# 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
That is to say, such an FCP has a certain tendency to be prevalent, but also has a certain tendency to be non-prevalent; we regard this as a kind of potential prevalent FCP. For example, in the real dataset 1, the FCP {A.Cu(M), C.Cu(M)} has a lower bound membership of 0.1567, an upper bound membership of 0.3513, and a prevalence tendency degree of 0.2636. We can find that when there is uncertainty in the fuzzy membership function, {A.Cu(M), C.Cu(M)} has a 26.36% tendency degree to be prevalent. However, in the traditional method based on type-1 fuzzy sets, the participation index of the FCP is 0.2384 when the prevalence threshold is 0.3; we can only find that it is non-prevalent and cannot find the influence of the uncertainty of the fuzzy membership function on its participation index and prevalence degree. When using type-1 fuzzy sets, we cannot find the potential association between the middle concentration of copper in the industrial area and the middle concentration of copper in the living area, even if the association is very significant. 3. We use the prevalence tendency degree to measure the prevalence degree of FCPs. This allows us to find the FCPs whose lowest possible value of participation index is higher than the prevalence threshold. These FCPs are still prevalent when there is uncertainty in the fuzzy membership function, so we define them as absolutely prevalent FCPs and we regard them as a kind of stable and prevalent FCP. However, in the traditional method based on type-1 fuzzy sets, it is difficult to find this kind of FCP according to the participation index and the prevalence threshold. For example, in the real dataset 1, an absolutely prevalent FCP {B.Cu(M), B.Zn(L), C.Cu(M), C.Zn(L)} has a lower bound membership of 0.3933 and an upper bound membership of 0.738. We can easily find that the pattern is stably prevalent and we can find the influence of the uncertainty of the fuzzy membership function on the prevalence degree of FCPs. However, in the traditional method based on type-1 fuzzy sets, the participation index of the FCP is 0.5726 and it is difficult to judge whether this pattern is prevalent when there is uncertainty in the fuzzy membership function. ## 7 Conclusions

In the work of traditional FCPs mining, the generation of table-instances of an FCP takes up most of the execution time. In this paper, we propose a method based on interval type-2 fuzzy sets and cliques. Firstly, cliques are generated from spatial datasets, and then we look for row-instances of the candidate FCP from the cliques, thus saving the time of generating row-instances of candidates through a large number of connection operations. Compared with type-1 fuzzy membership function, a type-2 fuzzy membership function more truly reflects the situation when there is uncertainty in the membership degree of fuzzy attributes. In this case, our method effectively measures the prevalence degree of fuzzy co-location patterns, and finds stable and prevalent fuzzy co-location patterns and fuzzy co-location patterns with a prevalence degree. It provides a reference for people to find the influence of the uncertainty of membership functions on the prevalence degree of fuzzy co-location patterns. At the same time, our method has some limitations, such as the time cost of the gradual method of adjusting the parameter of interval type-2 fuzzy membership function, which is high, and the efficiency is low when the number of instances or features is very large. For future work, we will consider using distributed and parallel mining to mine 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, which will lead to a tremendous time cost. Distributed and parallel mining can greatly improve the efficiency of mining fuzzy co-location patterns. In addition, we consider using interval type-3 fuzzy sets to deal with higher-level uncertainty.