Data Mining with Linguistic Thresholds

Transcription

1 Int. J. Contemp. Math. Science, Vol. 7, 22, no. 35, Data Mining with Linguitic Threhold Tzung-Pei Hong Department of Electrical Engineering National Univerity of Kaohiung Kaohiung, Taiwan, R.O.C. Ming-Jer Chiang Intitute of Information Engineering I-Shou Univerity Kaohiung, Taiwan, R.O.C. Shyue-Liang Wang Department of Computer Science New York Intitute of Technology New York, USA Abtract Data mining i the proce of extracting deirable knowledge or intereting pattern from exiting databae for pecific purpoe. In the pat, the minimum upport and minimum confidence were et at numerical value. Linguitic minimum upport and minimum confidence value are, however, more natural and undertandable for human being. Thi paper thu attempt to propoe a new mining approach for extracting intereting weighted aociation rule from tranaction, when the parameter needed in the mining proce are given in linguitic term. Item are alo evaluated by manager a linguitic term to reflect their importance, which are then tranformed a fuzzy et of weight. Fuzzy operation including fuzzy ranking are then ued to find weighted large itemet and aociation rule.

2 72 Tzung-Pei Hong, Ming-Jer Chiang and Shyue-Liang Wang Mathematic Subject Claification: 62-7 Keyword: aociation rule, data mining, weighted item, fuzzy et, fuzzy ranking Introduction Knowledge dicovery in databae (KDD) ha become a proce of coniderable interet in recent year a the amount of data in many databae h grown tremendouly large. The KDD proce generally conit of three phae: pre-proceing, data mining and pot-proceing [, 4, 5, 26]. Among them, data mining play a critical role to KDD. Recently, the fuzzy et theory ha been ued more and more frequently in intelligent ytem becaue of it implicity and imilarity to human reaoning [35, 36]. The theory ha been applied in field uch a manufacturing, engineering, diagnoi, economic, among other [7, 23, 25, 35]. Several fuzzy learning algorithm for inducing rule from given et of data h been deigned and ued to good effect with pecific domain [5, 7, 3, 6, 8-2, 29-3]. Strategie baed on deciion tree were propoed in [9, -2, 27-29, 32-33], and baed on verion pace were propoed in [3]. Fuzzy mining approache were propoed in [8, 22, 24, 34]. In the pat, the minimum upport and minimum confidence value were numerical. In thi paper, linguitic minimum upport and minimum confidence value are given, which are more natural and undertandable for human being. Alo, item may h different importance, which i evaluated by manager or expert a linguitic term. A novel mining algorithm i then propoed to find weighted linguitic aociation rule from tranaction data. It firt tranform linguitic weighted item, minimum upport and minimum confidence into fuzzy et, then filter weighted large itemet out by fuzzy operation. Weighted aociation rule with linguitic upport and confidence are then derived from the weighted large itemet. The remaining part of thi paper are organized a follow. Several approache for mining aociation rule are reviewed in Section 2. Fuzzy et and operation are introduced in Section 3. The notation ued in thi paper i defined in Section 4. The propoed weighted data-mining algorithm for linguitic minimum upport and confidence are decribed in Section 5. An example i given to illutrate the propoed mining algorithm in Section 6. Concluion and propoal of future work are given in Section 7. 2 Review of Mining Aociation Rule Agrawal and hi co-worker propoed everal mining algorithm baed on the concept of large itemet to find aociation rule in tranaction data [-4]. They divided the mining proce into two phae. In the firt phae, candidate itemet were generated and counted by canning the tranaction data. If the number of an itemet appearing in the tranaction wa larger than a pre-defined threhold value

