# 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
For the real dataset 2, 46 prevalent FCPs (11 absolutely prevalent FCPs and 35 FCPs with a prevalence tendency degree) were mined according to our measurement method, and 37 prevalent FCPs were mined according to the traditional measurement method. We find that our method finds more prevalent FCPs (including absolutely prevalent FCP and FCPs with a prevalence tendency degree) than the method of mining spatial FCPs based on type-1 fuzzy sets. In the method of mining FCPs based on type-1 fuzzy sets, the membership degree of the attribute of the spatial instance is a specific value because of the type-1 fuzzy membership function. Therefore, the participation index of the mined FCPs is also a specific value. We compare the participation index of FCPs with the prevalence threshold and we can divide these FCPs into prevalent FCPs and non-prevalent FCPs. For example, we apply the method of mining FCPs based on type-1 fuzzy sets to dataset 1. The prevalence threshold is set to 0.3 and the participation index of FCP [A.Cu(L), C.Cu(L)] is 0.2312, which is non-prevalent. The participation index of {B.Cu(L), B.Zn(L), C.Zn(L)} is 0.4342, which is prevalent. ## 6 Discussion

The membership of attributes of spatial instances is determined by the fuzzy membership function. However, the disadvantage of using type-1 fuzzy membership functions is that the type-1 fuzzy membership function itself has a large amount of uncertainty. The application of type-1 fuzzy sets in mining FCPs will result in large deviations. Therefore, we apply interval type-2 fuzzy sets to the process of mining FCPs in order to reduce the deviation caused by the uncertainty of type-1 fuzzy membership functions and find the influence of the uncertainty of the fuzzy membership function on the prevalence degree of FCPs. Moreover, our method is more efficient than the traditional FCPs mining method based on type-1 fuzzy sets [4]. Compared with the method of mining FCPs based on type-1 fuzzy sets, our method has the following advantages:

1. By constructing granular type-2 fuzzy membership functions, we reduce the deviation caused by the uncertainty of type-1 fuzzy membership functions. In this paper, the membership degrees of the attributes of spatial instances are expressed by interval values, and then the mined FCPs have an upper bound participation index and a lower bound participation index. We propose a method to measure the prevalence degree of FCPs when there is uncertainty in the fuzzy membership function, which \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Prevalent Fuzzy Co-Location Pattern** & \begin{tabular}{c} **Prevalence Tendency** \\ **Degree** \\ \end{tabular} & \
\begin{tabular}{c} **Lower Participation** \\ **Index and Upper** \\ **Participation Index** \\ \end{tabular} &
\begin{tabular}{c} **Participation Index for the** \\ **Traditional Method** \\ \end{tabular} \\ \hline \{A.Cu(M), A.Zn(M)] & 0.7965 & 0.2438, 0.5199 & 0.3661 \\ \hline \{A.Zn(H), D.Zn(H)] & 1 & 0.4114, 0.966 & 0.8528 \\ \hline \{A.Cu(M), D.Zn(H)] & 0.8208 & 0.2438, 0.5575 & 0.3661 \\ \hline \{E.Cu(L), E.Zn(L)] & 1 & 0.3657, 0.4952 & 0.4425 \\ \hline \{A.Cu(M), [PERSON](M), D.Cu(M)] & 0.7965 & 0.2438, 0.5199 & 0.3661 \\ \hline \{A.Cu(M), A.Zn(H), D.Zn(M)] & 0.9047 & 0.2828, 0.4633 & 0.3853 \\ \hline \{A.Cu(M), [PERSON](M), [PERSON](M), D.Zn(H)] & 0.7965 & 0.2438, 0.5199 & 0.3661 \\ \hline \{[PERSON](M), [PERSON](H), [PERSON](M), [PERSON](H)]\} & 0.9047 & 0.2828, 0.4633 & 0.3853 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Part of the mining results for real dataset 2.

provides a basis for judging the influence of the uncertainty of the fuzzy membership function on the results of mining FCPs. 2. Due to the uncertainty of fuzzy membership functions, we find FCPs whose lower bound participation index is less than the prevalence threshold but the upper bound participation index is higher than the threshold. This shows that in all the possible values of the participation index of these FCPs, some values are higher than the prevalence threshold and some values are lower.