Mining association rules from quantitative data

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/222034732 Mining association rules from quantitative data Article in Intelligent Data Analysis · November 1999 DOI: 10.1016/S1088-467X(99)00028-1 CITATIONS READS 199 392 3 authors, including: Tzung-Pei Hong National University of Kaohsiung 639 PUBLICATIONS 5,145 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Compact representation of frequent itemsets View project All content following this page was uploaded by Tzung-Pei Hong on 08 February 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. Mining Association Rules from Quantitative Data* Tzung-Pei Hong **, Chan-Sheng Kuo‡, and Sheng-Chai Chi‡ Department of Information Management ‡ Graduate School of Management Science I-Shou University Kaohsiung, 84008, Taiwan, R.O.C. e-mail: tphong@csa500.isu.edu.tw ABSTRACT Data mining is the process of extracting desirable knowledge or interesting patterns from existing databases for specific purposes. Most conventional data-mining algorithms identify the relationships among transactions using binary values, however, transactions with quantitative values are commonly seen in real-world applications. This paper thus proposes a new data-mining algorithm for extracting interesting knowledge from transactions stored as quantitative values. The proposed algorithm integrates fuzzy set concepts and the apriori mining algorithm to find interesting fuzzy association rules in given transaction data sets. Experiments with student grades at I-Shou University were also made to verify the performance of the proposed algorithm. Keywords: data mining, fuzzy set, association rule, transaction, quantitative value. -----------------------------------* This is a modified and expanded version of the paper "A data mining algorithm for transaction data with quantitative values," presented at The Eighth International Fuzzy Systems Association World Congress, 1999. **Corresponding author. 1 1. INTRODUCTION Knowledge discovery in databases (KDD) has become a process of considerable interest in recent years as the amounts of data in many databases have grown tremendously large. KDD means the application of nontrivial procedures for identifying effective, coherent, potentially useful, and previously unknown patterns in large databases [13]. The KDD process generally consists of the following three phases [12, 25]. (1) Pre-processing: This consists of all the actions taken before the actual data analysis process starts [12]. Famili et al think that it may be performed on the data for the following reasons: solving data problems that may prevent us from performing any type of analysis on the data, understanding the nature of the data, performing a more meaningful data analysis, and extracting more meaningful knowledge from a given set of data. (2) Data mining: This involves applying specific algorithms for extracting patterns or rules from data sets in a particular representation. (3) Post-processing: This translates discovered patterns into forms acceptable 2 for human beings. It may also make possible visualization of extracted patterns. Due to the importance of data mining to KDD, many researchers in database and machine learning fields are primarily interested in this new research topic because it offers opportunities to discover useful information and important relevant patterns in large databases, thus helping decision-makers easily analyze the data and make good decisions regarding the domains concerned. Data-mining is most commonly used in attempts to induce association rules from transaction data. Most previous studies have only shown, however, how binary valued transaction data may be handled. Transaction data in real-world applications do not usually consist of quantitative values, so designing a sophisticated data-mining algorithm able to deal with various types of data presents a challenge to workers in this research field. Fuzzy set theory is being used more and more frequently in intelligent systems because of its simplicity and similarity to human reasoning [22]. The theory has been applied in fields such as manufacturing, engineering, diagnosis, economics, among others [16, 22, 24, 36]. Several fuzzy