3 Data mining with linguitic threhold 73 (called minimum upport), the itemet wa conidered a large itemet. Itemet containing only one item were proceed firt. Large itemet containing only ingle item were then combined to form candidate itemet containing two item. Thi proce wa repeated until all large itemet had been found. In the econd phae, aociation rule were induced from the large itemet found in the firt phae. All poible aociation combination for each large itemet were formed, and thoe with calculated confidence value larger than a predefined threhold (called minimum confidence) were output a aociation rule. Cai et al. then propoed weighted mining to reflect different importance to different item [6]. Each item wa attached a numerical weight given by uer. Weighted upport and weighted confidence were then defined to determine intereting aociation rule. Yue et al. then extended their concept to fuzzy item vector [34]. In thi paper, the fuzzy concept are ued to repreent item importance, minimum upport and minimum confidence. Thee parameter are expreed in linguitic term, which are more natural and undertandable for human being. 3 Review of Related Fuzzy Concept Fuzzy et theory wa firt propoed by Zadeh and Goguen in 965 [36]. A function called the memberhip function, μ A (), x i defined for mapping a member x to a memebrhip degree between to. Triangular memberhip function are commonly ued and can be denoted by A = (a, b, c), where a b c (Figure ). The abcia b repreent the variable value with the maximal grade of memberhip value, i.e. μ A (b)=; a and c are the lower and upper bound of the available area. They are ued to reflect the fuzzine of the data. u A (x) a b c X Figure : A triangular memberhip function There are a variety of fuzzy et operation. Among them, three baic and commonly ued operation are complement, union and interection. Fuzzy ranking i ued to determine the order of fuzzy et and i thu quite important to actual application. Several fuzzy ranking method h been propoed in the literature. The ranking method uing gravitie i ued in the paper.

4 74 Tzung-Pei Hong, Ming-Jer Chiang and Shyue-Liang Wang 4 Notation The notation ued in thi paper i defined a follow. n: the total number of tranaction data; m: the total number of item; k: the total number of manager; A i : the i-th item, i m; W i,j : the tranformed fuzzy weight for importance of item A i, evaluated by the j-th manager; W i : the fuzzy rage weight for importance of item A i ; W : the fuzzy weight for importance of itemet ; connt i : the number of item A i appearing in the n tranaction; α : the predefined linguitic minimum upport value; β : the predefined linguitic minimum confidence value; I j : the j-th memberhip function of importance; I : the fuzzy rage weight of all poible linguitic term of importance; wup i : the fuzzy weighted upport of item A i; wconf R : the fuzzy weighted confidence of rule R ; minup: the tranformed fuzzy et from the linguitic minimum upport value α ; wminup: the fuzzy weighted et of minimum upport; minconf: the tranformed fuzzy et from the linguitic minimum confidence value β ; wminconf: the fuzzy weighted et of minimum confidence; C r : the et of candidate itemet with r item; L : the et of large itemet with r item. r 5 Weighted Data Mining for Linguitic Minimum Support and Minimum Confidence In thi ection, the fuzzy concept are ued to repreent item importance, minimum upport and minimum confidence. The propoed mining algorithm firt ue the et of memberhip function for importance to tranform manager linguitic evaluation of item importance into fuzzy weight. The fuzzy weight of an item from different manger are then raged. The algorithm then calculate the weighted count of all item from the tranaction data and according to the rage fuzzy weight of item. The given linguitic minimum upport value i alo tranformed into a fuzzy weighted et. All weighted large -itemet can thu be found by ranking the fuzzy weighted upport of each item with fuzzy weighted minimum upport. After that, candidate 2-itemet are formed from the weighted large -itemet and the ame procedure i ued to find all weighted large 2-itemet. Thi procedure i repeated until all weighted large itemet h been found. The fuzzy weighted confidence from large itemet are then calculated to find intereting aociation rule. Detail of the propoed mining algorithm are decribed below.

