# 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
After determining the number of features and the total number of instances, we determine the number of instances owned by each feature according to the Poisson distribution. In a grid of the region, the data density (number of edges) of all neighbor grids is controlled by a value called _clump_; we set the value of _clump_ to 1, in accordance with the general situation described in Ref. [6]. The number of edges in all instances is approximately equal to \(E\times clumpy\). When \(clump=1\), the number of edges of all instances is \(E\), where \(E\) is generated randomly. The effect of distance thresholds on the results

We investigate the influence of different distance thresholds on the execution time by changing the distance threshold of the neighborhood relationship between instances, as shown in Figure 10a. We find that with the increase of distance threshold, the execution time of the three methods increases because the number of instance neighbors will increase (that is, the number of edges will increase) and the average number of row-instances of each FCP will also increase. We also find that with the increase of distance threshold, the growth rate of the execution time of our method is much slower than that of Join-based and Joinless methods. This is mainly because our method searches cliques with all fuzzy features in candidate FCPs and uses them as row-instances of candidate FCPs instead of generating row-instances through connections. In the experiment, the Join-based FCP \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Phase 1**} & \multicolumn{1}{c}{**Phase 2**} & \multicolumn{1}{c}{**Phase 3**} \\ \hline Star Instances of & A & {A, B} & {A.Cu(M), B.Cu(M)} & {A.Cu(M), A.Zn(M)} \\ Feature A & A & & B.Cu(M) & A.Zn(M)] \\ \hline A.1, B.2, C.2 & A.1 & & \\ A.2, B.3, C.1 & A.2 & A.1, B.2 & \\ A.3, B.5, C.2 & A.3 & A.2, B.3 & A.1 \\ A.4, C.3 & A.4 & A.3, B.5 & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: An example of the extended Joinless algorithm. mining algorithm is represented by \"Join-based\", the Joinless FCP mining algorithm is represented by \"Joinless\", and the FCP mining algorithm based on interval type-2 fuzzy sets and cliques is represented by \"NDM\". With the increase of distance threshold, the trend of the pruning degree of the two pruning strategies is shown in Figure 10b. For a spatial dataset, the number of absolute table-instances is certain and will not change with the change of distance threshold. However, the average number of row-instances of FCPs will increase with the increase of distance threshold, so the pruning degree of the absolute table-instances pruning will decrease. In addition, with the increase of distance threshold, the average number of row-instances of each FCP will also increase. Therefore, in a candidate FCP, the number of fuzzy features whose upper bound participation ratio is below the prevalence threshold will decrease, the number of prevalent FCPs also increases, and the pruning rate of fuzzy features will also decrease. In the process of mining fuzzy co-location patterns, the combination of features, attributes, and fuzzy sets will generate a lot of fuzzy features and a large amount of candidate fuzzy co-location patterns. As the total number of row-instances of co-location patterns is certain, many candidate fuzzy co-location patterns have few row-instances, so the pruning degree of the fuzzy feature pruning strategy is very high. On average, for a candidate fuzzy co-location pattern, the number of absolute row-instances it contains is small, so the pruning degree of the absolute table-instance pruning strategy is low. The effect of prevalence threshold on the results

In this subsection, we study the influence of changing the prevalence threshold over a certain range on the execution time. As shown in Figure 11a, we find that the execution time of these three methods will decrease with the increase of the prevalence threshold.