learning algorithms for inducing rules from 3 given sets of data have been designed and used to good effect with specific domains [5-6, 8, 11, 15, 17-21, 30, 32-33]. Strategies based on decision trees [9] were proposed in [10, 26-27, 30, 34-35], and Wang et al. proposed a fuzzy version space learning strategy for managing vague information [32]. This paper integrates fuzzy-set concepts with the apriori mining algorithm [4] and uses the result to find interesting itemsets and fuzzy association rules in transaction data with quantitative values. A new mining algorithm, called the Fuzzy Transaction Data-mining Algorithm (FTDA) is proposed. It transforms quantitative values in transactions into linguistic terms, then filters them to find association rules by modifying the apriori mining algorithm [4] . The remaining parts of this paper are organized as follows. Agrawal et al.'s mining algorithms are reviewed in Section 2. Fuzzy-set concepts are introduced in Section 3. The proposed data-mining algorithm for quantitative values is described in Section 4. An example is given to illustrate the proposed algorithm in Section 5. Experiments to demonstrate the performance of the proposed data-mining algorithm are stated in Section 6. Conclusions and proposal of future work are given in Section 7. 4 2. Related Works The goal of data mining is to discover important associations among items such that the presence of some items in a transaction will imply the presence of some other items. To achieve this purpose, Agrawal and his co-workers proposed several mining algorithms based on the concept of large itemsets to find association rules in transaction data [1-4]. They divided the mining process into two phases. In the first phase, candidate itemsets were generated and counted by scanning the transaction data. If the number of an itemset appearing in the transactions was larger than a pre-defined threshold value (called minimum support), the itemset was considered a large itemset. Itemsets containing only one item were processed first. Large itemsets containing only single items were then combined to form candidate itemsets containing two items. This process was repeated until all large itemsets had been found. In the second phase, association rules were induced from the large itemsets found in the first phase. All possible association combinations for each large itemset were formed, and those with calculated confidence values larger than a predefined threshold (called minimum confidence) were output as association rules. 5 In addition to proposing methods for mining association rules from transactions of binary values, Agrawal et al. also proposed a method [31] for mining association rules from those with quantitative and categorical attributes. Their proposed method first determines the number of partitions for each quantitative attribute, and then maps all possible values of each attribute into a set of consecutive integers. It then finds large itemsets whose support values are greater than the user-specified minimum-support levels. These large itemsets are then processed to generate association rules, and rules of interest to users are output. Some other methods were also 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 [14]. They also proposed schemes to determine the conditions such that the confidence or support values of the rules are maximized. However, their schemes were only suitable for a single optimal region. Rastogi and Shim thus extended the problem for more than one optimal regions, and showed that the problem was NP-hard even for the case of one uninstantiated numeric attribute [28, 29]. Some works also used fuzzy set theory and data mining technology to solve classification problems [7, 23]. 6 In this paper, we use fuzzy set concepts to mine association rules from transactions with quantitative attributes. The mined rules are expressed in linguistic terms, which are more natural and understandable for human beings. 