k þ 1 to k. If there is only one (k 2 1)-dim fuzzy grid, then go to Step 5.
Table 1
The degrees which tp(1 # p # 12) belong to AAge3;1; AAge3;2 AAge3;3
tp AAge3;1 AAge3;2 AAge3;3
t1 0.95 0.05 0.00 t2 0.00 0.90 0.10 t3 0.45 0.55 0.00 t4 0.00 0.10 0.90 t5 0.00 1.00 0.00 t6 0.00 0.20 0.80 t7 0.90 0.10 0.00 t8 0.00 0.80 0.20 t9 0.35 0.65 0.00 t10 0.00 0.95 0.05 t11 0.65 0.35 0.00 t12 0.00 0.05 0.95 Table 2 Initial tabular FGTTFS Fuzzy grids FG TT FS Ax1 2;1 Ax2;21 A2;1x2 Ax2;22 t1 t2 Ax1 2;1 1 0 0 0 mx2;11ðt11Þ m x1 2;1ðt21Þ FSðA x1 2;1Þ Ax1 2;2 0 1 0 0 mx2;21ðt11Þ m x1 2;2ðt21Þ FSðA x1 2;2Þ Ax2 2;1 0 0 1 0 mx2;12ðt12Þ m x2 2;1ðt22Þ FSðA x2 2;1Þ Ax2 2;2 0 0 0 1 mx2;22ðt12Þ m x2 2;2ðt22Þ FSðA x2 2;2Þ
For any two unpaired rows, FGTTFS[u] and FGTTFS[v] (u – v), corresponding to large (k 2 1)-dim fuzzy grids do
4-1. From (FG[u ] OR FG[v ]) that corresponds to a candidate k-dim fuzzy grid c, if any two linguistic values are defined in the same linguistic variable, then discard c and skip Steps 4-2, 4-3 and 4-4. That is, c is invalid. 4-2. If FG[u ] and FG[v ] do not share (k 2 2) linguistic terms, then discard c and skip Steps 4-3 and 4.4. That is, c is invalid.
4-3. If there exist integers 1 # e1, e2, · · ·
, eksuch that (FG[u ] OR FG[v ])[e1] ¼ (FG[u ]
OR FG[v ])[e2] ¼ · · · ¼ (FG[u ] OR FG[v ])
[ek21] ¼ (FG[u ] OR FG[v ])[ek] ¼ 1, then
com-pute (TT[e1]· TT[e2]· · ·TT[ek]) and the fuzzy
support fs of c.
4-4. Add (FG[u ] OR FG[v ]) to table FG, (TT[e1]·
TT[e2]· · ·TT[ek]) to TT and fs to FS when fs is
larger than or equal to the min FS; otherwise, discard c.
End
Step 5. Check whether or not any large k-dim fuzzy grid is generated
If any large k-dim fuzzy grid is generated, then repeat by going to Step 4, else continue to execute the phase II. It is noted that that the final FGTTFS only stores large fuzzy grid. Phase II: Generate effective fuzzy association rules For two unpaired rows, FG[u] and FG[v] (u , v), corresponding to large fuzzy grids Luand Lvrespectively do
Step 1. Generate the antecedent part of the rule
1-1. Let temp be the number of nonzero elements in (FG[u ] AND FG[v ]).
1-2. If the number of nonzero elements in FG[u ] is equal to temp, then Lv, Luis hold, and the
antecedent part of one rule, say R, is generated as Lu; otherwise skip Steps 2 and 3.
Step 2. Generate the consequence of the rule
Use (FG[u ] XOR FG[v ]) to obtain the conse-quent part of R.
Step 3. Check or not whether rule R can be generated FCðRÞ ¼ FSðLvÞ=FSðLuÞ
If FC(R ) $ min FC, then R is effective. End
It should be noted that the design of FGBRMA follows that of the Apriori algorithm. In Section 5, a numerical example is used to demonstrate the effectiveness of FGBRMA.
