Mining fuzzy generalized association rules from quantitative data under fuzzy taxonomic structures

Mining fuzzy generalized association rules from quantitative

data under fuzzy taxonomic structures

Tzung-Pei Hong*

Department of Electrical Engineering National University of Kaohsiung

Kaohsiung, 811, Taiwan, R.O.C. E-mail: tphong@nuk.edu.tw

Fax: +886+7+5919138

Kuei-Ying Lin

Institute of Information Engineering I-Shou University

Kaohsiung, 840, Taiwan, R.O.C. E-mail: m883331m@isu.edu.tw

Fax: +886+7+6577711

Shyue-Liang Wang

Department of Information Management National University of Kaohsiung

Kaohsiung, 811, Taiwan, R.O.C. E-mail: slwang@nuk.edu.tw

Fax: +886+7+5919000

Abstract

Due to the increasing use of very large databases and data warehouses, mining

useful information and helpful knowledge from transactions has become an important

research area. Most conventional data-mining algorithms identify the relationships

among transactions using binary values and find rules at a single concept level.

Transactions with quantitative values and items with taxonomic relations are, however,

commonly seen in real-world applications. Besides, the taxonomic structures may not

be crisp. This paper thus proposes a fuzzy data-mining algorithm for extracting fuzzy

generalized association rules under given fuzzy taxonomic structures. The proposed

algorithm first generates expanded transactions according to given fuzzy taxonomic

structures. It then transforms each quantitative value into a fuzzy set in linguistic

terms. Each item uses only the linguistic term with the maximum cardinality in later

mining processes, thus making the number of fuzzy regions to be processed the same

as that of the original items. The mining process based on fuzzy counts is then

performed to find fuzzy generalized association rules from these items. The algorithm

can therefore focus on the most important linguistic terms and reduce its time

complexity.

Keywords: data mining, fuzzy set, fuzzy taxonomic structure, generalized association

1.

Introduction

The rapid development of computer technology, especially increased capacities

and decreased costs of storage media, has led businesses to store huge amounts of

external and internal information in large databases at low cost. Mining useful

information and helpful knowledge from these large databases has thus evolved into

an important research area [2][4]. Years of effort in data mining has produced a

variety of efficient techniques. Depending on the types of databases to be processed,

mining approaches may be classified as working on transactional databases, temporal

databases, relational databases, and multimedia databases, among others. Depending

on the classes of knowledge sought, mining approaches may be classified as finding

association rules, classification rules, clustering rules, and sequential patterns, among

others [4].

Deriving association rules from transaction databases is most commonly seen in

data mining [1][2][4]. It discovers relationships among items such that the presence of

certain items in a transaction tends to imply the presence of certain other items. Most

previous studies concentrated on showing how binary-valued transaction data on a

single level of items may be handled. However, transaction data in real-world

applications usually consist of quantitative values and items are often organized in a

quantitative data on multiple levels of items presents a challenge to workers in this

research field.

In the past, Agrawal and his co-workers proposed several mining algorithms for

finding association rules in transaction data based on the concept of large itemsets

[1-2, 13]. They also proposed a method [14] for mining association rules from data

sets using quantitative and categorical attributes. Their proposed method first

determined the number of partitions for each quantitative attribute, and then mapped

all possible values of each attribute onto a set of consecutive integers. Other methods

have also been proposed to handle numeric attributes and to derive association rules.

Fukuda et al. introduced the optimized association-rule problem and permitted

association rules to contain single uninstantiated conditions on the left-hand side [5].

They also proposed schemes for determining conditions under which rule confidence

or support values were maximized. However, their schemes were suitable only for

single optimal regions. Rastogi and Shim extended the approach to more than one

optimal region, and showed that the problem was NP-hard even for cases involving

one uninstantiated numeric attribute [13][14].

