# 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 the increase in the number of features in the candidate FCPs, the number of fuzzy features whose upper bound participation degree \(\overline{PR}\) is less than the prevalence threshold will also increase. Therefore, the pruning rate of the fuzzy feature pruning strategy also increases. When the number of features increases, the number of prevalent FCPs decreases, but the number of FCPs pruned by the absolute table-instances pruning strategy decreases more slowly, so the pruning degree of absolute table-instances continues to rise. The effect of the number of instances on the results

In this subsection, we examine the effect of increasing the number of instances on execution time. As shown in Figure 13a, we find that with the increase of the number of instances, the execution time of the three methods increases. When the number of instances is 50,000, the efficiency of the FCP mining algorithm based on interval type-2 fuzzy sets and cliques is much faster than the Join-based (or Joinless) approach. This is because when the data in an area becomes denser, the average number of row-instances of FCPs will be more, and the Join-based (or Joinless) method needs to spend a lot of time to connect and generate (or filter) row-instances. The method based on interval type-2 fuzzy sets and cliques first generates cliques and then searches the row-instances of candidates from cliques, which saves a lot of time cost caused by connection or filtering operations. Therefore, when the instances in the region are dense, the efficiency of the method based on interval type-2 fuzzy sets and cliques is obviously higher than that of the Join-based (or Joinless) approach. With the increase of the number of instances, the trend of the pruning degree of the two proposed pruning strategies is shown in Figure 13b. As the number of instances increases, the average number of row-instances of each FCP will also increase. Therefore, the pruning rate of the fuzzy feature pruning strategy will decrease. The number of FCPs pruned by the absolute table-instance pruning strategy increases, but the number of prevalent

Figure 12: (**a**) Effect of the number of features; (**b**) Pruning degree of the two strategies. Figure 13: (**a**) Effect of the number of instances; (**b**) Pruning degree of the two strategies. FCPs increase faster, so the pruning degree of the absolute table-instance pruning strategy decreases. (5) The effect of the number of attributes on the results

In this paper, we take the content of the heavy metals, copper and zinc, in topsoil as an example and construct the granular type-2 fuzzy membership function for copper and zinc, respectively, so there are two attributes. Here, we only keep the zinc attribute of the instances and examine the influence of the number of attributes on the execution time, as shown in Figure 14a. We find that when the number of attributes decreases, the number of fuzzy features will also decrease, and then the generated candidate FCPs and prevalent FCPs will also decrease a lot--and the execution time greatly reduces. The trend of the pruning degree for the two proposed pruning strategies is shown in Figure 14b when there is one attribute. This is similar to the trend of the pruning rate when there are two attributes. With one attribute, with the increase of the number of features, the number of fuzzy features whose upper bound participation ratio \(\overline{PR}\) is less than the prevalence threshold will also increase, so the pruning rate of the fuzzy feature pruning strategy will also increase. When the number of features increases, the number of prevalent FCPs decrease, but the number of FCPs pruned by the absolute table-instance strategy decrease more slowly, so the pruning degree of the absolute table-instance pruning strategy continues to rise. ### Results for Real Datasets

In this section, we examine the validity of our proposed approach through the use of two real datasets.