5. Numerical example
A database relation EMP with 10 tuples tp(1 # p # 10)
is shown asTable 3. The purpose is to employ FGBRMA to Table 4
An initial FG obtained from EMP Fuzzy grids FG
AAge3;1 AAge3;2 AAge3;3 AMarried
2;1 AMarried2;2 ANumcars2;1 ANumcars2;2 AIncome3;1 · · · ACareer4;3 ACareer4;4
AAge3;1 1 0 0 0 0 0 0 0 · · · 0 0 AAge3;2 0 1 0 0 0 0 0 0 · · · 0 0 AAge3;3 0 0 1 0 0 0 0 0 · · · 0 0 AMarried2;1 0 0 0 1 0 0 0 0 · · · 0 0 AMarried 2;2 0 0 0 0 1 0 0 0 · · · 0 0 ANumcars 2;1 0 0 0 0 0 1 0 0 · · · 0 0 ANumcars 2;2 0 0 0 0 0 0 1 0 · · · 0 0 AAge3;1 0 0 0 0 0 0 0 1 · · · 0 0 AIncome 3;2 0 0 0 0 0 0 0 0 · · · 0 0 AIncome 3;3 0 0 0 0 0 0 0 0 · · · 0 0 ACareer 4;1 0 0 0 0 0 0 0 0 · · · 0 0 ACareer 4;2 0 0 0 0 0 0 0 0 · · · 0 0 ACareer 4;3 0 0 0 0 0 0 0 0 · · · 1 0 ACareer 4;4 0 0 0 0 0 0 0 0 · · · 0 1 Table 3 Relation EMP
tp Age Married Numcars Income Career
t1 19 N 0 20 000 Student t2 35 Y 1 50 000 Teacher t3 23 N 1 33 000 Engineer t4 33 Y 1 35 000 Engineer t5 45 Y 1 50 000 Trader t6 56 Y 1 45 000 Trader t7 18 N 0 25 000 Student t8 20 N 1 30 000 Engineer t9 33 Y 1 35 000 Engineer t10 35 Y 1 45 000 Trader
find the fuzzy association rules from EMP. For simplicity, some columns or rows of the subsequent tables are omitted by ‘· · ·’.
Phase I. Generate large fuzzy grids † Perform fuzzy partition
Since both ‘Age’ and ‘Income’ are quantitative attri-butes, K ¼ 3 is considered for these two attributes for simplicity. Also, suppose the domain interval of ‘Age’ is [0,60], and that of Income is [15 000,60 000]. The linguistic values used in this section are described as follows:
AAge3;1 : young; AAge3;2 : medium; AAge3;3 : old; AIncome 3;1 : low;
AIncome3;2 : medium; AIncome3;3 : high; AMarried2;1 : Married;
AMarried2;2 : Unmarried; ANumcars2;1 : Owns zero car; ANumcars2;2 : Owns one car; ACareer
4;1 : student; ACareer4;2 : teacher; ACareer4;3 :
engineer; ACareer4;4 : trader.
† Construct the initial table FGTTFS
After scanning EMP, the initial FGTTFS shown as
Tables 4 and 5 is built, from which we can see that all candidate 1D fuzzy grids are generated.
† Generate large 1-dim fuzzy grids
Suppose the user-specified minimum FS is 0.3, those 1-dim fuzzy grids whose fuzzy supports are smaller than the user-specified minimum FS can be removed from FGTTFS. For simplifying the table structure, FGTTFS is recon-structed as shown inTables 6 and 7, respectively.
† Generate large 2-dim fuzzy grids
FromTable 6, we can see that rows 1, 2, 3, 4, 5, 6 and 7 correspond to the large 1D fuzzy grids AAge3;2; AMarried2;1 ;
AMarried2;2 ; ANumcars2;2 ; A3Income;2 ; ACareer4;3 ; and ACareer4;4 ; respectively.
The invalid fuzzy grids such as AMarried2;1 £ AMarried2;2 cannot be
inserted into FGTTFS.
To show how a large 2-dim fuzzy grid is generated, we select AAge3;2 and AMarried
2;1 as an example. It is clear that FG[1]
and TT[1] corresponding to AAge3;2 are (1, 0, 0, 0, 0, 0, 0) and (0.6333, 0.8333, 0.7667, 0.9000, 0.5000, 0.1333, 0.6000, 0.6667, 0.9000, 0.8333), respectively. Moreover, FG[2] and TT[2] correspond to AMarried2;1 are (0, 1, 0, 0, 0, 0, 0) and (0.0,
1.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 1.0, 1.0), respectively. A candidate 2D fuzzy grid AAge3;2 £ AMarried
2;1 is thus generated by
computing (FG[1] OR FG[2]) ¼ (1, 1, 0, 0, 0, 0, 0). The fuzzy support (i.e. 0.4100) of AAge3;2 £ AMarried
2;1 can be further
obtained by computing (TT[1]·TT[2]) ¼ (0.0000, 0.8333, 0.0, 0.9000, 0.5000, 0.1333, 0.0000, 0.0000, 0.9000, 0.8333). Since 0.4100 is larger than 0.3, AAge3;2 £ AMarried
2;1 is inserted to
FGTTFS. Large 2D fuzzy grids are sequentially inserted to the bottom of FGTTFS. Those large 2-dim fuzzy grids that can be inserted to FGTTFS are shown asTables 8 and 9. We can also see that AAge3;2 £ AMarried
2;1 is the eighth row of FGTTFS.