5 Data mining with linguitic threhold 75 The algorithm: INPUT: A et of n tranaction data, a et of m item with their importance evaluated by k manager, three et of memberhip function repectively for importance, minimum upport and minimum confidence, a pre-defined linguitic minimum upport value α, and a pre-defined linguitic minimum confidence value β. OUTPUT: A et of weighted aociation rule. STEP : Tranform each linguitic term of importance for item A i, i m, which i evaluated by the j-th manager into a fuzzy et W i,j of weight, uing the given memberhip function of item importance. STEP 2: Calculate the fuzzy rage weight addition a: W of each item A i by fuzzy i k W i = k j= W i, j. STEP 3: Calculate the number (count i ) of each item A i appearing in the et of tranaction. STEP 4: Calculate the fuzzy weighted upport wup i of each item A i a: counti Wi wup i =, n where n i the number of tranaction. STEP 5: Tranform the given linguitic minimum upport value α into a fuzzy et minup of minimum upport, uing the given memberhip function of minimum upport. STEP 6: Calculate the fuzzy weighted et (wminup) of the given minimum upport value a: wminup = minup (the gravity of I ), where I = m I j j= with I j being the j-th memberhip function of importance. I repreent the fuzzy rage weight of all poible linguitic term of importance. STEP 7: Check whether the weighted upport (wup i ) of each item A i i larger than or equal to the fuzzy weighted minimum upport (wminup) by fuzzy ranking. Any fuzzy ranking approach can be applied here a long a it can generate a crip rank. If wup i i equal to or greater than wminup, put A i in the et of large -itemet L. STEP 8: Set r =, where r i ued to repreent the number of item kept in the current large itemet. STEP 9: Generate the candidate et C r+ from L r in a way imilar to that in the apriori algorithm [4]. That i, the algorithm firt join L r and L r auming that r- item in the two itemet are the ame and the other one i different. It then keep in C r+ the itemet, which h all their m,

6 76 Tzung-Pei Hong, Ming-Jer Chiang and Shyue-Liang Wang ub-itemet of r item exiting in L r. STEP : Do the following ubtep for each newly formed (r+)-itemet with item (, 2,, r+ ) in C r+ : (a) Calculate the fuzzy weight W W of itemet a: = W Λ W ΛK ΛW, 2 r+ where W i i the fuzzy rage weight of item i, calculated in STEP 2. If the minimum operation i ued for the interection, then: W r+ = MinW i= (b) Calculate the number ( count ) of itemet appearing in the tranaction data. (c) Calculate the weighted upport wup of itemet a: W count wup =, n where n i the number of tranaction. (d) Check whether the weighted upport wup of itemet i greater than or equal to the fuzzy weighted minimum upport wminup by fuzzy ranking. If wup i greater than or equal to wminup, put in the et of large (r+)-itemet L r+. STEP : IF L r+ i null, then do the next tep; otherwie, et r = r+ and repeat Step 9-. STEP 2: Tranform the given linguitic minimum confidence value β into a fuzzy et minconf of minimum confidence, uing the given memberhip function of minimum confidence. Step 3: Calculate the fuzzy weighted et (wminconf) of the given minimum confidence value a: wminconf = minconf (the gravity of I ), where I i the ame a that calculated in Step 6. STEP 4: Contruct the aociation rule from each weighted large q-itemet with item (, 2,, q ), q 2, uing the following ubtep: (a) Form all poible aociation rule a follow: Λ... Λ j Λ j + Λ... Λ q j, j = to q. (b) Calculate the weighted confidence value wconf R of each poible aociation rule R a: i wconf R count = count j W. (c) Check whether the weighted confidence wconf R of aociation rule R i greater than or equal to the fuzzy weighted minimum confidence wminconf by fuzzy ranking. If wconf R i greater than or equal to wminconf, keep rule R in the intereting rule et. STEP 5: For each rule R with weighted upport wup R and weighted confidence