3. Review of Fuzzy Set Concepts Fuzzy set theory was first proposed by Zadeh and Goguen in 1965 [37]. Fuzzy set theory is primarily concerned with quantifying and reasoning using natural language in which words can have ambiguous meanings. This can be thought of as an extension of traditional crisp sets, in which each element must either be in or not in a set. Formally, the process by which individuals from a universal set X are determined to be either members or non-members of a crisp set can be defined by a characteristic or discrimination function [37]. For a given crisp set A, this function assigns a value µ A (x) to every x ∈ X such that 1 0 µ A (x) = if and only if x∈A if and only if x∉A. 7 Thus, the function maps elements of the universal set to the set containing 0 and 1. This kind of function can be generalized such that the values assigned to the elements of the universal set fall within specified ranges, referred to as the membership grades of these elements in the set. Larger values denote higher degrees of set membership. Such a function is called the membership function, µ A ( x) , by which a fuzzy set A is usually defined. This function is represented by µ A : X → [ 0, 1] , where [0, 1] denotes the interval of real numbers from 0 to 1, inclusive. The function can also be generalized to any real interval instead of [0,1]. A special notation is often used in the literature to represent fuzzy sets. Assume that x1 to xn are the elements in fuzzy set A, and µ1 to µn are, respectively, their grades of membership in A. A is then usually represented as follows: A = µ1 / x1 + µ2 / x2 +...+ µn / xn . 8 An α-cut of a fuzzy set A is a crisp set Aα that contains all elements in the universal set X with membership grades in A greater than or equal to a specified value of α. This definition can be written as: Aα = {x ∈ X︱ µ A ( x) ≥ α }. The scalar cardinality of a fuzzy set A defined on a finite universal set X is the summation of the membership grades of all the elements of X in A. Thus, A = ∑ µ A(x) x∈X . Among operations on fuzzy sets are the basic and commonly used complementation, union and intersection, as proposed by Zadeh. (1) The complementation of a fuzzy set A is denoted by ¬ A, and the membership function of ¬ A is given by: µ ¬A ( x) = 1 − µ ¬A ( x) , ∀ x ∈X . (2) The intersection of two fuzzy sets A and B is denoted by A I B, and the membership function of A I B is given by: 9 µA IB { } ( x) = min µ A ( x) , µ B ( x) , ∀ x ∈X . (3) The union of fuzzy sets A and B is denoted by A U B, and the membership function of A U B is given by: µA ∪B { } ( x) = max µ A ( x) , µ B ( x) , ∀ x ∈X . 4. The fuzzy data-mining algorithm for quantitative values In this section, the fuzzy concepts are used in the apriori data-mining algorithm to discover useful association rules among quantitative values. The following notation is used in this paper. n: the total amount of transaction data; m: the total number of attributes; D (i ) : the i-th transaction datum, 1 ≤ i ≤ n; Aj : the j-th attribute, 1 ≤ j ≤ m; Aj : the number of fuzzy regions for A j ; R jk : the k-th fuzzy region of A j , 1 ≤ k ≤ A j ; 10 v (ji ) : the quantitative value of A j in D (i ) ; f j(i) : the fuzzy set converted from v (ji ) ; f jk(i ) : the membership value of v (ji ) in Region R jk ; count jk : the summation of f jk( i ) , i=1 to n; count max : the maximum count value among count jk values, k=1 to A j ; j R max : the fuzzy region of A j with count max ; j j α : the predefined minimum support level; λ : the predefined minimum confidence value; Cr: the set of candidate itemsets with r attributes (items); Lr : the set of large itemsets with r attributes (items). The proposed fuzzy mining algorithm first uses membership functions to transform each quantitative value into a fuzzy set in linguistic terms. The algorithm then calculates the scalar cardinalities of all linguistic terms in the transaction data. Each attribute uses only the linguistic term with the maximum cardinality in later mining processes, thus keeping the number of items the same as that of the original attributes. The algorithm is therefore focused on the most important linguistic terms, which reduces its time complexity. The mining process using fuzzy counts is then performed to find fuzzy association rules. Details of the proposed mining algorithm 11 are described below. The FTDA algorithm: INPUT: A body of n transaction data, each with m attribute values, a set of membership functions, a predefined minimum support