† Generate large 3-dim fuzzy grids
To show how a large 3-dim fuzzy grid is generated, we select AAge3;2 £ AMarried
2;1 and A Age 3;2 £ A
Numcars
2;2 as an example. It
is clear that FG[8] and TT[8] corresponding to AAge3;2 £ AMarried2;1 are (1, 1, 0, 0, 0, 0, 0) and (0.0000, 0.8333, 0.0000,
0.9000, 0.5000, 0.1333, 0.0000, 0.0000, 0.9000, 0.8333), respectively. Moreover, FG[9] and TT[9] corresponding to AAge3;2 £ ANumcars
2;2 are (1, 0, 0, 1, 0, 0, 0) and (0.0000, 0.8333,
0.7667, 0.9000, 0.5000, 0.1333, 0.0000, 0.6667, 0.9000, 0.8333), respectively. A candidate 3-dim fuzzy grid AAge3;2 £ AMarried2;1 £ ANumcars2;2 can be generated by computing (FG[8]
OR FG[9]) ¼ (1, 1, 0, 1, 0, 0, 0). Since AAge3;2 £ AMarried 2;1 and
Table 5
An initial TTFS obtained from EMP
TT FS t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 0.3667 0.0000 0.2333 0.0000 0.0000 0.0000 0.4000 0.3333 0.0000 0.0000 0.1333 0.6333 0.8333 0.7667 0.9000 0.5000 0.1333 0.6000 0.6667 0.9000 0.8333 0.6767 0.0000 0.1667 0.0000 0.1000 0.5000 0.8667 0.0000 0.0000 0.1000 0.1667 0.1900 0.0000 1.0000 0.0000 1.0000 1.0000 1.0000 0.0000 0.0000 1.0000 1.0000 0.6000 1.0000 0.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.4000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.2000 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 1.0000 1.0000 1.0000 0.8000 0.7778 0.0000 0.2000 0.1111 0.0000 0.0000 0.5556 0.3333 0.1111 0.0000 0.2089 0.2222 0.4444 0.8000 0.8889 0.4444 0.6667 0.4444 0.6667 0.8889 0.6667 0.6133 0.0000 0.5556 0.0000 0.0000 0.5556 0.3333 0.0000 0.0000 0.0000 0.3333 0.1778 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.2000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.4000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 0.3000 Table 6 Reconstruction of FG Fuzzy grids FG AAge3;2 AMarried
2;1 AMarried2;2 ANumcars2;2 AIncome3;2 ACareer4;3 ACareer4;4
AAge3;2 1 0 0 0 0 0 0 AMarried 2;1 0 1 0 0 0 0 0 AMarried 2;2 0 0 1 0 0 0 0 ANumcars 2;2 0 0 0 1 0 0 0 AIncome 3;2 0 0 0 0 1 0 0 ACareer 4;3 0 0 0 0 0 1 0 ACareer 4;4 0 0 0 0 0 0 1
AAge3;2 £ ANumcars 2;2 share A
Age
3;2 and no any two linguistic values
are defined in the same linguistic variable, therefore, we are sure that c is valid. The fuzzy support (i.e. 0.3277) of AAge3;2 £ AMarried2;1 £ ANumcars2;2 can be further obtained by computing
(TT[1]·TT[2]·TT[4]) ¼ (0.0000, 0.6944, 0.0000, 0.8100, 0.2500, 0.0178, 0.0000, 0.0000, 0.8100, 0.6944). Since 0.3277 is larger than 0.3, AAge3;2 £ AMarried
2;1 £ A Numcars 2;2 is
inserted to FGTTFS. Large 3-dim fuzzy grids are also sequentially inserted to the bottom of FGTTFS. Those large
3-dim fuzzy grids that can be inserted to FGTTFS are shown asTables 10 and 11.
† Generate large 4-dim fuzzy grids
It is clear that no any large 4-dim fuzzy grid can be generated, we thus stop to generate large fuzzy grids and continue to execute the phase II.
