# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets
## 3 Related Definitions and Theorems
### Lemma and Theorem  
**Lemma 1**.: _The upper bound participation index (or lower bound participation index) of an FCP and the upper bound participation ratio (or lower bound participation ratio) of fuzzy features in a pattern decrease monotonically with the increase of the size of the pattern._  
Proof.: If instance \(s_{1}\) of fuzzy feature \(\textbf{Fuzz\_f}\)1s is included in the row-instances of fuzzy pattern \(\textbf{Fuzz\_c}\), then when fuzzy pattern \(\textbf{Fuzz\_c}^{\prime}\subseteq\textbf{Fuzz\_c}\), instance \(s_{1}\textbf{Fuzz\_c}^{\prime}\) must also be included in the row-instances of \(\textbf{Fuzz\_c}^{\prime}\). According to Definitions 8 and 9, the denominator of the upper bound participation ratio (or lower bound participation ratio) of the same fuzzy feature in the pattern and pattern \(\textbf{Fuzz\_c}\) is the same, and the numerator decreases with the increase of the size of the pattern. Therefore, the upper bound participation ratio (or lower bound participation ratio) of the fuzzy features decreases monotonically. According to the definition of the upper bound participation index and lower bound participation index, the upper bound participation index (or lower bound participation index) is the minimum of the upper bound participation ratio (or lower bound participation ratio) of all fuzzy features in an FCP. Therefore, the upper bound participation index (or lower bound participation index) decreases monotonically with the increase of the size of the pattern.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \
\begin{tabular}{c} **Functional** \\ **Area** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **Sampling** \\ **Points** \\ \end{tabular} & **Cu(L)** & **Cu(M)** & **Cu(H)** & **Zn(L)** & **Zn(M)** & **Zn(H)** \\ \hline
**A** & 1 & 0 & [0.4403, 0.7456] & [0.0104, 0.0831] & 0 & 0 & 1 \\ \hline
**A** & 2 & 0 & [0.3614, 0.6848] & 0 & 0 & [0.5414, 0.8298] & 0 \\ \hline
**A** & 3 & 0 & [0.8, 0.9591] & 0 & 0 & [0.5191, 0.8146] & 0 \\ \hline
**B** & 1 & 0 & 0 & [0.283, 0.5931] & 0 & 0 & [0.3905, 0.7023] \\ \hline
**B** & 2 & 0 & [0.3152, 0.6386] & 0 & 0 & [0.7138, 0.9252] & 0 \\ \hline
**B** & 3 & 0 & [0.7633, 0.9455] & 0 & 0 & [0.2495, 0.5656] & 0 \\ \hline
**B** & 4 & 0 & [0.5786, 0.8991] & 0 & 0 & [0.8315, 0.9695] & 0 \\ \hline
**C** & 1 & 0 & [0.4069, 0.717] & [0.0174, 0.1141] & 0 & 0 & 1 \\ \hline
**D** & 1 & 0 & [0.9169, 0.9896] & 0 & 0 & [0.1702, 0.4586] & [0.0335, 0.1698] \\ \hline \hline \end{tabular}
\end{table}
Table 2: The interval membership of the fuzzy attribute of the sampling point.  
**Theorem 1**.: _If a \(k\)-1 size pattern is absolutely non-prevalent, then any \(k\) size patterns that it contains are also absolutely non-prevalent._  
Proof.: For a \(k\)-1 size pattern \(\textbf{Fuzz\_c}\), if a \(k\) size pattern \(\textbf{Fuzz\_c}^{\prime}\supseteq\textbf{Fuzz\_c}\), according to **Lemma 1**, \(\overline{\text{PI}}(\textbf{Fuzz\_c})\leq\overline{\text{PI}}(\textbf{Fuzz\_c})\). If this \(k\)-1 size pattern \(\textbf{Fuzz\_c}\) is absolutely non-prevalent, then \(\overline{\text{PI}}(\textbf{Fuzz\_c})<min\_prev\). Therefore, if \(\overline{\text{PI}}(\textbf{Fuzz\_c})\leq\overline{\text{PI}}(\textbf{Fuzz\_c})< min\_prev\), then \(k\) size pattern \(\textbf{Fuzz\_c}^{\prime}\) is also absolutely non-prevalent. \(\square\)