Recently, the fuzzy set theory has been used more and more frequently in

intelligent systems because of its simplicity and similarity to human reasoning [11].

designed and used to good effect with specific domains. Strategies based on decision

trees were proposed in [12, 18-19]. Wang et al. proposed a fuzzy version-space

learning strategy for managing vague information [17]. Hong et al. also proposed a

fuzzy mining algorithm for managing quantitative data [6].

In [10], we proposed a data-mining algorithm able to deal with quantitative data

under a crisp taxonomic structure. In that approach, each item definitely belongs to

only one ancestor in the taxonomic structure. The taxonomic structures may, however,

not be crisp in real-world applications. An item may belong to different classes in

different views. This paper thus proposes a new fuzzy data-mining algorithm for

extracting fuzzy generalized association rules under given fuzzy taxonomic structures.

The proposed algorithm first generates expanded transactions according to given

fuzzy taxonomic structures. It then transforms each quantitative value into a fuzzy set

in linguistic terms. Each item uses only the linguistic term with the maximum

cardinality in later mining processes, thus making the number of fuzzy regions to be

processed the same as that of the original items. The mining process based on fuzzy

counts is then performed to find fuzzy generalized association rules from these items.

The algorithm can therefore focus on the most important linguistic terms and reduce

its time complexity. The rules mined are expressed in linguistic terms, which are more

The remaining parts of this paper are organized as follows. Data mining with

fuzzy taxonomic structures is described in Section 2. A novel data-mining algorithm

for quantitative values under fuzzy taxonomic structures is proposed in Section 3. An

example to illustrate the proposed algorithm is given in Section 4. Experimental

results are shown in Section 5. Conclusions and future works are stated in Section 6.

2. Data mining with a fuzzy taxonomic structure

Previous studies on data mining focused on finding association rules on a

single-concept level. Mining multiple-concept-level rules may, however, lead to

discovery of more general and important knowledge from data. Relevant taxonomies

of data items are thus usually predefined in real-world applications. An item may,

however, belong to different classes in different views. When taxonomic structures are

not crisp, hierarchical graphs can be used to represent them. Terminal nodes on the

hierarchical graphs represent the items actually appearing in transactions; internal

nodes represent classes or concepts formed by lower-level nodes. A simple example is

Figure 1. An example of fuzzy taxonomic structures

In this example, vegetarian diet falls into two classes: fruit and vegetable. Fruit

can be further classified into apple and tomato. Similarly, assume vegetable is divided

into tomato and cabbage. Note that tomato belongs to both fruit and vegetable with

different membership degrees. It is thought of as fruit with 0.9 membership value and

as vegetable with 0.7. The membership value of tomato belonging to vegetarian diet

can be calculated using the max-min product combination. Since both fruit and

vegetable belong to vegetarian diet with membership value 1, the membership value

of tomato belonging to vegetarian diet is then max(min(1, 0.9), min(1, 0.7))=0.9. Only

the terminal items (apple, tomato, cabbage, pork and beef) can appear in transactions.

The membership degrees of ancestors for each terminal node are shown in Table 1.

Vegetarian Diet Meat

Fruit Vegetable

Apple Tomato Cabbage

Pork Beef 1 1

1 1

Table 1. The membership degrees of ancestors for each terminal node in this example Terminal node Membership values of ancestors

Apple 1/Fruit, 1/ Vegetarian-Diet

Tomato 0.9/Fruit, 0.7/ Vegetable, 0.9/ Vegetarian-Diet Cabbage 1/Vegetable, 1/ Vegetarian-Diet

Pork 1/Meat Beef 1/Meat

Wei and Chen proposed a method to find generalized association rules under

fuzzy taxonomic structures [16]. The items to be processed in their approach are

binary. Their mining process first calculated the membership values of ancestors for

each terminal node in the manner mentioned above. It then added the ancestors of

items to the given transactions as Srikant and Agrawal’s did [15]. Candidate itemsets

were generated and counted by scanning the expanded transaction data. If the number

of an itemset appearing in the expanded transactions was larger than a pre-defined

threshold value (called the minimum support), the itemset was considered a large

itemset. Itemsets containing only single items were first processed. The 1-large

itemsets derived were then combined to form candidate itemsets containing two items.