Phase II: Generate effective fuzzy association rules When all large fuzzy grids are generated, fuzzy association rules can be easily generated. If the minimum Table 7 Reconstruction of TTFS TT FS t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 0.6333 0.8333 0.7667 0.9000 0.5000 0.1333 0.6000 0.6667 0.9000 0.8333 0.6767 0.0000 1.0000 0.0000 1.0000 1.0000 1.0000 0.0000 0.0000 1.0000 1.0000 0.6000 1.0000 0.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.4000 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 1.0000 1.0000 1.0000 0.8000 0.2222 0.4444 0.8000 0.8889 0.4444 0.6667 0.4444 0.6667 0.8889 0.6667 0.6133 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.4000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 0.3000 Table 8
FG with large 2D fuzzy grids
Fuzzy grids FG
AAge3;2 AMarried
2;1 AMarried2;2 ANumcars2;2 AIncome3;2 ACareer4;3 ACareer4;4
AAge3;2 £ AMarried 2;1 1 1 0 0 0 0 0 AAge3;2 £ ANumcars 2;2 1 0 0 1 0 0 0 AAge3;2 £ AIncome 3;2 1 0 0 0 0 0 0 AAge3;2 £ ACareer 4;3 1 0 0 0 0 1 0 AMarried2;1 £ ANumcars 2;2 0 1 0 1 0 0 0 AMarried 2;1 £ AIncome3;2 0 1 0 0 0 0 0 AMarried 2;1 £ ACareer4;4 0 1 0 0 0 0 1 ANumcars 2;2 £ AIncome3;2 0 0 0 1 1 0 0 ANumcars 2;2 £ ACareer4;3 0 0 0 1 0 1 0 ANumcars2;2 £ ACareer 4;4 0 0 0 1 0 0 1 AIncome 3;2 £ ACareer4;3 0 0 0 0 1 1 0 Table 9
TTFS with large 2D fuzzy grids
TT FS t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 0.0000 0.8333 0.0000 0.9000 0.5000 0.1333 0.0000 0.0000 0.9000 0.8333 0.4100 0.0000 0.8333 0.7667 0.9000 0.5000 0.1333 0.0000 0.6667 0.9000 0.8333 0.5533 0.1407 0.3703 0.6134 0.8000 0.2222 0.0889 0.2666 0.4445 0.8000 0.5556 0.4302 0.0000 0.0000 0.7667 0.9000 0.0000 0.0000 0.0000 0.6667 0.9000 0.0000 0.3233 0.0000 1.0000 0.0000 1.0000 1.0000 1.0000 0.0000 0.0000 1.0000 1.0000 0.6000 0.0000 0.4444 0.0000 0.8889 0.4444 0.6667 0.0000 0.0000 0.8889 0.6667 0.4000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 0.3000 0.0000 0.4444 0.8000 0.8889 0.4444 0.6667 0.0000 0.6667 0.8889 0.6667 0.5467 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.4000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 0.3000 0.0000 0.0000 0.8000 0.8889 0.0000 0.0000 0.0000 0.6667 0.8889 0.0000 0.3245
fuzzy confidence is 0.75, then fuzzy association rules with individual fuzzy confidences that are extracted from EMP are shown asTable 12. Obviously, the larger minimum fuzzy confidence, the smaller number of fuzzy association rules.
The main purpose of this numerical example is to demonstrate the effectiveness and usefulness of FGBRMA. In fact, the meaning of the fuzzy terms can be changed by linguistic hedge, as discussed in Section 6.
6. Discussions and analysis
In this paper, we propose the FGBRMA. As we have explained above, FGBRMA consists of two phases: one to generate the large fuzzy grids, and the other to generate the fuzzy association rules. It seems that the proposed algorithm is an efficient algorithm since it scans a database only once and applies Boolean operations on tables to generate large fuzzy grids and fuzzy association rules.
However, some significant topics must be discussed as follows.
6.1. Use the linguistic hedge to change the meaning of the fuzzy terms
The meaning of the linguistic values of a quantitative attribute, say xm, can be changed by a linguistic hedge [20,21], such as ‘very’ or ‘more or less’. For example, ðAxm K;imÞ 0¼ very Axm K;im ¼ ðA xm K;imÞ 2 ð15Þ ðAxm K;imÞ 00¼ more or less Axm K;im ¼ ðA xm K;imÞ 1=2 ð16Þ
The membership functions, shown asFig. 7, of ðAxm
K;imÞ 0and ðAxm K;imÞ 00are ½mxm K;imðxÞ 2 and ½mxm K;imðxÞ 1=2; respectively.
It seems that these use of linguistic hedge will provide usefully linguistic values, which will make the fuzzy association rules discovered from the database more flexible for the users.
