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FIGURE 7. The derived membership functions. ‘STEP 12: These centroids generated from previous step are then used as the center of each interval in the membership function. The centroid forms the center of each fuzzy inter- val. At last, the generated membership functions are shown in Fig. 7.

Figure 7 The derived membership functions. ‘STEP 12: These centroids generated from previous step are then used as the center of each interval in the membership function. The centroid forms the center of each fuzzy inter- val. At last, the generated membership functions are shown in Fig. 7.

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TABLE 1. Comparison of the existing membership function turning approaches.
TABLE 1. Comparison of the existing membership function turning approaches.
TABLE 3. Notation for the first phase. include an additional half of its width on either side, creating an overlap. In the same way, the values are extended to include an additional quarter of its width on either side to create an overlap in the GNAAROQ.
TABLE 3. Notation for the first phase. include an additional half of its width on either side, creating an overlap. In the same way, the values are extended to include an additional quarter of its width on either side to create an overlap in the GNAAROQ.
STEP 3: The information density similarity (DS) of Ce and a is calculated using following substeps. ~ SUBSTEP 2.2: The width of each interval is then extended to include a quarter of it’s neighbouring intervals. The inter- vals become C} = {20, 22, 23, 25, 27, 30, 32,39}, C? = {39, 45, 53, 61, 64}, cS = {61, 64, 67, 71, 76, 83, 83} and cy = {76, 83, 83, 87, 88, 88, 90, 100} respectively. The width of each interval becomes 30, which is represented as i i B B 30. The number of instances in each partition becomes: hi 8, no 5, hy 7, ny 8. After extending, the intervals are shown in Fig. 2.
STEP 3: The information density similarity (DS) of Ce and a is calculated using following substeps. ~ SUBSTEP 2.2: The width of each interval is then extended to include a quarter of it’s neighbouring intervals. The inter- vals become C} = {20, 22, 23, 25, 27, 30, 32,39}, C? = {39, 45, 53, 61, 64}, cS = {61, 64, 67, 71, 76, 83, 83} and cy = {76, 83, 83, 87, 88, 88, 90, 100} respectively. The width of each interval becomes 30, which is represented as i i B B 30. The number of instances in each partition becomes: hi 8, no 5, hy 7, ny 8. After extending, the intervals are shown in Fig. 2.
TABLE 6. The quantitative transactions. TABLE 7. The quantitative values for item A.
TABLE 6. The quantitative transactions. TABLE 7. The quantitative values for item A.
Step 8: STEPs 3 to 7 are repeated until only one interval remains. The final set of intervals for each level is shown in Figs. 3-6.
Step 8: STEPs 3 to 7 are repeated until only one interval remains. The final set of intervals for each level is shown in Figs. 3-6.
TABLE 8. The threshold values of all intervals.
TABLE 8. The threshold values of all intervals.
TABLE 9. Formula 0 « ek + (1-0) # nk used to decide on the most suitable level.
TABLE 9. Formula 0 « ek + (1-0) # nk used to decide on the most suitable level.
FIGURE 8. Distribution of quantitative values. Some of the customers of this online retail store are whole- salers who purchase items in bulk; thus, the quantitative values in this dataset are much higher: the quantitative values fall in the range of | to 4,800 (excluding 3 outliers). However, the majority of the customers is not wholesalers and is much more likely to buy items in smaller quantities. Therefore, if the number of times that a purchase is made for each quantitative value is counted, the dataset is highly unbalanced in favor of the lower end of the quantitative range. In total, there are 73,314 instances where a customer purchases a single item (a quantitative value of 1), whereas there are only 4 instances when a customer purchases bulk items (with a quantitative value of 4,800). This point is illustrated in Fig. 8.
FIGURE 8. Distribution of quantitative values. Some of the customers of this online retail store are whole- salers who purchase items in bulk; thus, the quantitative values in this dataset are much higher: the quantitative values fall in the range of | to 4,800 (excluding 3 outliers). However, the majority of the customers is not wholesalers and is much more likely to buy items in smaller quantities. Therefore, if the number of times that a purchase is made for each quantitative value is counted, the dataset is highly unbalanced in favor of the lower end of the quantitative range. In total, there are 73,314 instances where a customer purchases a single item (a quantitative value of 1), whereas there are only 4 instances when a customer purchases bulk items (with a quantitative value of 4,800). This point is illustrated in Fig. 8.
FIGURE 10. Membership functions used in the online retail dataset.
FIGURE 10. Membership functions used in the online retail dataset.
FIGURE 9. Membership functions used in the synthetic dataset. of the items. The previous approach of using predefined membership functions is chosen for comparison [4]. The membership functions used in each of the two datasets are shown in Figs. 9 and 10, respectively. The previous approach is referred to as the PMF (predefined membership function) approach in the following sections.
FIGURE 9. Membership functions used in the synthetic dataset. of the items. The previous approach of using predefined membership functions is chosen for comparison [4]. The membership functions used in each of the two datasets are shown in Figs. 9 and 10, respectively. The previous approach is referred to as the PMF (predefined membership function) approach in the following sections.