This process was repeated until all large itemsets had been found. After that, all

possible generalized association rules were induced from the large itemsets. The rules

with confidence values larger than a predefined threshold (called the minimum

and interesting rules were output. An interesting rule must satisfy at least one of the

following three interest requirements:

1.a rule with no ancestor rules (by replacing the items in a rule with their

ancestors in the taxonomy) mined out,

2.the support value of a rule being R-time larger than the expected support

values of its ancestor rules, and

3.the confidence value of a rule being R-time larger than the expected

confidence values of its ancestor rules.

Wei and Chen’s concepts will be used in our approach to mine fuzzy interesting

generalized association rules from quantitative transaction data. The rules mined are

expressed in linguistic terms, which are more natural and understandable for human

beings.

3. Mining fuzzy generalized association rules from

quantitative data under fuzzy taxonomic structure

The proposed generalized mining algorithm integrates fuzzy-set concepts and

generalized data mining technologies to find cross-level interesting rules from

taxonomy. The knowledge derived is represented by fuzzy linguistic terms, and thus

easily understandable by human beings.

The proposed fuzzy mining algorithm first calculates the membership values of

ancestors for each terminal node as Wei and Chen’s approach did [16]. It then forms

expanded transactions and uses membership functions to transform each quantitative

value into a fuzzy set in linguistic terms. It adopts an iterative search approach to

finding large itemsets. Each item uses only the linguistic term with the maximum

cardinality in later mining processes, thus making the number of fuzzy regions to be

processed the same as the number of original items. The algorithm therefore focuses

on the most important linguistic terms, which reduces its time complexity. A mining

process using fuzzy counts is performed to find fuzzy multiple-level association rules.

Fuzzy interest requirements are then checked to remove uninteresting rules. Details of

the proposed fuzzy mining algorithm are stated below.

The fuzzy generalized mining algorithm for fuzzy taxonomic structures:

INPUT: A body of n quantitative transaction data, a set of membership functions, a

fuzzy taxonomic structure, a minimum support valueα , a minimum

confidence valueλ, and an interest threshold R.

OUTPUT: A set of fuzzy generalized association rules.

given fuzzy taxonomic structure.

STEP 2: Calculate the quantitative value vik of each ancestor item Ak in transaction

datum Di (I=1 to n) as:

) ( * ∑ = i j k D in T ij A j ik v T v µ ,

where Tj is a terminal item appearing in Di, vij is the quantitative value of Tj ,

and µAk(Tj) is the membership value of item Tj belonging to ancestor Ak.

STEP 3: Add ancestors of appearing items to transactions with their quantities

calculated from STEP 2.

STEP 4: Transform the quantitative value vij of each expanded item name Ij (terminal

item or ancestor item) appearing in transaction datum Di (I=1 to n) into a

fuzzy set fij represented as 

       + + + j j jh ijh j ij j ij R f R f R f .... 2 2 1 1

using the given

membership functions, where hj is the number of fuzzy regions for Ij, Rjl is

the l-th fuzzy region of Ij, 1≤l≤hj, and fijlis vij’s fuzzy membership value

in region Rjl.

STEP 5: Calculate the scalar cardinality countjl of each fuzzy region Rjl in the

transaction data as:

∑

STEP 6: Find max-

(

)

h l j MAXcount count j 1 =

will be used to represent the fuzzy characteristic of item Ij in later mining

processes.

STEP 7: Check whether the value max-countj of a region max-Rj, j = 1 to m, is larger

than or equal to the predefined minimum support value α . If a region

max-Rj is equal to or greater than the minimum support value, put it in the

large 1-itemsets (L ). That is, 1

{

}

L₁ = max− _jmax− _j ≥

α

STEP 8: Generate the candidate set C2 from L1. Each 2-itemset in C2 must not include

items with ancestor or descendant relations in the fuzzy taxonomy.

