# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets
## 3 Related Definitions and Theorems  
### Definitions  
Traditional co-location patterns can reflect the spatial association of instances but ignore the influence of other information--besides location information--on the mining results. In order to articulate our method, we give the following definitions:  
Definition 1 (Data fuzzification): The most important step of attribute data fuzzification is to determine the fuzzy membership function. According to the fuzzy membership function, the attribute data is expressed as a fuzzy membership. Supposing the fuzzy set over the domain is \(F\), then the fuzzy membership function is \(U_{F}\). We use membership function \(U_{F}\) to describe the fuzzy set \(F\), that is, \(U_{F}\): \(x\rightarrow[0,1]\). In this paper, we use the upper boundary and lower boundary of the footprint of uncertainty (FOU) to describe the type-2 fuzzy membership function. That is, we introduce the upper bound membership function \(\overline{U}_{F}(x)\) and the lower bound membership function \(U_{F}(x)\). Definition 2 (Granular evaluation data): We use the expert evaluation method to evaluate the interval membership degrees of the interval values under a fuzzy set to form granular evaluation data, so as to reduce the error when the specific membership degree of a specific value under a fuzzy set is evaluated. Definition 3 (Fuzzy feature): Fuzzy features refer to different types of things with fuzzy attributes in a space. A fuzzy feature set refers to the set of fuzzy features, expressed as \(\textbf{Fuzz\_Fea}=\{\textit{Fuzz\_f1},\textit{Fuzz\_f2},\ldots,\textit{Fuzz\_ fn}\}\). A fuzzy feature is represented as \(\textit{Fuzz\_fi}\) (\(1\leq i\leq\textit{n}\)). The instances of a fuzzy feature are the sampling points of the fuzzy feature in the space and the fuzzy membership degree for the fuzzy feature when it is greater than 0. **Definition 4** (Neighborhood relationship). : _A spatial dataset refers to the set of instances with type (feature), location information, and attribute information in a space. Suppose **Fuzz_Fea** is a set of features and **S** is the set of their instances, for any two instances, \(s_{1}\) and \(s_{2}\), if the Euclidean distance between the two instances is less than the threshold \(min\_dist\) given by the user--that is, distance(\(s_{1}\), \(s_{2}\)) \(\leq\)\(min\_dist\)--then the two instances satisfy the neighborhood relationship, which is represented as \(R(s_{1},s_{2})\)._  
**Definition 5** (Fuzzy co-location pattern). : _A fuzzy co-location pattern (FCP) **Fuzz_c** is a subset of the set of fuzzy features, that is, **Fuzz_c**\(\subseteq\)**Fuzz_Fea**. The size of an FCP is the number of fuzzy features of the pattern, which is expressed as |Fuzz_c|._  
**Definition 6** (Row-instance and table-instance). : _Suppose there is a subset of spatial instances **S** ={\(s_{1}\), \(s_{2}\),...,\(s_{k}\)} where any two instances are neighbors. If **S** contains all the fuzzy features of an FCP **Fuzz_c** and there is no subset of **S** that can contain all the fuzzy features of **Fuzz_c**, then **S** is called a row-instance of **Fuzz_c**. All row-instances of **Fuzz_c** constitute the table-instance of **Fuzz_c**._  
**Definition 7** (Absolute row-instance and absolute table-instance). : _Suppose there is a spatial instances\({}_{i}\) (i\(\in\){1, 2,... \(k\)}). If \(s_{i}\) is an instance of the fuzzy feature \(Fuzz\_f_{1}\) and also an instance of the fuzzy feature \(Fuzz\_f_{2}\), then\({}_{i}\) is called an absolute row-instance of a fuzzy co-location pattern {Fuzz_f_1, _Fuzz_f_2}. For a fuzzy co-location pattern, all its absolute row-instances constitute its absolute table-instance._  
**Definition 8** (Upper bound participation ratio and lower bound participation ratio).