value α , and a predefined confidence value λ . OUTPUT: A set of fuzzy association rules. STEP 1: Transform the quantitative value v (ji ) of each transaction datum D (i ) , i=1 to n, for each attribute Aj, j=1 to m, into a fuzzy set f j(i ) represented as (i )   f (i ) f (i )  j1 + j2 + .... + f jl   R R j2 R jl  using the given membership functions, where Rjk  j1  is the k-th fuzzy region of attribute Aj, f jk(i ) is v (ji ) ’s fuzzy membership value in region R jk , and l (= A j ) is the number of fuzzy regions for Aj. STEP 2: Calculate the scalar cardinality of each attribute region Rjk in the transaction data: count jk = ∑ f jk(i ) . n i =1 = MAX(countjk ) , for j= 1 to m, where A j is the number of fuzzy STEP 3: Find countmax j Aj k =1 max regions for A j . Let R max be the region with countmax for attribute A j . R j j j will be used to represent this attribute in later mining processing. max STEP 4: Check whether the countmax of each R j , j=1 to m, is larger than or equal to j 12 the predefined minimum support value α . If R j max { is equal to or greater than } the minimum support value, put it in the set of large 1-itemsets (L1). That is, ≥ α, 1 ≤ j ≤ m . Countmax L1 = Rmax j j STEP 5: Set r=1, where r is used to represent the number of items kept in the current large itemsets. STEP 6: Generate the candidate set Cr+1 from Lr in a way similar to that in the apriori algorithm [4]. 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. It then keeps in Cr+1 the itemsets, which have all their sub-itemsets of r items existing in Lr. STEP 7: Do the following substeps for each newly formed (r+1)-itemset s with items (s1 , s2 , ..., sr +1 ) in Cr+1: (a) Calculate the fuzzy value of each transaction data D (i ) in s as f s( i ) = f s1(i )Λ f s(2i ) Λ ... Λ f s(r i+)1 , where f s(ji ) is the membership value of D (i ) in region sj. If the minimum operator is used for the intersection, (i ) (i ) then f s = Min f s j . r +1 j =1 (b) Calculate the scalar cardinality of s in the transaction data as: counts = ∑ f s( i ) . n i =1 13 (c) If counts is larger than or equal to the predefined minimum support value α , put s in Lr+1. STEP 8: IF Lr+1 is null, then do the next step; otherwise, set r=r+1 and repeat STEPs 6 to 8. STEP 9: Construct the association rules for all large q-itemset s with items (s1 , ) s 2 , ..., s q , q ≥ 2, using the following substeps: (a) Form all possible association rules as follows: s1 Λ ... Λ s k −1 Λ s k +1 Λ ... Λ s q → s k , k=1 to q. (b) Calculate the confidence values of all association rules using: (i ) ∑ fs n ∑( n i =1 f s(1i ) Λ ... Λ i =1 f s(ki−) 1 , f s(ki+) 1 Λ ... Λ . f s(qi ) ) STEP 10: Output the rules with confidence values larger than or equal to the predefined confidence threshold λ . After STEP 10, the rules output can serve as meta-knowledge concerning the given transactions. Since each attribute uses only the linguistic term with the maximum cardinality in the mining process, the number of items is thus the same as that of the original attributes. Instead, if in Step 4, all the regions with support values larger than the threshold are considered, the algorithm will generate more rules, but 14 will take much more time than the proposed method. Trade-off thus exists between the rule completeness and the time complexity. 5. An Example In this section, an example is given to illustrate the proposed data-mining algorithm. This is a simple example to show how the proposed algorithm can be used to generate association rules for course grades according to historical data concerning students' course scores. The data set includes 10 transactions, as shown in Table 1. Table 1:The set of students’ course scores in the example Case No. OOP DB ST DS MIS 1 86 77 86 71 68 2 61 87 89 77 80 3 84 89 86 79 89 4 73 86 79 84 62 5 70 85 87 72 79 6 65 67 86 61 87 7 71 87 75 71 80 8 86 69 64 84 88 9 75 65 86 86 79 10 83 68 65 85 89 Each case consists of five course scores: Object-Oriented Programming (denoted OOP), Database (denoted DB), Statistics (denoted ST), Data Structure (denoted DS), 15 and Management Information System (denoted MIS). Each course is thought of as an attribute in the mining process. Assume the fuzzy membership functions for the course scores are as