7 Data mining with linguitic threhold 77 wconf R in the intereting rule et, find the linguitic minimum upport region S i and the linguitic minimum confidence region C j with wminup i- wup R < wminup i and wminconf j- wconf R < wminconf j by fuzzy ranking, where: wminup i = minup i (the gravity of I ), wminconf j = minconf j (the gravity of I ), minup i i the given memberhip function for S i and minconf j i the given memberhip function for C j. Output rule R with linguitic upport value S i and linguitic confidence value C j. The rule output after tep 5 can erve a meta-knowledge concerning the given tranaction. 6 An Example In thi ection, an example i given to illutrate the propoed data-mining algorithm. Thi i a imple example to how how the propoed algorithm can be ued to generate weighted aociation rule from a et of tranaction. The data et include ten tranaction, a hown in Table. Table : The data et ued in thi example TID Item ABC 2 ACE 3 ABF 4 BCD 5 BCE 6 ABDE 7 ABCE 8 ABCEF 9 ABCEF BCDEF Each tranaction i compoed of a tranaction identifier and item purchaed. There are ix item, repectively being A, B, C, D, E and F, to be purchaed. The importance of the item i evaluated by three manager a hown in Table 2. Table 2: The item importance evaluated by three manager Item Manager Manger2 Manager3 A Important Ordinary Ordinary B Very Important Important Important C Ordinary Important Important D Unimportant Unimportant Very Important E Important Important Important F Unimportant Unimportant Ordinary

8 78 Tzung-Pei Hong, Ming-Jer Chiang and Shyue-Liang Wang Aume the memberhip function for item importance are given in Figure 2. In Figure 2, item importance i divided into five fuzzy region: Very Unimportant, Unimportant, Ordinary, Important and Very Important. Each fuzzy region i repreented by a memberhip function. For implicity, triangular memberhip function are ued here. The memberhip function in Figure 2 can be repreented a follow: Very Unimportant (VU): (,,.25), Unimportant (U): (,.25,.5), Ordinary (O): (.25,.5,.75), Important (I): (5, 75, ), and Very Important (VI): (.75,, ). Unimportant Important Very Unimportant Ordinary Memberhip value Very Important Weight Figure 2: The memberhip function of item importance ued in thi example For the training data given in Table, the propoed mining algorithm proceed a follow. Step : The linguitic term for item importance given in Table 2 are tranformed into fuzzy et by the memberhip function in Figure 2. For example, item A i evaluated to be important by Manager, and can then be tranformed a a triangular fuzzy et (.5,.75, ) of weight. The tranformed reult for Table 2 are hown in Table 3. Table 3: The fuzzy weight tranformed from the item importance in Table 2. Item Manager Manger2 Manager3 A (.5,.75,) (.25,.5,.75) (.25,.5,.75) B (.75,, ) (.5,.75,) (.5,.75,) C (.25,.5,.75) (.5,.75,) (.5,.75,) D (,.25,.5) (,.25,.5) (.75,, ) E (.5,.75,) (.5,.75,) (.5,.75,) F (,.25,.5) (,.25,.5) (.25,.5,.75) Step 2: The rage weight of each item i calculated by fuzzy addition. The reult are hown in Table 4.

9 Data mining with linguitic threhold 79 Table 4: The rage fuzzy weight of all the item Very Low Low Middle High Very High Memberhip value Item Average Fuzzy Weight A (.333,.583,.833) B (.583,.833, ) C (.47,.667,.97) D (,.67,.47) E (.5,.75, ) F (.83,.333,.583) Minimum upport Step 3: The appearing number (count) of each item i counted from the tranaction in Table. Reult for all the item are hown in Table 5. Table 5: The count of all the item Item Count A 7 B 9 C 8 D 3 E 7 F 4 Step 4: The weighted upport of each item i calculated. Reult for all the item are hown in Table 6. Table 6: The weighted upport of all the item Item Weighted Support A (.233,.48,.583) B (.525,.75,.9) C (.333,.533,.733) D (,.5,.25) E (.35,.525,.7) F (.33,.33,.233) Step 5: The given linguitic minimum upport value i tranformed into a fuzzy et of minimum upport. Aume the memberhip function for minimum upport are given in Figure 3.