FIGURE 12. Relationship between number of rules and minimum support on the online retail dataset. FIGURE 11. Relationship between number of rules and minimum support on the synthetic dataset.
FIGURE 12. Relationship between number of rules and minimum support on the online retail dataset. FIGURE 11. Relationship between number of rules and minimum support on the synthetic dataset.
FIGURE 13. Relationship between number of rules and minimum confidence on the synthetic dataset.
FIGURE 13. Relationship between number of rules and minimum confidence on the synthetic dataset.
Fig. 14 shows that—for the same reasons outlined in the previous section—the PME approach outperforms the three proposed approaches in terms of the number of rules gener- ated. Additionally, for the same reasons described previously, the generated rules refer to notably large fuzzy regions, which essentially make them useless. Below, we selected two rules discovered by PMF and the proposed approach and show them in Fig. 15 and Fig. 16 to explain the differences.
Fig. 14 shows that—for the same reasons outlined in the previous section—the PME approach outperforms the three proposed approaches in terms of the number of rules gener- ated. Additionally, for the same reasons described previously, the generated rules refer to notably large fuzzy regions, which essentially make them useless. Below, we selected two rules discovered by PMF and the proposed approach and show them in Fig. 15 and Fig. 16 to explain the differences.
FIGURE 17. Number of rules extracted by the proposed approaches with different theta values on the synthetic dataset. At last, because the weighting parameter 6 that is utilized to identify the optimum level for generating membership functions will influence the final result, the experiments were conducted to show the number of rules mined by the proposed approaches with different 9 values. The results on the two datasets are shown in Fig. 17 and Fig. 18.
FIGURE 17. Number of rules extracted by the proposed approaches with different theta values on the synthetic dataset. At last, because the weighting parameter 6 that is utilized to identify the optimum level for generating membership functions will influence the final result, the experiments were conducted to show the number of rules mined by the proposed approaches with different 9 values. The results on the two datasets are shown in Fig. 17 and Fig. 18.
FIGURE 16. A fuzzy temporal association rule extracted by the proposed approach.
FIGURE 16. A fuzzy temporal association rule extracted by the proposed approach.
TZUNG-PEI HONG (Senior Member, IEEE) received the B.S. degree in chemical engineering from National Taiwan University, in 1985, and the Ph.D. degree in computer science and information engineering from National Chiao Tung University, in 1992. ae. SREAGR | SESSA nei Sere ae from National Taiwan University, in 1985, and the Ph.D. degree in computer science and information engineering from National Chiao Tung University, in 1992. He served at the Department of Computer Science, Chung Hua Polytechnic Institute, from 1992 to 1994, and at the Department of Information Management, I-Shou University, from 1994 to 2001. He was in charge of the whole computerization and library planning in preparation at the National University of Kaohsiung, from 1997 to 2000, where he also served as the First Director of the Library and Computer Center, National University of Kaohsiung, from 2000 to 2001, as the Dean of Academic Affairs, from 2003 to 2006, as the Administrative Vice President, from 2007 to 2008, and as the Academic Vice President, in 2010. He is currently a Distinguished Professor at the Department of Computer Science and Information Engineering and at the Department of Electrical Engineering, National University of Kaohsiung, and a Joint Professor at the Department of Computer Science and Engineering, National Sun Yat-sen University, Taiwan. He has published more than 500 research articles in international/national journals and conferences and has planned more than 50 information systems. He is also the board member of more than 40 journals and a program committee member of more than 500 conferences. His current research interests include knowledge engineering, data mining, soft computing, management information systems, and www applications. He got the first National Flexible Wage Award from the Ministry of Education in Taiwan.
TZUNG-PEI HONG (Senior Member, IEEE) received the B.S. degree in chemical engineering from National Taiwan University, in 1985, and the Ph.D. degree in computer science and information engineering from National Chiao Tung University, in 1992. ae. SREAGR | SESSA nei Sere ae from National Taiwan University, in 1985, and the Ph.D. degree in computer science and information engineering from National Chiao Tung University, in 1992. He served at the Department of Computer Science, Chung Hua Polytechnic Institute, from 1992 to 1994, and at the Department of Information Management, I-Shou University, from 1994 to 2001. He was in charge of the whole computerization and library planning in preparation at the National University of Kaohsiung, from 1997 to 2000, where he also served as the First Director of the Library and Computer Center, National University of Kaohsiung, from 2000 to 2001, as the Dean of Academic Affairs, from 2003 to 2006, as the Administrative Vice President, from 2007 to 2008, and as the Academic Vice President, in 2010. He is currently a Distinguished Professor at the Department of Computer Science and Information Engineering and at the Department of Electrical Engineering, National University of Kaohsiung, and a Joint Professor at the Department of Computer Science and Engineering, National Sun Yat-sen University, Taiwan. He has published more than 500 research articles in international/national journals and conferences and has planned more than 50 information systems. He is also the board member of more than 40 journals and a program committee member of more than 500 conferences. His current research interests include knowledge engineering, data mining, soft computing, management information systems, and www applications. He got the first National Flexible Wage Award from the Ministry of Education in Taiwan.
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Computer ScienceAssociation Rule LearningSimilarity Geometry
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