shown in Figure 1. Membership value Low 1 Middle High 59 63 69 73 78 85 90 100 X Figure 1: The membership function used in this example In this example, each attribute has three fuzzy regions: Low, Middle, and High. Thus, three fuzzy membership values are produced for each course score according to the predefined membership functions. For the transaction data in Table 1, the proposed mining algorithm proceeds as follows. STEP 1: Transform the quantitative values of each transaction datum into fuzzy sets. Take the OOP score in Case 1 as an example. The score “86” is converted into a fuzzy set ( 0.0 Low + 0 .0 0 .7 + ) Middle High using the given membership functions. This step is repeated for the other cases and courses, and the results are shown in Table 2. 16 Table 2: The fuzzy sets transformed from the data in Table 1 Case No. OOP DB L M H L 1 0.0 0.0 0.7 0.0 2 0.8 0.0 0.0 3 0.0 0.1 4 0.0 5 H DS L M H L 0.7 0.0 0.0 0.0 0.7 0.0 0.0 0.8 0.0 0.0 0.5 0.0 0.0 0.9 0.0 1.0 0.0 0.0 0.0 0.7 0.0 0.7 0.0 0.0 6 0.4 0.2 0.0 7 0.0 0.8 8 0.0 9 10 Count M ST H L 0.0 0.8 0.0 0.1 0.5 0.0 0.9 0.0 0.7 0.0 0.0 0.4 0.2 0.0 0.7 0.0 0.5 0.1 0.0 0.0 0.9 0.0 0.5 0.1 0.0 0.1 0.5 0.7 0.0 0.0 0.0 0.6 0.0 0.0 0.8 0.0 0.9 0.0 0.0 0.5 0.1 0.2 0.4 0.0 0.0 0.0 0.7 0.8 0.0 0.0 0.0 0.0 0.8 0.0 0.0 0.0 0.8 0.0 0.8 0.0 0.0 0.8 0.0 0.0 0.4 0.2 0.0 0.7 0.0 0.6 0.0 0.5 0.1 0.0 0.0 0.1 0.5 0.0 0.0 0.8 0.0 0.8 0.0 0.4 0.2 0.0 0.0 0.0 0.7 0.0 0.0 0.7 0.0 0.5 0.1 0.0 0.2 0.4 0.1 0.5 0.0 0.4 0.2 0.0 0.0 0.0 0.6 0.0 0.0 0.9 1.2 3.8 2.3 0.7 1.7 2.4 0.8 2.3 4.0 3.8 0.9 1.6 4.6 0.8 M MIS 3.9 M H STEP 2: Calculate the scalar cardinality of each attribute region in the transactions as the count value. Take the region OOP.Low as an example. Its scalar cardinality = (0.0 + 0.8 + 0.0 + 0.0 + … + 0.0) = 1.2. This step is repeated for the other regions, and the results are shown in the bottom line of Table 2. STEP 3: Find the region with the highest count among the three possible regions for each attribute. Take the course OOP as an example. The count is 1.2 for Low, 3.8 for Middle, and 2.3 for High. Since the count for Middle is the highest among the three counts, the region Middle is thus used to represent the course OOP in later mining processing. This step is repeated for the other regions, and “High” is chosen 17 for DB, ST and MIS, and “Middle” is chosen for OOP and DS. The number of items chosen are thus the same as that of the original attributes, meaning the algorithm is focused on the important items, and the time complexity could be reduced. STEP 4: Check whether the count of any region (item) kept in STEP 3 is larger than or equal to the predefined minimum support value α . Assume in this example, α is set at 2.5. Since the count values of OOP.Middle, DB.High, ST.High, DS.Middle and MIS.High are all larger than 2.5, these items are put in L1 (Table 3). Table 3: The set of large 1-itemsets L1 for this example Itemset count OOP.Middle DB.High ST.High DS.Middle MIS.High 3.8 3.8 4.6 3.9 4.0 STEP 5: Set r=1. STEP 6: Generate the candidate set Cr+1 from Lr. C2 is first generated from L1 as follows: (OOP.Middle, DB.High), (OOP.Middle, ST.High), (OOP.Middle, DS.Middle), (OOP.Middle, MIS.High), (DB.High, ST.High), (DB.High, DS.Middle), (DB.High, MIS.High), (ST.High, DS.Middle), (ST.High, MIS.High), and (DS.Middle, MIS.High). 18 STEP 7: Do the following substeps for each newly formed candidate itemset. (a) Calculate the fuzzy membership value of each transaction datum. Here, the minimum operator is used for the intersection. Take (OOP.Middle, DB.High) as an example. The derived membership value for Case 1 is calculated as: min(0.0, 0.0)=0.0. The results for the other cases are shown in Table 4. Table 4: The membership values for OOP.Middle Λ DB.High Case 1 2 3 4 5 6 7 8 9 10 OOP.Middle 0.0 0.0 0.1 1.0 0.7 0.2 0.8 0.0 0.8 0.2 DB.High 0.0 0.8 0.9 0.7 0.6 0.0 0.8 0.0 0.0 0.0 OOP.Middle ∩ DB.High 0.0 0.0 0.1 0.7 0.6 0.0 0.8 0.0 0.0 0.0 The results for the other 2-itemsets can be derived in similar fashion. (b) Calculate the scalar cardinality (count) of each candidate 2-itemset in the transaction data. Results for this example are shown in Table 5. 