# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets
## 3 Related Definitions and Theorems
### Definitions
: _Suppose there is an FCP **Fuzz_c** = {Fuzz_f_1, _Fuzz_f_2,..., \(Fuzz\_f_{k}\)}, the lower bound participation ratio of the fuzzy feature \(Fuzz\_f_{i}\)(i\(\in\){1, 2,... \(k\)}) is expressed as \(\underline{PR}(\textbf{Fuzz\_c}\), \(Fuzz\_f_{i})\), which is defined as the ratio of the sum of the lower bound membership degrees of the instances that appear non-repeatedly in the table-instance of the FCP **Fuzz_c** to the sum of the lower bound membership degrees of all the instances of \(Fuzz\_f_{i}\). The upper participation ratio is expressed as \(\overline{PR}\) (**Fuzz_c**, \(Fuzz\_f_{i}\)), which is defined as the ratio of the sum of the upper bound membership degrees of the instances that appear non-repeatedly in the table-instance of the FCP **Fuzz_c** to the sum of the upper bound membership degrees of all the instances of \(Fuzz\_f_{i}\), namely:_  
\[\underline{PR}\Big{(}\textbf{Fuzz\_c},\textbf{Fuzz\_f}_{i}\Big{)}=\sum\underline {u}\Big{(}\pi_{Fuzz\_f_{i}}\;table\_instance(Fuzz\_c)\Big{)}/\sum\underline{u} (table\_instance(\{Fuzz\_f_{i}\}));\]  
\[\overline{PR}(\textbf{Fuzz\_c},\textbf{Fuzz\_f}_{i}\Big{)}=\sum\overline{u} \Big{(}\pi_{Fuzz\_f_{i}}\;table\_instance(Fuzz\_c)\Big{)}/\sum\overline{u}(table \_instance(\{Fuzz\_f_{i}\}))\]  
_where \(\pi\) is the relational projection operation, \(u(x)\)denotes the lower bound membership of \(x\), and \(\overline{u}(x)\)denotes the upper bound membership of \(x\)._  
**Definition 9** (Upper bound participation index and lower bound participation index). : _The upper bound participation index of FCP **Fuzz_c** is expressed as \(\overline{PI}(\textbf{Fuzz\_c})\), which is defined as the minimum of the upper bound participation ratio of all fuzzy features \(Fuzz\_f_{i}\) in **Fuzz_c**, which is:_  
\[\overline{PI}(\textbf{Fuzz\_c})=min_{i=1}^{k}\big{\{}\overline{PR}\big{(} \textbf{Fuzz\_c},\textbf{Fuzz\_f}_{i}\big{)}\big{\}}.\]  
The lower bound participation index of FCP **Fuzz_c** is expressed as \(\underline{PI}(\textbf{Fuzz\_c})\), which is defined as the minimum of the lower bound participation ratio of all fuzzy features \(Fuzz\_f_{i}\) in **Fuzz_c**, which is:_  
\[\underline{PI}(\textbf{Fuzz\_c})=min_{i=1}^{k}\big{\{}\underline{PR}\big{(} \textbf{Fuzz\_c},\textbf{Fuzz\_f}_{i}\big{)}\big{\}}.\]  
**Definition 10** (Prevalent fuzzy co-location pattern). : _According to Definition 9, an FCPFuzz_c has an upper bound participation index and a lower bound participation index. For an FCPFuzz_c when given a prevalence threshold min_prev, if \(\overline{PI}(\textbf{Fuzz\_c})<\text{min\_prev}\), **Fuzz_c** is called an absolutely non-prevalent pattern. If \(\underline{PI}(\textbf{Fuzz\_c})\geq\text{min\_prev}\), **Fuzz_c** is called an absolutely prevalent pattern. If \(\underline{PI}(\textbf{Fuzz\_c})<\text{min\_prev}\)\(\leq\)\(\overline{PI}\) (\(\textbf{Fuzz\_c}\)), **Fuzz_c** is called an FCPF with prevalence tendency degree, and the prevalence tendency degree of the fuzzy pattern is:_  
\[\theta=\left(\overline{PI}\left(\textbf{Fuz\_c}\right)-\text{minprev}\right)/ \left(\overline{PI}(\textbf{Fuz\_c})-\underline{PI}(\textbf{Fuz\_c})\right)\]  
We call the absolutely prevalent FCPs and FCPs with a prevalence tendency degree prevalent FCPs. **Example 1**. : We take the content of heavy metals, copper and zinc, in the topsoil of an area as an example. We divide the areas into the agricultural area (functional area A), residential area (functional area B), industrial area (functional area C), and commercial area (functional area D). Then, we sample the surface soil of each functional area and then measure the heavy metal content at each sampling point. The number of sampling points of functional areas, spatial location of sampling point, copper content, and zinc content are shown in Table 1. The spatial position of the sampling points are shown in Figure 1.