10 72 Tzung-Pei Hong, Ming-Jer Chiang and Shyue-Liang Wang Very Low Low Middle High Very High Memberhip value Minimum upport Figure 3: The memberhip function of minimum upport Alo aume the given linguitic minimum upport value i High. It i then tranformed into a fuzzy et of minimum upport, (.5,.75, ), according to the given memberhip function in Figure 3. Step 6: The fuzzy rage weight of all poible linguitic term of importance in Figure 3 i firt calculated a: I = [(,,.25) + (,.25,.5) + (.25,.5,.75) + (.5,.75, ) +(.75,, )]/5 = (.3,.5,.7). The gravity of I i then ( )/3, which i.5. The fuzzy weighted et of minimum upport for High i then (.5,.75, ).5, which i (.25,.375,.5). Step 7: The weighted upport of each item i compared with the fuzzy weighted minimum upport by fuzzy ranking. Aume the gravity ranking approach i adopted in thi example. Take Item A a an example. The rage height of the weighted upport for item A i ( )/3, which i.48. The rage height of the fuzzy weighted minimum upport i ( )/3, which i.375. Since.48 >.375, A i thu a large weighted -itemet. Similarly, B, C and E are large weighted -itemet. Thee -itemet are put in L. Step 8: r i et at, where r i ued to tore the number of item kept in the current itemet. Step 9: The candidate et C r+ i generated from L r. C 2 i then firt generated from L a follow: (A, B), (A, C), (A, E), (B, C), (B, E) and (C, E). Step : The following ubtep are done for each newly formed candidate itemet in C 2. (a) The fuzzy weight of all the itemet in C 2 are calculated. Here, the minimum operator i ued for interection. Take Itemet (A, B) a an example. It fuzzy weighted upport value i calculated a min((.333,.583,.833), (.583,.833, )), which i (.333,.583,.833). The reult for all the 2-itemet are hown in Table 7.

11 Data mining with linguitic threhold 72 Table 7: The fuzzy weight of the 2-itemet in C 2. Itemet Fuzzy Weight (A, B) (.333,.583,.833) (A, C) (.333,.583,.833) (A, E) (.333,.583,.833) (B, C) (.47,.667,.97) (B, E) (.5,.75, ) (C, E) (.47,.667,.97) (b) The calar cardinality (count) of each candidate 2-itemet i counted from the tranaction data. Reult for thi example are hown in Table 8. Table 8: The count of the itemet in C 2. Itemet Count (A, B) 6 (A, C) 5 (A, E) 5 (B, C) 7 (B, E) 5 (C, E) 6 (c) The weighted upport of each candidate 2-itemet i calculated. Take (A, B) a an example. The minimum fuzzy weight of (A, B) i (.333,.583,.833) and the count i 6. It weighted upport i then (.333,.583,.833)*6/, which i (.2,.35,.5). All the weighted upport of the candidate 2-itemet are hown in Table 9. Table 9: The weighted upport of the 2-item Itemet Fuzzy Weight (A, B) (.2,.35,.5) (A, C) (.67,.292,.47) (A, E) (.67,.292,.47) (B, C) (.292,.467,.642) (B, E) (.25,.375,.5) (C, E) (.25,.4,.55) (d) The weighted upport of each candidate 2-itemet i compared with the fuzzy weighted minimum upport by fuzzy ranking. A mentioned above, the gravity ranking approach i adopted in thi example. (B, C), (B, E) and (C, E) are then found to be large weighted 2-itemet. Thee 2-itemet are put in L 2. Step : Since L 2 i not null in the example, r = r + = 2. Step 9 to are repeated to find L 3. C 3 i then generated from L 2. Only the 3-itemet (B, C, E) i formed in thi example. The rage height of it fuzzy weighted upport i.34, maller than.375. It i not put in L 3, and L 3 i thu an empty et.