19 Table 5: The fuzzy counts of the itemsets in C2 Itemset (OOP.Middle, DB.High) (OOP.Middle, ST.High) (OOP.Middle, DS.Middle) (OOP.Middle, MIS.High) (DB.High, ST.High) (DB.High,DS.Middle) (DB.High, MIS.High) (ST.High, DS.Middle) (ST.High, MIS.High) (DS.Middle, MIS.High) count 2.2 1.8 1.7 0.9 2.2 2.7 1.4 2.8 1.8 1.1 (c) Check whether these counts are larger than or equal to the predefined minimum support value 2.5. Two itemsets, (DB.High, DS.Middle) and (ST.High, DS.Middle), are thus kept in L2 (Table 6). Table 6: The itemsets and their fuzzy counts in L2 Itemset count (DB.High, and DS.Middle) (ST.High, and DS.Middle) 2.7 2.8 STEP 8: IF Lr+1 is null, then do the next step; otherwise, set r=r+1 and repeat STEPs 6 to 8. Since L2 is not null in the example above, r=r+1=2. STEPs 6 to 8 are then repeated to find L3. C3 is first generated from L2, and only the itemset (DB.High, ST.High, DS.Middle) is formed. Its count is calculated as 1.9, smaller than 2.5. It is not put in L3, and L3 is thus an empty set. STEP 9 then begins. 20 STEP 9: Construct the association rules for each large itemset using the following substeps. (a) Form all possible association rules. The following four association rules are possible: If DB = High, then DS = Middle; If DS = Middle, then DB = High; If ST = High, then DS = Middle; If DS = Middle, then ST = High. (b) Calculate the confidence factors for the above association rules. Assume the given confidence threshold λ is 0.7. Take the first association rule as an example. The fuzzy count of DB.High ∩ DS.Middle is calculated as shown in Table 7. 21 Table 7: The fuzzy counts for DB.High ∩ DS.Middle Case 1 2 3 4 5 6 7 8 9 10 Count DB.High 0.0 0.8 0.9 0.7 0.6 0.0 0.8 0.0 0.0 0.0 3.8 DS.Middle DB.High ∩ DS.Middle 0.8 0.0 0.7 0.7 0.5 0.5 0.1 0.1 0.9 0.6 0.0 0.0 0.8 0.8 0.1 0.0 0.0 0.0 0.0 0.0 3.9 2.7 The confidence factor for the association rule "If DB = High, then DS = Middle" is then: ∑ ( DB . High ∩ DS . Middle ) 10 i =1 ∑ ( DB . High ) 10 i =1 = 2.7 =0.71. 3.8 Results for the other three rules are shown below. "If DS = Middle, then DB = High" has a confidence factor of 0.69; "If ST = High, then DS = Middle" has a confidence factor of 0.61; "If DS = Middle, then ST = High" has a confidence factor of 0.72. STEP 10: Check whether the confidence factors of the above association rules 22 are larger than or equal to the predefined confidence threshold λ . Since the confidence λ was set at 0.7 in this example, the following two rules are thus output to users: 1. If the Database score is high, then the Data Structure score is middle, with a confidence factor of 0.71; 2. If the Data Structure score is middle, then the Statistics score is high, with a confidence factor of 0.72. The two rules above are thus output as meta-knowledge concerning the given transactions. 6. Experimental Results Student score data from the Department of Information Management at I-Shou University, Taiwan, were used to show the feasibility of the proposed mining algorithm. A total of 260 transactions were included in the data set. Each transaction consisted of scores that a student had gotten. Execution of the mining algorithm was performed on a Pentium-PC. 23 In the experiments, we set α =70 and λ =0.6. A total of 31 rules were mined out. Three rule mined out are shown below as an example. 1. If the Management Information Systems score is middle, then the Business Data Communication score is middle, with a confidence factor of 0.73. 2. If the Operation Systems score is middle, then the System Analysis and Design score is High, with a confidence factor of 0.70. 3. If the Operations Research score is low, then the managerial mathematics score is low, with a confidence factor of 0.61. Experiments were also made to show the relationships between numbers of large itemsets and minimum support values. Results are shown in Figure 2, where um r o r It ms ts λ =0.6. ot l . . . . inimum upport Figure 2. The relationship between numbers of large itemsets and minimum support values. 