STEP 9: For each newly formed candidate 2-itemset s with items (s1, s2) in C2:

(a) Calculate the fuzzy value of s in each transaction datum Di as fis = fis1Λfis2 ,

where

j is

f is the membership value of Di in region sj. If the minimum

operator is used for the intersection, then fis =min(fis1,fis2). (b) Calculate the scalar cardinality of s in the transaction data as:

counts =

∑

(c) If counts is larger than or equal to the predefined minimum support value

α

put s in the large 2-itemsets (L2).

STEP 10: IF L2 is null, then exit the algorithm; otherwise, do the next step.

current large itemsets.

STEP 12: Generate the candidate set Cr+1 from Lr in a way similar to that in the

apriori algorithm [1]. That is, the algorithm first joins Lr and Lr assuming

that r-1 items in the two itemsets are the same and the other one is different.

Store in Cr+1 itemsets having all their sub-r-itemsets in Lr.

STEP 13: For each newly formed (r+1)-itemset s with items

(

s

,

s

,

...,

s

)

Cr+1:

(a) Calculate the fuzzy value of s in each transaction datum Di as

1 2 1Λ Λ...Λ + = is is is_r is f f f f , where j is

f is the membership value of Di in

region sj. If the minimum operator is used for the intersection, then

j is r j is Minf f 1 1 + = = .

(b) Calculate the scalar cardinality of s in the transaction data as:

counts =

∑

(c) If counts is larger than or equal to the predefined minimum support value

α

put s in Lr+1.

STEP 14: If Lr+1 is null, then do the next step; otherwise, set r=r+1 and repeat STEPs

12 to 14.

STEP 15: Construct the association rules for all the large q-itemset s containing items

(a) Form all possible association rules thusly:

s1Λ...Λs_k−1Λs_k+1Λ...Λs_q → s_k, k=1 to q.

(b) Calculate the confidence values of all association rules using the formula:

∑ ∑ = = Λ Λ Λ Λ _{− +} n i is is is is n i is q K K f f f f f 1 1 ) ... , ... ( 1 1 1 .

STEP 16: Keep the rules with confidence values larger than or equal to the predefined

confidence thresholdλ.

STEP 17: Output the rules without ancestor rules (by replacing the items in a rule with

their ancestors in the taxonomy) to users as interesting rules.

STEP 18: For each remaining rule s (represented as s₁Λs₂Λ...Λs_r →s_(r+1)), find the

close ancestor rule t (represented as t₁Λt₂Λ...Λt_r →t_(r+1)) and calculate

the support interest measure Isupport (s) of s as:

( )

and the confidence interest measure Iconfidence (s) of s as:

( )

where confidences and confidencet are respectively the confidence values

of rules s and t; output the rules with their support interest measures or

threshold R to users as interesting rules.

Note that in Step 17, an ancestor of a fuzzy rule is formed by replacing the items in the rule with their ancestors in the fuzzy taxonomy, but the linguistic terms in both the rules must be the same. The rules output from Steps 17 and 18 can then serve as meta-knowledge concerning the given transactions.

4. An example

In this section, an example is given to illustrate the proposed fuzzy generalized mining algorithm. This is a simple example to show how the proposed algorithm generates fuzzy generalized association rules from quantitative transactions under a fuzzy taxonomic structure. The data set includes the six transactions shown in Table 2.