Table 10
FG with large 3D fuzzy grids
Fuzzy grids FG
AAge3;2 AMarried
2;1 AMarried2;2 ANumcars2;2 AIncome3;2 ACareer4;3 ACareer4;4
AAge3;2 £ AMarried 2;1 £ ANumcars2;2 1 1 0 1 0 0 0 AAge3;2 £ ANumcars 2;2 £ AIncome3;2 1 0 0 1 1 0 0 AAge3;2 £ ANumcars 2;2 £ ACareer4;3 1 0 0 1 0 1 0 AMarried 2;1 £ ANumcars2;2 £ AIncome3;2 0 1 0 1 1 0 0 AMarried 2;1 £ ANumcars2;2 £ ACareer4;4 0 1 0 1 0 0 1 ANumcars2;2 £ AIncome 3;2 £ ACareer4;3 0 0 0 1 1 1 0 Table 11
TTFS with large 3D fuzzy grids
TT FS t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 0.0000 0.8333 0.0000 0.9000 0.5000 0.1333 0.0000 0.0000 0.9000 0.8333 0.4100 0.0000 0.3703 0.6134 0.8000 0.2222 0.0889 0.0000 0.4445 0.8000 0.5556 0.3895 0.0000 0.0000 0.7667 0.9000 0.0000 0.0000 0.0000 0.6667 0.9000 0.0000 0.3233 0.0000 0.4444 0.0000 0.8889 0.4444 0.6667 0.0000 0.0000 0.8889 0.6667 0.4000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 0.3000 0.0000 0.0000 0.8000 0.8889 0.0000 0.0000 0.0000 0.6667 0.8889 0.0000 0.3245 Table 12
Fuzzy association rules from EMP
Fuzzy association rules FC
AAge3;2 ) ANumcars 2;2 0.8176 ACareer 4;3 ) AAge3;2 0.8083 AMarried 2;1 ) ANumcars2;2 1.0 ANumcars 2;2 ) AMarried2;1 0.75 ACareer 4;4 ) AMarried2;1 1.0 AIncome3;2 ) ANumcars 2;2 0.8914 ACareer4;3 ) ANumcars 2;2 1.0 ACareer 4;4 ) ANumcars2;2 1.0 ACareer 4;3 ) AIncome3;2 0.8113 AAge3;2 £ AMarried 2;1 ) ANumcars2;2 0.7993 AMarried 2;1 £ AIncome3;2 ) ANumcars2;2 1.0 AMarried 2;1 £ ACareer4;4 ) ANumcars2;2 1.0 ANumcars2;2 £ ACareer 4;4 ) AMarried2;1 1.0 ANumcars 2;2 £ ACareer4;3 ) AIncome3;2 0.8113 AIncome3;2 £ ACareer 4;3 ) ANumcars2;2 1.0
6.2. Define different number of linguistic values in each quantitative attribute
The number of linguistic values defined in each quantitative attribute need not be equal to K. For example, ‘Age’ can be divided into three fuzzy sets, and ‘Income’ can be divided into four fuzzy sets. Thus, a fuzzy association rule such as AAge3;2 £ AIncome
4;1 ) A Height 5;3
may be generated. In fact, decision makers may specify possible linguistic values for one quantitat-ive attribute by using their preferences or domain knowledge.
6.3. Other topics
We do not restrict the shapes of the membership functions defined in the quantitative attributes. That is, trapezoid functions can also be used. In Lin and Lee[22], and Jang[23], the adjustment of the membership functions by learning from examples was proposed. Therefore, it seems to be possible to refine the membership functions of linguistic values by using various machine learning techniques.
On the other hand, database mining problems involving classification can be viewed within a common framework of rule discovery[24]. Based on FGBRMA, it is feasible to develop effective algorithms to discover the fuzzy classifi-cation rules.
7. Conclusions
Fuzzy association rules described by the natural language are well suitable for the thinking of human subjects. Thus, fuzzy association rules will be helpful to increase the flexibility for the users in making any decisions or designing the fuzzy systems. Furthermore, it seems that the goal of knowledge acquisition can be achieved for users by checking the fuzzy classification rules. Therefore, finding fuzzy association rules is necessary.
In this paper, the definitions of fuzzy support and fuzzy confidence for determining large fuzzy grids and effective fuzzy association rules, respectively, are proposed. An effective algorithm named FGBRMA consisting of two phases is further proposed. From the numerical example
described in Section 5, we can see that the proposed algorithm is effective and useful for finding fuzzy association rule.
As we have mentioned in discussions, we can also use various partition methods or the linguistic hedge to make the fuzzy association rules discovered from the database more flexible to decision makers.
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