# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets
## 4 Methods
### A Mining Method Based on Spatial Cliques
#### 4.1.2 Generating FCP Mining Cliques (FCPM-Cliques)
**Definition 14** (Complete clique set). : _Given a set of cliques scl of a spatial dataset materialized by spatial neighborhood relationships and the set containing all cliques of the spatial dataset is represented as \(\mathrm{ascl}\), if the following formula is satisfied, \(\mathrm{scl}\) is a set of complete cliques:_  
\(\forall\)\(\mathrm{cl}\)\(\in\)\(\mathrm{ascl}\), \(\exists\)\(\mathrm{cl}^{\prime}\)\(\in\)\(\mathrm{scl}\)\(\Rightarrow\)\(\mathrm{cl}\)\(\in\)\(\mathrm{cl}^{\prime}\). **Lemma 3**. : _N-cliques are complete._  
Proof. : Given a clique \(\mathrm{cl}=\{s_{1},\)\(s_{2},\)\(\ldots,\)\(s_{k}\}\) where \(s_{i}\) is an instance, if \(\mathrm{cl}\) is a maximal clique, \(\forall\)s \(\
ot\in\)\(\mathrm{cl}\), \(s\), and \(\mathrm{cl}\) cannot form a new clique. It is not difficult to see that the NDM can generate all the maximal cliques. According to Definition 14, the set containing all maximal cliques is complete. Thus, N-cliques are complete. \(\square\)  
The four strategies (Strategy 1-Strategy 4) can make decisions concerning which node can be added to a clique to form a new N-clique, and, according to **Lemma 3**, we know that N-cliques are complete. Therefore, the cliques generated by the NDM are correct and complete. In the process of mining spatial co-location patterns, we calculate the participation ratio of features according to the number of instances in a spatial clique. However, in the process of mining fuzzy spatial co-location patterns, we calculate the participation ratio of fuzzy features according to the membership degree of the attributes of instances in a spatial clique. Therefore, we pay more attention to the association between attributes of instances in the spatial clique, and we propose a kind of FCPM-clique and make the following definition: In a spatial clique, if there is an association between fuzzy attributes of instances, the fuzzy attributes of these instances can form FCPM-cliques. We find that, in a clique, there is an association between fuzzy attributes of different instances, so they can form FCPM-cliques. There is also an association between fuzzy attributes with different attributes in an instance, so these can also form FCPM-cliques. For example, for two neighbor instances A.1 and B.1, the fuzzy attribute Cu(M) of A.1 and the fuzzy attribute Cu(H) of B.1 can form an FCPM-clique {A.1.Cu (M), B.1.Cu (H)}, which can be used as a row-instance of FCP {A.Cu(M), B.Cu (H)}--as shown in Figure 8. For instance A.2, its fuzzy attributes Cu (H) and Zn (H) can form an FCPM-clique {A.2.Cu (H), A.2.Zn (H)}, which can be used as a row-instance of FCP {A.Cu(H), A.Zn (H)}--as shown in Figure 9.  
Figure 8: An example of constituting FCPM-cliques. For a candidate FCP, first we find the co-location pattern consisting of all the non-recurring features in the fuzzy candidate pattern. Then, we find the row-instances that contain all the fuzzy features of the fuzzy candidate pattern from the table-instance of the co-location pattern. These row-instances are the row-instances of the candidate fuzzy pattern. Through the FCPM-clique, we can find these spatial cliques containing all fuzzy features of the candidate FCPs.