Table 2. Six transactions in this example

Transaction ID Items

1 (Apple, 3) (Tomato, 4) (Beef, 2) 2 (Tomato, 7) (Cabbage, 7) (Beef, 7) 3 (Tomato, 2) (Cabbage, 10) (Beef, 5) 4 (Cabbage, 9) (Beef, 10)

5 (Apple, 7) (Pork, 8) 6 (Apple, 2) (Tomato, 8)

Each transaction includes a transaction ID and some purchased items. Each item

transaction consists of nine units of cabbage and ten units of beef. Assume the

predefined fuzzy taxonomy is as shown in Figure 1. For convenience, the simple

symbols in Table 3 are used to represent the items and groups.

Table 3. Items and groups are represented by simple symbols for convenience Items Symbol Groups Symbol

Apple A Fruit T1

Tomato B Vegetable T2

Cabbage C Vegetarian Diet T3

Pork D Meat T4

Beef E

Also assume that the fuzzy membership functions are the same for all the items

and are as shown in Figure 2.

Figure 2. The membership functions used in this example

In the example, amounts are represented by three fuzzy regions: Low, Middle

and High. Thus, three fuzzy membership values are produced for each item amount

Low Middle High

1 0 Membership Value Item Amount 0 1 6 11

Low Middle High

1 0 Membership Value Item Amount 0 1 6 11

according to the predefined membership functions. For the transaction data in Table 2,

all the expanded transactions are shown in Table 4.

Table 4. The expanded transactions

Transaction ID Expanded Items

1 (A, 3)(B, 4)(E, 2)(T1, 6.6)(T2, 2.8)(T3, 6.6)(T4, 2) 2 (B, 7)(C, 7)(E, 7)(T1, 6.3)(T2, 11.9)(T3, 13.3)(T4, 7) 3 (B, 2)(C, 10)(E, 5)(T1, 1.8)(T2, 11.4)(T3, 11.8)(T4, 5) 4 (C, 9)(E, 10)(T2, 9)(T3, 9)(T4, 10) 5 (A, 7)(D, 8)(T1, 7) (T3, 7)(T4, 8) 6 (A, 2)(B, 8) (T1, 9.2)(T2, 5.6)(T3, 9.2)

The quantitative values of the expanded items are represented using fuzzy sets,

which are shown in Table 5, where the notation item.term is called a fuzzy region.

Table 5. The fuzzy sets transformed from the data in Table 4

TID Fuzzy set

1 ) . 2 . 0 . 8 . 0 )( . 12 . 0 . 88 . 0 )( . 36 . 0 . 64 . 0 )( . 12 . 0 . 88 . 0 )( . 2 . 0 . 8 . 0 )( . 6 . 0 . 4 . 0 )( . 4 . 0 . 6 . 0 ( 4 4 3 3 2 2 1

1MiddleTHighTLow TMiddleTMiddleTHighTLow TMiddle

T Middle E Low E Middle B Low B Middle A Low A + + + + + + + 2 ) . 2 . 0 . 8 . 0 )( . 0 . 1 )( . 0 . 1 )( . 06 . 0 . 94 . 0 )( . 2 . 0 . 8 . 0 )( . 2 . 0 . 8 . 0 )( . 2 . 0 . 8 . 0 ( 4 4 3 2 1

1MiddleTHighTHighTHighTMiddleTHigh

T High E Middle E High C Middle C High B Middle B + + + + + 3 ) . 8 . 0 . 2 . 0 )( . 0 . 1 )( . 0 . 1 )( . 16 . 0 . 84 . 0 )( . 8 . 0 . 2 . 0 )( . 8 . 0 . 2 . 0 )( . 2 . 0 . 8 . 0 ( 4 4 3 2 1

1Low TMiddleTHighTHigh TLow TMiddle

T Middle E Low E High C Middle C Middle B Low B + + + + + 4 ) . 8 . 0 . 2 . 0 )( . 6 . 0 . 4 . 0 )( . 6 . 0 . 4 . 0 )( . 8 . 0 . 2 . 0 )( . 6 . 0 . 4 . 0 ( 4 4 3 3 2