24 From Figure 2, it is easily seen that the numbers of large itemsets decreased along with an increase in minimum support values. This is quite consistent with our intuition. The curve of the numbers of large 1-itemsets was also smoother than that of the numbers of large 2-itemsets, meaning that the minimum support value had a larger influence on itemsets with more items. Experiments were then made to show the relationships between numbers of association rules and minimum support values along with different minimum um r o Asso i tion ul s confidence values. Results are shown in Figure 3. . . . . on . . on . . on . . inimum upport Figure 3. The relationship between numbers of association rules and minimum support values. From Figure 3, it is easily seen that the numbers of association rules decreased along with the increase in minimum support values. This is also quite consistent with our intuition. Also, the curve of numbers of association rules with larger minimum 25 confidence values was smoother than that of those with smaller minimum confidence values, meaning that the minimum support value had a large effect on the number of association rules derived from small minimum confidence values. The relationship between numbers of association rules and minimum um r o Asso i tion ul s confidence values along with various minimum support values is shown in Figure 4. support support support support . . . . . Con i n Figure 4. The relationship between numbers of association rules and minimum confidence values. From Figure 4, it is easily seen that the numbers of association rules decreased along with an increase in minimum confidence values. This is also quite consistent with our intuition. The curve of numbers of association rules with larger minimum support values was smoother than that for smaller minimum support values, meaning that the minimum confidence value had a larger effect on the number of association rules when smaller minimum support values were used. All of the various curves 26 however converged to 0 as the minimum confidence value approached 1. Experiments were made to measure the accuracy of the proposed mining A ur y algorithm. Results for different amounts of transaction data are shown in Figure 5. sup. on . . sup. on . . sup. on . . um r o tr ns tions Figure 5: The relationship between accuracy and numbers of transactions. From Figure 5, it is easily seen that the accuracy increased along with an increase in the number of transactions, meaning that a larger set of sampling data yielded a better mining result. A large minimum support value also yielded a higher accuracy than a small minimum support value. In this experiment, the association rules mined can actually be used to help the faculty in the Department of Information Management evaluate course programs and 27 understand the students' learning interests and capacities in the courses. 7. Conclusion and future work In this paper, we have proposed a generalized data-mining algorithm, called FTDA, which can process transaction data with quantitative values and discover interesting patterns among them. The rules thus mined exhibit quantitative regularity for large databases and can be used to provide some suggestions to appropriate supervisors. The proposed algorithm can also solve conventional transaction-data problems by using degraded membership functions. Experimental results with the students’ scores in the Department of Information Management at I-Shou University, Taiwan, show the feasibility of the proposed mining algorithm. When compared with the traditional crisp-set mining methods for quantitative data, our approach can get smooth mining results due to the fuzzy membership characteristics. Also, when compared with the fuzzy mining methods, which take all the fuzzy regions into consideration, our method can get a good time complexity. Trade-off exists between the rule completeness and the time complexity. Although the proposed method works well in data mining for quantitative values, 28 it is just a beginning. There is still much work to be done in this field. Our method assumes that the membership functions are known in advance. In [17-19], we also proposed some fuzzy learning methods to automatically derive the membership functions. In the future, we will attempt to dynamically adjust the membership functions in the proposed mining algorithm to avoid the bottleneck of membership function acquisition. We will also attempt to design specific data-mining models for various problem domains. 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