# 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
\(P_{1}=\textbf{Fuzz\_Fea};k=1\); 3. while(\(P_{k}\
eq\varnothing\)){ 4. \(C_{k+1}=\) gen_candidate(\(P_{k}\)); 5. for \(i\) in 1 ton 6. for \(j\) in \(cgf_{i}=cgf_{1}\), \(cgf_{1}\) is the first feature of \(C_{k+1}\) 7. \(SI_{k+1}=\) gen_star_instance(\(C_{k+1},x\)); 8. end for 9. end for 10. if \(t=1\) (\(t\) is the number of non recurring features in a candidate fuzzy pattern) 11. \(PI_{k+1}\) = The instances of the only feature that have all fuzzy feature of the candidate fuzzy co-location pattern; 12. \(CI_{k+1}=PI_{k+1}\); 13. if \(t=2\) 14. \(PI_{k+1}\)= The row - instances in \(SI_{k+1}\) which have all fuzzy features of the candidate fuzzy co-location pattern; 15. \(CI_{k+1}=PI_{k+1}\); 16. else do 17. \(PI_{k+1}\)= The star instances \(SI_{t}\) that have all fuzzy features of the candidate fuzzy co-location pattern; 18. \(C_{k+1}=\) filter_coarse_prevalent_colocations(\(C_{k+1}\), \(PI_{k+1}\), _minprev_); 19. \(CI_{k+1}=\) gen_clique_instances(\(C_{k+1}\), \(PI_{k+1}\)); 20. end do 21. \(P_{k+1}\)= filter_prevalent_colocations(\(C_{k+1}\), \(CI_{k+1}\), _min_prev_); 22. \(k=k+1\); 23. \(\textbf{Fre\_P}=\textbf{Fre\_P}\cup P_{k+1}\); 24. } ```

**Algorithm 5**The extended Joilness algorithm. Our expansion includes: For a candidate FCP, first, we find the pattern consisting of all the non-recurring features in the candidate FCP; then, we find the star instances that have all the fuzzy features of the candidate FCP from the star instances of the co-location pattern. These star instances are the star instances of the FCP. Finally, we find the clique instances of the FCP from the star instances (at this time, the clique instances are the row-instances of the FCP) (Steps 17-19). When the number of features in a candidate FCP is 1, the instances of the only feature that have all fuzzy features in the candidate FCP simultaneously are selected as the row-instances of the candidate FCP (Steps 10-12). When the number of features in a candidate FCP is 2, the size-2 star instances that have all the fuzzy features in the candidate are selected as the size-2 row-instances of the candidate (Steps 13-15). For example, suppose there is a candidate FCP {A.Cu(M), B.Cu(M)}, feature A is the first feature of the candidate FCP, according to Steps 13-16. From this, we obtain all the star-instances of feature A (Phase 1). Then, we get all the row-instances of co-location pattern {A, B} from the star-instances of feature A (Phase 2). Then, from these row-instances, we filter the row-instances with fuzzy features A.Cu(M) and B.Cu(M), which are the row-instances of the candidate FCP. According to Steps 10-12, we filter the instances that have fuzzy features A.Cu(M) and A.Zn(M) from the instances of feature A; these instances are the row-instances of the candidate {A.Cu(M), A.Zn(M)} (Phase 3), as can be seen in Table 5.

## 5 Results

In this section, we evaluate our methods using synthetic and real datasets, and a wide range of experiments have been carried out. We used a method similar to Ref. [2] to generate synthetic datasets and compared the performance of the three algorithms in terms of the number of features, the number of instances, the prevalence thresholds, the distance thresholds, and the size of the fuzzy sets. We used two real datasets to evaluate our methods: one is the heavy metal content of the topsoil sampling points in Quanzhou City, and the other is the heavy metal content of the topsoil sampling points near the industrial area of Jiayuguan. All our experiments are carried out using C#. The Intel PC we use has the Windows 10 operating system, Intel Core [EMAIL_ADDRESS] GHz, and 8 GB memory. ### Results for Synthetic Datasets

A method similar to Ref. [6] is used to randomly generate synthetic data. First, the size of the region of our instances distribution is set to \(D\times D\), and the whole region is divided into several grids with the size \(min\_dist\times min\_dist\), where \(min\_dist\) is the distance threshold of the instance neighbors.