2Middle T High T Middle T High T Middele T High

T High E Middle E High C Middle C + + + + + 5 ) . 4 . 0 . 6 . 0 )( . 2 . 0 . 8 . 0 )( . 2 . 0 . 8 . 0 )( . 4 . 0 . 6 . 0 )( . 2 . 0 . 8 . 0 ( 4 4 3 3 1

1MiddleTHighTMiddle THighTMiddleTHigh

T High D Middle D High A Middle A + + + + + 6 ) . 64 . 0 . 36 . 0 )( . 92 . 0 . 08 . 0 )( . 64 . 0 . 36 . 0 )( . 4 . 0 . 6 . 0 )( . 2 . 0 . 8 . 0 ( 3 3 2 2 1

1Middle THigh TLow TMiddleTMiddle THigh

T High B Middle B Middle A Low A + + + + +

and T4, and "High" is chosen C, T2 and T3.

The count of each region selected is then checked against the predefined

minimum support value α . Assume in this example,α is set at 1.5. L1 is shown in

Table 6.

Table 6. The set of large 1-itemsets in this example

Itemset

count

Similarly, L2 includes (E.Middle, T2.High), (E.Middle, T3.High), (T1. Middle,

T4.Middle), (T2.High, T4.Middle) and (T3.High, T4.Middle). L3 is null.

Assume the given confidence thresholdλ is set at 0.7. The following three rules

are thus kept:

If E = Middle, then T2 = High, with a support value of 1.8 and a

confidence value of 0.9;

If E = Middle, then T3 = High, with a support value of 1.92 and a

confidence value of 0.96;

If T4 = Middle, then T3 = High, with a support value of 2.12 and a

confidence value of 0.82

out, it is thus output as an interesting rule. Assume the given interest threshold R is set at 3. Since the support and the confidence interest measures of the remaining two rules are less than 3, they are thus not interesting rules.

5. Experimental Results

The section reports on experiments made to show the effect of the proposed

approach. They were implemented in C on a Pentium-III 700 Personal Computer. The

number of levels was set at 3. There were 64 purchased items (terminal nodes) on

level 3, 16 generalized items on level 2, and 4 generalized items on level 1. Each

non-terminal node had four branches. Only the terminal nodes could appear in

transactions. Totally 10000 transactions with an average of 12 purchased items in each

transaction are generated in each run. The relationships between numbers of rules

mined and minimum support values for minimum confidence value set at 0.7 are

Figure 3. The relationships between numbers of rules mined and minimum supports

From Figure 3, it is easily seen that numbers of rules mined decrease along with

increase of minimum support values. This is quite consistent with our intuition.

6. Conclusions and future works

In this paper, we have proposed a fuzzy generalized mining algorithm for

processing transaction data with quantitative values and discovering interesting

generalized association rules among them. The rules thus mined out exhibit

quantitative regularity under given fuzzy taxonomic structures and can be used to

provide suggestions to appropriate supervisors. Compared to conventional crisp-set

mining methods for quantitative data, our approach gets smoother mining results due

to its fuzzy membership characteristics.

0 200 400 600 800 1000 1200 1400 1600 1800 300 400 500 600 700 800 900 1000 min-support nu m be r of r ul es 0 200 400 600 800 1000 1200 1400 1600 1800 300 400 500 600 700 800 900 1000 min-support nu m be r of r ul es

Although the proposed method works well in data mining for quantitative values,

it is just a beginning. There is still much work to be done in this field. Our method

assumes that membership functions are known in advance. In [7-9], we proposed

some fuzzy learning methods to automatically derive membership functions. We will

therefore attempt to dynamically adjust the membership functions in the proposed

fuzzy generalized mining algorithm to avoid the bottleneck of membership function

acquisition. We will also attempt to design specific data-mining models for various

problem domains.

Acknowledgment

The authors would like to thank the anonymous referees for their very

constructive comments. This research was supported by the National Science Council

of the Republic of China under contract NSC 90-2213-E-390-001.

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