# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets
## 4 Methods
### Extending Traditional Algorithms to Discover Prevalent FCPs
#### 4.2.2 An Extended Joinless Algorithm for Mining Prevalent FCPs
With higher prevalence thresholds, the Join-based (or Joinless) FCP mining algorithm can prune a lot of candidates, so when the prevalence threshold increases, the time cost will be reduced rapidly. For our method based on interval type-2 fuzzy sets and cliques, the prevalence threshold only participates in filtering the absolutely prevalent FCPs and FCPs with prevalent tendency degrees, so the execution time will decrease slowly with the increase of the prevalence threshold. When the prevalence threshold is higher, the time cost of the Join-based (or Joinless) FCP mining algorithm is less than our method. This is because the number of prevalent FCPs with higher \(\overline{PI}\) and \(\underline{PI}\) is far less than that with lower \(\overline{PI}\) and \(\underline{PI}\). In other words, when the prevalence threshold changes within a certain range, the time cost of the method based on interval type-2 fuzzy sets and cliques is more stable, which is more conducive to finding prevalent FCPs. Figure 10: **(a) Effect of the neighbor distance thresholds; (b) Pruning degree of the two strategies. **

The effect of increasing the prevalence threshold on the pruning degree of the two pruning strategies is shown in Figure 11b. With the increase of the prevalence threshold, the number of absolutely non-prevalent patterns also increases, so the pruning rate of the fuzzy feature pruning increases. When the prevalence threshold increases, the number of absolute table-instances is unchanged, but the number of absolutely prevalent patterns and patterns with a prevalent tendency degree will decrease, and the number of pruned candidates for the absolute table-instance pruning strategy will also decrease. When the speed of the number of pruned candidates for the absolute table-instance pruning strategy is less than the speed of the reduction of the number of prevalent patterns, the pruning degree of absolute table-instances will increase. The effect of the number of fuzzy features on the results

In this subsection, we examine the effect of increasing the number of fuzzy features on the execution time. The number of fuzzy features is the number of features multiplied by the number of fuzzy attributes multiplied by the number of fuzzy sets. In this experiment, the number of fuzzy attributes is set to 2, the number of fuzzy sets is set to 3, and the number of features we select are 3, 6, 9, 12, and 15, respectively. As shown in Figure 12a, we find that the execution time of the three methods first increases and then decreases because when the number of features increases from 3 to 6, the number of candidate FCPs also increases rapidly; the number of operations that generate or examine row-instances of candidate FCPs also increases. When the number of fuzzy features continues to increase, the average number of row-instances of candidates will decrease because the three methods satisfy the downward closure property, so the pruning strategy in this paper can be used to effectively prune absolutely non-prevalent candidates in the process of generating prevalent FCPs. In addition, for the method based on interval type-2 fuzzy sets and cliques, the number of edges also affects its execution time. When the number of fuzzy features increases from 18 to 36, the number of edges increases. When the number of fuzzy features increases from 36 to 90, the number of edges decreases. When the number of fuzzy features is 72, the efficiency of the Joinless FCP mining method is higher than the other two methods; this is because our pruning strategies effectively prune many candidates. In other words, when the number of fuzzy features is small, our method based on interval type-2 fuzzy sets and cliques is more efficient than the other two methods. Figure 11: (**a**)Effect of the prevalence threshold; (**b**) Pruning degree for the two strategies. The effect of increasing the prevalence threshold on the trend of the pruning degree of the two pruning strategies is shown in Figure 12b.