12 722 Tzung-Pei Hong, Ming-Jer Chiang and Shyue-Liang Wang Step 2: The given linguitic minimum confidence value i tranformed into a fuzzy et of minimum confidence. Aume the memberhip function for minimum confidence value are hown in Figure 4, which are imilar to thoe in Figure 3. Memberhip value Very Low Low Middle High Very High Minimum confidence Figure 4: The memberhip function of minimum confidence Alo aume the given linguitic minimum confidence value i High. It i then tranformed into a fuzzy et of minimum confidence, (.5,.75, ), according to the given memberhip function in Figure 4. Step 3: The fuzzy rage weight of all poible linguitic term of importance i the ame a that found in Step 5. It gravity i thu.5. The fuzzy weighted et of minimum confidence for High i then (.5,.75, ).5, which i (.25,.375,.5). Step 4: The aociation rule from each large itemet are contructed by uing the following ubtep. (a) All poible aociation rule are formed a follow: If B, then C; If C, then B; If B, then E; If E, then B; If C, then E; If E, then C. (b) The weighted confidence factor for the above poible aociation rule are calculated. Take the firt poible aociation rule a an example. The count of B and B C are repectively.9 and.7, and the fuzzy weight of B C i (.47,.667,.97). The weighted confidence factor for the aociation rule If B, then C i then:.7.9 (.47,.667,.97) = (.324,.59,.73). (c) The weighted confidence of each aociation rule i compared with the fuzzy weighted minimum confidence by fuzzy ranking. A. In thi example, the following ix rule are put in the intereting rule et:. If B i bought then C i bought; 2. If C i bought then B i bought; 3. If B i bought then E i bought; 4. If E i bought then B i bought; 5. If C i bought then E i bought;

13 Data mining with linguitic threhold If E i bought then C i bought. STEP 5: The linguitic upport and confidence value are found for each rule R. Take the third intereting aociation rule "If B, then E a an example. It weighted upport i (.25,.375,.5) and weighted confidence i (.278,.47,.556). Since the memberhip function for linguitic minimum upport region High i (.5,.75, ) and for Very High i (.75,, ). The weighted fuzzy et for thee two region are (.25,.375,.5) and (.375,.5,.5). Since (.25,.375,.5) (.25,.375,.5) < (.375,.5,.5) by fuzzy ranking, the linguitic upport value for Rule R i then High. Similarly, the linguitic confidence value for Rule R i High. All the ix intereting aociation rule are then output a:. If B i bought then C i bought, with a very high upport and a very high confidence; 2. If C i bought then B i bought, with a very high upport and a very high confidence; 3. If B i bought then E i bought, with a high upport and a high confidence; 4. If E i bought then B i bought, with a high upport and a very high confidence; 5. If C i bought then E i bought, with a very high upport and a very high confidence; 6. If E i bought then C i bought, with a very high upport and a very high confidence. The ix rule above are thu output a meta-knowledge concerning the given weighted tranaction. 7 Concluion and Future Work In thi paper, we h propoed a new weighted data-mining algorithm to find intereting aociation rule with linguitic upport and confidence. Item are evaluated by manager a linguitic term, which are then tranformed and raged a fuzzy et of weight. Fuzzy operation including fuzzy ranking are ued to find weighted large itemet and aociation rule. Compared to previou mining approache, the propoed one ha linguitic input and output, which are more natural and undertandable for human being. Acknowledgement. Thi reearch wa upported by the National Science Council of the Republic of China under contract NSC E Reference [] R. Agrawal, T. Imielinki and A. Swami, Mining aociation rule between et of item in large databae, The 993 ACM SIGMOD Conference, Wahington DC, USA, 993. [2] R. Agrawal, T. Imielinki and A. Swami, Databae mining: a performance perpective, IEEE Tranaction on Knowledge and Data Engineering, Vol. 5, No. 6, 993,

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