# 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
The first real dataset is the spatial distribution data for heavy metals in the topsoil of Jiayuguan City, and is shown in Figure 15a. It is divided into four functional areas: industrial area, agricultural area, living area, and Gobi area. According to the proportion of the region, these data are projected on to a space of \(850*670\). The second real dataset is the spatial distribution data for heavy metals in the urban topsoil of Quanzhou City, and is shown in Figure 15b. It is divided into five functional areas: traffic area, commercial area, urban green area, residential area, and industrial area, and we project these data on to a space of \(700*500\) according to the proportion of the region. Figure 14: (**a**) Effect of the number of attributes; (**b**) Pruning degree of the two strategies. For dataset 1, we set the distance threshold \(min\_dist\) of the neighborhood relationship to 50, prevalence threshold to 0.3, and mined 112 prevalent FCPs. For dataset 2, we set the distance threshold \(min\_dist\) of the neighborhood relationship to 50, prevalence threshold to 0.3, and mined 46 prevalent FCPs. Table 6 shows part of the mining results for dataset 1 and Table 7 shows a part of the mining results for dataset 2. In Table 6, A, B, C, and D, respectively, represent the industrial area, agricultural area, residential area, and Gobi area. For example, an FCP {A.Cu(L), C.Cu(L)} indicates that an industrial area with a low copper content and a residential area with a low copper content often tend to be located together, and the degree of this trend is 0.6125. An FCP {B.Cu(M), B.Zn(L), C.Cu(M), C.Zn(L)} indicates that agricultural areas with a low zinc content and middle copper content, and residential areas with a low zinc content and middle copper content often tend to be located together--and the degree of this tendency is 1. In Table 7, A, B, C, D, and E, respectively, represent the traffic area, commercial area, urban green area, residential area, and industrial area. For example, an FCP {A.Zn(H), D.Zn(H)} indicates that traffic areas with a high zinc content and residential areas with a high zinc content often tend to be located together, and the degree of this tendency is 1. An FCP {A.Cu(M), A.Zn(H), D.Cu(M)} indicates that traffic areas with a middle copper content and high zinc content, and residential areas with a middle copper content often tend to be located together, and the degree of this tendency is 0.9047. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Prevalent Fuzzy Co-Location Pattern** & **Prevalence Tendency Degree** & \
\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(L), C.Cu(L)} & 0.6125 & 0.0709, 0.6621 & 0.2312 \\ \hline \{A.Cu(L), D.Cu(L)} & 0.2328 & 0.036, 0.3801 & 0.1169 \\ \hline \{A.Cu(M), C.Cu(M)} & 0.2636 & 0.1567, 0.3513 & 0.2384 \\ \hline \{D.Cu(M), D.Zn(H)} & 1 & 0.4781, 0.9135 & 0.6599 \\ \hline \{D.Cu(L), D.Zn(M)} & 0.9096 & 0.2414, 0.8896 & 0.6681 \\ \hline \{A.Zn(L), A.Cu(M), C.Zn(L)} & 0.2636 & 0.1567, 0.3513 & 0.2384 \\ \hline \{A.Zn(L), C.Zn(L), D.Cu(M)} & 0.2167 & 0.1749, 0.3346 & 0.2492 \\ \hline \{B.Cu(L), B.Zn(L), C.Zn(L)} & 0.8586 & 0.2597, 0.5448 & 0.4342 \\ \hline \{B.Cu(M), B.Zn(L), C.Cu(M), C.Zn(M)} & 0.7039 & 0.1804, 0.5844 & 0.3358 \\ \hline \{B.Cu(M), B.Zn(L), C.Cu(M), C.Zn(L)} & 1 & 0.3933, 0.738 & 0.5726 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Part of the mining results for real dataset 1. Figure 15: (**a**) Real dataset 1; (**b**) Real dataset 2. For the real dataset 1, 112 prevalent FCPs (26 absolutely prevalent FCPs and 86 FCPs with a prevalence tendency degree) were mined according to our measurement method, and 63 prevalent FCPs were mined according to the traditional measurement method.