## 3 Results and Discussion
### Performance Comparison  
All of the experiments in this paper were executed in MATLAB R2014a on a personal computer running a Windows Operating System with an Intel Core i5 processor, 4 GB of RAM, and a clock speed of 3.40 GHz. For evaluation purposes, the performance results of mining NAARs are for the following objectives: (1) rate of anomalies; (2) number of final rules; (3) processing time. We compared our proposed method in several scenarios (i.e., reduce only circular rules (CR), reduce only subsumption rules (SR), and reduce both anomalous rules (AR)) with the traditional method, the results of which are shown in Figures 3 and 4. Figure 3: The performance comparison of level of anomalies (**left**) and number of final rules (**right**). Figure 3 depicts a comparison of level of anomalies and the total number of final rules generated, with different support threshold values. It can be observed that the number of final rules from the traditional method increased significantly as the support threshold decreased. The proposed method could generate a four to five times lower number of rules compared with the traditional method. This condition occurred because of the nature of traditional method which consider all possible combination of items to generate rules. Moreover, considering all the possible combination of items might lead to an increased number of redundancies or anomalies contained in those rule sets. Therefore, as shown in Figure 3, it also produces the highest level of anomalies, which reaches around twice the anomalous rate of those rules produced by the proposed method. On the other hand, in our proposed method, we verify each of the extracted rules with respect to a set of rules which contains all possible subsets or superset of the corresponding rule to be verified in order to find anomalous rules. Hence, the proposed method produces fewer rules. In addition, the level of anomalies reduces with the increase of support values because at a higher support threshold, the total number of generated rules also decreases and leads to fewer potential anomalous rules being found. In addition, the difference of total number of rules after reducing only circularity only between the traditional and proposed-CR is not huge, yet may still contain subsumption rules. Meanwhile, proposed-SR shows that there are more rules being reduced due to subsumption rules which created a significant difference number of final rules generated with the traditional method. However, similarly to the proposed-CR, it might still contain another anomaly (i.e., circular rules). Therefore, it is better to reduce all possible anomalies, such as the proposed-AR. In comparison with our proposed approach (i.e., proposed-AR), one can see that the total number of rules generated decreases drastically compared with the traditional approach since it removes both anomalies: subsumption and circular rules, in which subsumption rules contribute more in level of anomalous than circular rules. As mentioned above, since the traditional method considers all possible combinations of items to generate rules, it is obvious that in terms of processing time, the traditional method will take a significantly longer time than the proposed approach. In contrary, our proposed method generates rules based on a depth-first search method and a closed framework, thus, not all possible combinations of item sets will be traversed and evaluated. Therefore, the proposed method could trim searching time and generate rules more efficiently. According to Figure 4, it seems that proposed-SR computation time is the lowest. However, note that our proposed-AR considers both anomalies to be found, while the proposed-SR only reduces subsumption reduction regardless the existence of possible circular rules. Thus, the former took a slightly longer time than proposed-SR. Toward the end, we can conclude that  
Figure 4: The performance comparison of processing time.